17735:
4168:
12736:
15480:
3959:
12516:
355:. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs to be more attentive to the domain issue in the unbounded case. This is explained below in more detail.
11474:
16033:
We may now state the classification result for self-adjoint operators: Two self-adjoint operators are unitarily equivalent if and only if (1) their spectra agree as sets, (2) the measures appearing in their direct-integral representations have the same sets of measure zero, and (3) their spectral
14584:
7824:
The spectral theorem in general can be expressed similarly as the possibility of "diagonalizing" an operator by showing it is unitarily equivalent to a multiplication operator. Other versions of the spectral theorem are similarly intended to capture the idea that a self-adjoint operator can have
13320:
operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the
Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one
7425:
In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an
10555:
applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric (specifically, an essentially self-adjoint) operator that has an orthonormal basis of eigenvectors. Consider the complex
Hilbert space
1972:
15315:
3192:
7408:
11591:
11012:
In the case where the
Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying
4163:{\displaystyle \Vert (A-\lambda )x\Vert \geq {\frac {|\langle (A-\lambda )x,x\rangle |}{\Vert x\Vert }}=\left|\left\langle A{\frac {x}{\Vert x\Vert }},{\frac {x}{\Vert x\Vert }}\right\rangle -\lambda \right|\cdot \Vert x\Vert \geq d(\lambda )\cdot \Vert x\Vert .}
15824:
10528:
15048:
12731:{\displaystyle \operatorname {Dom} \left({\hat {H}}^{*}\right)=\left\{{\text{twice differentiable functions }}f\in L^{2}(\mathbb {R} )\left|\left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}f}{dx^{2}}}-x^{4}f(x)\right)\in L^{2}(\mathbb {R} )\right.\right\}.}
8233:
10419:
15219:
11003:
Although the distinction between a symmetric operator and a (essentially) self-adjoint operator is subtle, it is important since self-adjointness is the hypothesis in the spectral theorem. Here we discuss some concrete examples of the distinction.
15615:
15229:
The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators
959:
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in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint. In one dimension, for example, the operator
11358:
13798:
10242:
9823:
14439:
13520:
of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions. Physically, these boundary conditions are related to reflections of the particle at the origin.
12414:
is not essentially self-adjoint on the space of smooth, rapidly decaying functions. In this case, the failure of essential self-adjointness reflects a pathology in the underlying classical system: A classical particle with a
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self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain a self-adjoint operator.
1303:
1074:
1855:
15714:
14320:
6063:
3509:
15475:{\displaystyle L_{\mu }^{2}\left(\mathbf {R} ,\mathbf {H} _{n}\right)=\left\{\psi :\mathbf {R} \to \mathbf {H} _{n}:\psi {\text{ measurable and }}\int _{\mathbf {R} }\|\psi (t)\|^{2}d\mu (t)<\infty \right\}}
5479:
10988:
in quantum mechanics. As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a
Hilbert space basis is that it has a compact inverse.
3898:
8005:
7046:
14844:
3626:
3567:
1715:
616:
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8517:
5924:
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5772:
3947:
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7570:. (Physicists would say that the eigenvectors are "non-normalizable.") Physicists would then go on to say that these "generalized eigenvectors" form an "orthonormal basis in the continuous sense" for
13624:
9528:
4584:
9369:
5421:
3675:
2638:
347:
The structure of self-adjoint operators on infinite-dimensional
Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are
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prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitable
450:
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has eigenvectors with pure imaginary eigenvalues, which is impossible for a symmetric operator. This strange occurrence is possible because of a cancellation between the two terms in
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1587:
506:
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10099:, both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using the
6426:
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8117:
2904:
13131:
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In this case, we can understand the implications of the domain issues for the spectral theorem. If we use the first choice of domain (with no boundary conditions), all functions
9734:
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1029:
16169:
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13093:
12932:
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12478:
is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension.)
10157:
10129:
876:
14603:
13963:
would still be symmetric and would now, in fact, be essentially self-adjoint. This change of boundary conditions gives one particular essentially self-adjoint extension of
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735:
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12510:
on the space of smooth, rapidly decaying functions, the adjoint will be "the same" operator (i.e., given by the same formula) but on the largest possible domain, namely
12508:
12476:
9238:
9209:
228:
13019:
10321:
7270:
1113:
12198:
9154:
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2551:
2487:
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2414:
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2224:
11063:
7752:
6649:
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5175:
5014:
2978:
2691:
1650:
13269:
12443:
2382:
1847:
1433:
15093:
13574:
13518:
13296:
13239:
11789:
11629:
11504:
9265:
8417:
8390:
7779:
7719:
5646:
5142:
1762:
1480:
1383:
809:
14331:
13465:
9128:
9059:
8782:
1356:
15858:
8013:
5019:
5536:
2685:
13906:
13886:
13866:
13491:
12896:
11762:
11742:
11722:
9571:
9551:
9177:
9099:
9079:
9030:
9010:
8975:
8951:
8931:
8911:
8770:
8740:
8720:
8634:
8614:
8285:
8265:
7931:
7819:
7799:
7156:
6264:
6110:
5556:
5315:
4321:
3294:
3268:
3215:
3091:
2998:
2862:
2815:
2662:
2507:
2189:
2079:
2039:
2019:
1995:
1782:
1735:
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1613:
1453:
1403:
1323:
1165:
986:
868:
659:
636:
539:
470:
405:
381:
11017:. In mathematical terms, choosing the boundary conditions amounts to choosing an appropriate domain for the operator. Consider, for example, the Hilbert space
11469:{\displaystyle \operatorname {Dom} \left(A^{\mathrm {cl} }\right)=\left\{{\text{functions }}f{\text{ with two derivatives in }}L^{2}\mid f(0)=f(1)=0\right\},}
6901:
15921:
Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of
14041:
12004:
15665:
14579:{\displaystyle D^{\alpha }={\frac {1}{i^{|\alpha |}}}\partial _{x_{1}}^{\alpha _{1}}\partial _{x_{2}}^{\alpha _{2}}\cdots \partial _{x_{n}}^{\alpha _{n}}.}
12204:
has no eigenvectors at all. If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for
15238:
are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the
13827:
is not essentially self-adjoint, but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings
10179:
9760:
11301:
is symmetric. This operator is not essentially self-adjoint, however, basically because we have specified too many boundary conditions on the domain of
9582:
17624:
13313:
11140:
8980:
More complete versions of the spectral theorem exist as well that involve direct integrals and carry with it the notion of "generalized eigenvectors".
8889:
The spectral theorem holds for both bounded and unbounded self-adjoint operators. Proof of the latter follows by reduction to the spectral theorem for
2232:
236:
7694:
Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the
Fourier transform, which allows a general
9859:
8523:
6516:
14163:
17011:
15982:
9954:
4706:
3745:
3001:
13848:
In this case, the failure of essential self-adjointenss is due to an "incorrect" choice of boundary conditions in the definition of the domain of
5320:
16857:
17460:
10041:
4921:
1967:{\displaystyle \langle x,Ax\rangle ={\overline {\langle Ax,x\rangle }}={\overline {\langle x,Ax\rangle }}\in \mathbb {R} ,\quad \forall x\in H.}
13717:
13321:
attempts to find self-adjoint extensions of the
Hamiltonian corresponding to different types of boundary conditions or conditions at infinity.
1263:
1034:
17287:
17016:
Spectral Theory and
Quantum Mechanics:Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation
14740:
More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If
14254:
5999:
3449:
5426:
989:
17450:
3819:
3443:
the operator is self-adjoint. This implies, for example, that a non-self-adjoint operator with real spectrum is necessarily unbounded.
16753:
Bonneau, Guy; Faraut, Jacques; Valent, Galliano (2001). "Self-adjoint extensions of operators and the teaching of quantum mechanics".
7936:
6992:
3187:{\displaystyle \rho (A)=\left\{\lambda \in \mathbb {C} \,:\,\exists (A-\lambda I)^{-1}\;{\text{bounded and densely defined}}\right\}.}
14753:
7403:{\displaystyle A^{*}x=y\Leftrightarrow (A^{*}-\lambda I)x=y-\lambda x\Leftrightarrow (A-\lambda I)x=y-\lambda x\Leftrightarrow Ax=y.}
3572:
3513:
1675:
544:
17878:
17577:
17432:
16052:
8478:
5884:
17001:
16802:
Bebiano, N.; da Providência, J. (2019-01-01). "Non-self-adjoint operators with real spectra and extensions of quantum mechanics".
15928:
11586:{\displaystyle \operatorname {Dom} \left(A^{*}\right)=\left\{{\text{functions }}f{\text{ with two derivatives in }}L^{2}\right\}.}
17771:
17408:
9374:
5722:
9737:
3905:
3302:
16912:
14866:
10743:
12200:, and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary conditions),
9450:
4539:
17233:
17137:
17111:
17076:
17031:
9297:
7781:. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by
5373:
3631:
17:
2599:
2053:
refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked.
12347:
15855:
Any self-adjoint operator on a separable
Hilbert space is unitarily equivalent to multiplication by the function λ ↦ λ on
8329:
8290:
6302:
5594:
2556:
814:
13582:
12449:
self-adjoint, but it does admit self-adjoint extensions obtained by specifying "boundary conditions at infinity". (Since
5239:
4452:
16121:
As remarked above, the
Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the
9656:
4843:
4634:
17300:
17049:
16979:
15819:{\displaystyle \bigoplus _{1\leq \ell \leq \omega }L_{\mu _{\ell }}^{2}\left(\mathbf {R} ,\mathbf {H} _{\ell }\right).}
420:
10523:{\displaystyle f\left(H_{\text{eff}}\right)=\int dE\left|\Psi _{E}\right\rangle f(E)\left\langle \Psi _{E}^{*}\right|}
3401:
3360:
1488:
17895:
17389:
17280:
16944:
16734:
16238:
16218:
13629:
defined on the space of continuously differentiable complex-valued functions on , satisfying the boundary conditions
10998:
10841:
6841:
5821:
5485:
4888:
2452:
1787:
14732:
is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function
7197:
3022:
2088:
17659:
13970:
742:
51:
16243:
11669:
11600:
itself, just a less stringent smoothness assumption. Meanwhile, since there are "too many" boundary conditions on
6595:
17304:
16804:
14027:
This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators
13307:
10985:
10879:
9180:
8639:
7872:
7054:
3708:
198:
11744:
are needed to cancel out the boundary terms in the integration by parts. Thus, any sufficiently smooth function
10569:
6192:
5180:
1118:
15043:{\displaystyle P^{\mathrm {*form} }\phi =\sum _{\alpha }D^{\alpha }\left({\overline {a_{\alpha }}}\phi \right)}
12336:
A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from
12097:
10535:
10250:
5777:
4368:
3272:
3014:
2136:
2130:
16262:
The reader is invited to perform integration by parts twice and verify that the given boundary conditions for
7650:
6469:
6353:
4326:
3680:
17843:
17455:
17129:
16265:
13330:
8228:{\displaystyle \operatorname {Dom} T_{h}:=\left\{\psi \in L^{2}(X,\mu )\;|\;h\psi \in L^{2}(X,\mu )\right\},}
4589:
3950:
1543:
479:
12211:
12149:
11075:
17910:
17848:
17738:
17511:
17445:
17273:
16174:
6381:
4399:
174:
13635:
11634:
10414:{\displaystyle H_{\text{eff}}^{*}\left|\Psi _{E}^{*}\right\rangle =E^{*}\left|\Psi _{E}^{*}\right\rangle }
2867:
17764:
17475:
16755:
15214:{\displaystyle \operatorname {dom} P^{*}=\left\{u\in L^{2}(M):P^{\mathrm {*form} }u\in L^{2}(M)\right\}.}
13098:
13024:
9704:
7878:
7573:
7535:
664:
15610:{\displaystyle \int _{\mathbf {R} }|\lambda |^{2}\ \|\psi (\lambda )\|^{2}\,d\mu (\lambda )<\infty .}
13241:
are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential
11994:
The problem with the preceding example is that we imposed too many boundary conditions on the domain of
6115:
4501:
17951:
17864:
17720:
17674:
17598:
17480:
13914:
13708:
11958:
11926:
11894:
11858:
11319:
10935:
10627:
7480:
7161:
6157:
5964:
5929:
5297:
The arguments made thus far hold for any symmetric operator. It now follows from self-adjointness that
4807:
4772:
4217:
954:{\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,\quad \forall x\in \operatorname {Dom} A.}
14664:{\displaystyle P(\operatorname {D} )\phi =\sum _{\alpha }c_{\alpha }\operatorname {D} ^{\alpha }\phi }
11794:
8429:
7836:
6269:
4176:
2909:
1001:
18005:
17715:
17531:
16146:
15240:
15068:
is a restriction of the distributional extension of the formal adjoint to an appropriate subspace of
10035:
9838:
8989:
7436:
6805:
6431:
1250:{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,\quad \forall x,y\in \operatorname {Dom} A.}
17905:
15833:, the corresponding measures are equivalent in the sense that they have the same sets of measure 0.
13807:. One can show that each one of these solution spaces is 1-dimensional, generated by the functions
13136:
13062:
12901:
12845:
12809:
12744:
11855:
is not essentially self-adjoint. After all, a general result says that the domain of the adjoint of
11287:{\displaystyle \operatorname {Dom} (A)=\left\{{\text{smooth functions}}\,f\mid f(0)=f(1)=0\right\},}
10134:
10106:
18000:
17971:
17900:
17567:
17465:
17368:
16034:
multiplicity functions agree almost everywhere with respect to the measure in the direct integral.
15829:
This representation is unique in the following sense: For any two such representations of the same
12937:
12280:
7614:
711:
13908:
is symmetric. If we replaced the boundary conditions given above by the single boundary condition
13415:
13172:
10675:
9270:
6068:
4679:
17995:
17664:
17440:
10662:
8240:
6795:{\displaystyle \operatorname {Dom} (A-\lambda I)\subsetneq \operatorname {Dom} (A^{*}-\lambda I)}
4255:
3220:
2455:
352:
12780:
12484:
12452:
12337:
9214:
9185:
6720:{\displaystyle H=\operatorname {Im} (A-\lambda I)\subseteq \operatorname {Im} (A^{*}-\lambda I)}
2784:{\displaystyle \left\|A\right\|=\sup \left\{|\langle x,Ax\rangle |:x\in H,\,\|x\|\leq 1\right\}}
204:
197:
are represented by self-adjoint operators on a Hilbert space. Of particular significance is the
17869:
17838:
17757:
17695:
17639:
17603:
13967:. Other essentially self-adjoint extensions come from imposing boundary conditions of the form
12985:
11065:(the space of square-integrable functions on the interval ). Let us define a momentum operator
10300:
8424:
7236:
1079:
92:
12177:
9133:
7105:
5712:{\displaystyle \left(\operatorname {Im} R_{\lambda }\right)^{\perp }=\ker R_{\bar {\lambda }}}
2827:
2524:
2460:
2419:
2387:
2316:
2197:
17874:
17817:
17402:
14423:{\displaystyle x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{n}^{\alpha _{n}}.}
13394:, defined initially on smooth compactly supported functions, is essentially self-adjoint for
11020:
10561:
10294:
8878:{\displaystyle UAU^{-1}\psi (x)=h(x)\psi (x),\quad \forall \psi \in U\operatorname {Dom} (A)}
7724:
5561:
5151:
4990:
2951:
1622:
337:
17398:
13244:
12418:
11596:
That is to say, the domain of the closure has the same boundary conditions as the domain of
8101:{\displaystyle T_{h}\psi (x)=h(x)\psi (x),\quad \forall \psi \in \operatorname {Dom} T_{h},}
2352:
1826:
1408:
17803:
17678:
17221:
17171:
17019:
16967:
16866:
16823:
16774:
15925:(or equivalently its sets of measure 0) is uniquely determined and the measurable function
15071:
14154:
13690:
13537:
13496:
13274:
13217:
11767:
11607:
11482:
10726:
10160:
9243:
8395:
8368:
7757:
7697:
7645:
6151:
5120:
1740:
1458:
1361:
787:
415:
108:
39:
17265:
17156:
13441:
9104:
9035:
1332:
8:
17799:
17644:
17582:
17296:
13316:, self-adjoint operators are precisely the infinitesimal generators of unitary groups of
11195:
is not symmetric (because the boundary terms in the integration by parts do not vanish).
10324:
7910:
2641:
517:
166:
120:
17211:
17175:
17023:
16971:
16870:
16827:
16778:
6982:{\displaystyle \operatorname {Im} (A-{\bar {\lambda }}I)^{\perp }=\ker(A^{*}-\lambda I)}
5518:
2667:
17669:
17536:
17195:
17121:
16813:
16764:
16223:
14124:{\displaystyle N_{\pm }=\left\{u\in L^{2}(M):P_{\operatorname {dist} }u=\pm iu\right\}}
13891:
13871:
13851:
13476:
12881:
12077:{\displaystyle \operatorname {Dom} (A)=\{{\text{smooth functions}}\,f\mid f(0)=f(1)\}.}
11747:
11727:
11707:
10031:
9747:
9698:
9556:
9536:
9162:
9084:
9064:
9015:
8995:
8960:
8936:
8916:
8896:
8755:
8725:
8705:
8619:
8599:
8270:
8250:
7916:
7804:
7784:
7141:
6249:
6095:
5541:
5300:
4306:
3279:
3253:
3200:
3076:
2983:
2847:
2800:
2647:
2492:
2174:
2064:
2024:
2004:
1980:
1767:
1720:
1655:
1598:
1438:
1388:
1308:
1150:
971:
853:
644:
621:
524:
455:
408:
390:
366:
341:
100:
16889:
16852:
14593:(D) defined on the space of infinitely differentiable functions of compact support on
17925:
17822:
17794:
17649:
17239:
17229:
17213:
Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators
17187:
17143:
17133:
17107:
17072:
17055:
17045:
17027:
16975:
16950:
16940:
16894:
16839:
16790:
16730:
16228:
16126:
10540:
7431:
1326:
182:
170:
135:
104:
17199:
3435:), though non-self-adjoint operators with real spectrum exist as well. For bounded (
17935:
17812:
17654:
17572:
17541:
17521:
17506:
17501:
17496:
17179:
17132:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
16921:
16884:
16874:
16831:
16782:
10552:
10237:{\displaystyle I=\int dE\left|\Psi _{E}\right\rangle \left\langle \Psi _{E}\right|}
10100:
9818:{\displaystyle T=\int _{-\infty }^{+\infty }\lambda d\operatorname {E} (\lambda ).}
8890:
7420:
2192:
781:
772:
348:
330:
322:
190:
127:
116:
17333:
17251:
14035:. They are determined by the unitary maps between the eigenvalue spaces
10829:π; the well-known orthogonality of the sine functions follows as a consequence of
9643:{\displaystyle \operatorname {E} (\lambda )=\mathbf {1} _{(-\infty ,\lambda ]}(T)}
17915:
17516:
17470:
17418:
17413:
17384:
17086:
16233:
15842:
15244:
11181:{\displaystyle \operatorname {Dom} (A)=\left\{{\text{smooth functions}}\right\},}
8954:
8773:
7609:
5145:
3436:
2303:{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle ,\quad \forall x,y\in H.}
965:
509:
473:
311:{\displaystyle {\hat {H}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi ,}
143:
17343:
13888:
is a first-order operator, only one boundary condition is needed to ensure that
7721:
function to be expressed as a "superposition" (i.e., integral) of the functions
17705:
17557:
17358:
16926:
16907:
14943:
13317:
10894:
9938:{\displaystyle f(H)=\int dE\left|\Psi _{E}\rangle f(E)\langle \Psi _{E}\right|}
9850:
9834:
8586:{\displaystyle UAU^{-1}\xi =B\xi ,\quad \forall \xi \in \operatorname {Dom} B.}
6583:{\displaystyle \{A-\lambda I,A-{\bar {\lambda }}I\}:\operatorname {Dom} A\to H}
2346:
194:
151:
14226:{\displaystyle P\left({\vec {x}}\right)=\sum _{\alpha }c_{\alpha }x^{\alpha }}
11306:
17989:
17808:
17780:
17710:
17634:
17363:
17348:
17338:
17243:
17207:
17191:
17147:
17059:
17044:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
16843:
16794:
16019:{\displaystyle \lambda \mapsto \operatorname {dim} \left(H_{\lambda }\right)}
10006:{\displaystyle H\left|\Psi _{E}\right\rangle =E\left|\Psi _{E}\right\rangle }
4762:{\displaystyle \Vert R_{\lambda }x\Vert \geq d(\lambda )\cdot \Vert x\Vert ,}
3806:{\displaystyle \Vert (A-\lambda )x\Vert \geq d(\lambda )\cdot \Vert x\Vert ,}
3065:
2082:
384:
159:
46:
16954:
15911:{\displaystyle \int _{\mathbf {R} }^{\oplus }H_{\lambda }\,d\mu (\lambda ).}
15841:
The spectral multiplicity theorem can be reformulated using the language of
13529:
12445:
potential escapes to infinity in finite time. This operator does not have a
11851:
Since the domain of the closure and the domain of the adjoint do not agree,
7477:, for example, physicists would say that the eigenvectors are the functions
5364:{\displaystyle x\in \operatorname {Dom} A=\operatorname {Dom} R_{\lambda },}
5111:{\displaystyle \|x_{n}-x_{m}\|\leq {\frac {1}{d(\lambda )}}\|y_{n}-y_{m}\|,}
17700:
17353:
17323:
16898:
16879:
13312:
In quantum mechanics, observables correspond to self-adjoint operators. By
12273:
is self-adjoint is a compromise: the domain has to be small enough so that
17095:
Methods of Mathematical Physics: Vol 2: Fourier Analysis, Self-Adjointness
15720:
is unitarily equivalent to the operator of multiplication by the function
11998:. A better choice of domain would be to use periodic boundary conditions:
10984:
This Hamiltonian has pure point spectrum; this is typical for bound state
10084:{\displaystyle \left|\Psi _{E}\right\rangle \left\langle \Psi _{E}\right|}
8776:
Hilbert space is unitarily equivalent to a multiplication operator, i.e.,
4980:{\displaystyle y_{n}=R_{\lambda }x_{n}\in \operatorname {Im} R_{\lambda }}
17930:
17629:
17619:
17526:
17328:
17090:
16769:
14245:
13793:{\displaystyle {\begin{aligned}-iu'&=iu\\-iu'&=-iu\end{aligned}}}
13438:
has a counterpart in the classical dynamics of a particle with potential
10925:. The Hamiltonian for the harmonic oscillator has a quadratic potential
10096:
10092:
9533:
The goal of functional calculus is to extend this idea to the case where
2818:
147:
31:
1298:{\displaystyle \operatorname {Dom} A^{*}\supseteq \operatorname {Dom} A}
1069:{\displaystyle \operatorname {Dom} A\subseteq \operatorname {Dom} A^{*}}
17562:
17394:
17183:
16988:
7828:
7825:"eigenvectors" that are not actually in the Hilbert space in question.
3395:
2794:
178:
81:
16835:
16786:
11069:
on this space by the usual formula, setting the Planck constant to 1:
17106:. Boston, Mass.: McGraw-Hill Science, Engineering & Mathematics.
15709:{\displaystyle \left\{\mu _{\ell }\right\}_{1\leq \ell \leq \omega }}
14315:{\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}
10091:. In the Dirac notation, (projective) measurements are described via
9837:
is used as combined expression for both the spectral theorem and the
6058:{\displaystyle \{(R_{\lambda }x,x)\mid x\in \operatorname {Dom} A\}.}
3504:{\displaystyle S=\{x\in \operatorname {Dom} A\mid \Vert x\Vert =1\},}
3247:
412:
14856:
are (not necessarily constant) infinitely differentiable functions.
13493:
for a particle moving on a half-line. Nevertheless, the Hamiltonian
16818:
16404:
5474:{\displaystyle y=R_{\lambda }x\in \operatorname {Im} R_{\lambda }.}
186:
17069:
Methods of Modern Mathematical Physics: Vol 1: Functional analysis
16294:
ensure that the boundary terms in the integration by parts vanish.
16125:
of the Laplacian −Δ since as an operator it is non-negative; (see
13301:
10860:
then has a countable family of eigenvectors which are complete in
17749:
16428:
12331:
11724:
vanishes at both ends of the interval, no boundary conditions on
3893:{\displaystyle \textstyle d(\lambda )=\inf _{r\in }|r-\lambda |.}
155:
16524:
8000:{\displaystyle T_{h}:\operatorname {Dom} T_{h}\to L^{2}(X,\mu )}
7041:{\displaystyle \operatorname {Im} (A-{\bar {\lambda }}I)\neq H.}
16655:
16619:
14839:{\displaystyle P\phi (x)=\sum _{\alpha }a_{\alpha }(x)\left(x)}
11604:, there are "too few" (actually, none at all in this case) for
9371:
can be defined as the unique bounded operator with eigenvalues
3621:{\displaystyle \textstyle M=\sup _{x\in S}\langle Ax,x\rangle }
3562:{\displaystyle \textstyle m=\inf _{x\in S}\langle Ax,x\rangle }
3398:. The spectrum of a self-adjoint operator is always real (i.e.
1710:{\displaystyle \operatorname {Dom} A=\operatorname {Dom} A^{*}}
611:{\displaystyle G(A)=\{(x,Ax)\mid x\in \operatorname {Dom} A\}.}
16380:
11305:, which makes the domain of the adjoint too big (see also the
11134:, which amounts to choosing boundary conditions. If we choose
10844:
inverse, meaning that the corresponding differential equation
16111:{\displaystyle \Delta =\sum _{i=1}^{n}\partial _{x_{i}}^{2}.}
13467:: The classical particle escapes to infinity in finite time.
8512:{\displaystyle U\operatorname {Dom} A=\operatorname {Dom} B,}
5919:{\displaystyle R_{\lambda }\colon \operatorname {Dom} A\to H}
15968:{\displaystyle \lambda \mapsto \mathrm {dim} (H_{\lambda })}
13327:
The one-dimensional Schrödinger operator with the potential
11297:
then using integration by parts, one can easily verify that
10852:
is solved by some integral (and therefore compact) operator
17295:
16488:
16308:
12717:
9437:{\displaystyle f(\lambda _{j}):=e^{-it\lambda _{j}/\hbar }}
9032:
is a self-adjoint operator, we wish to define the operator
5767:{\displaystyle {\bar {\lambda }}\in \mathbb {C} \setminus }
16966:, Graduate Texts in Mathematics, vol. 267, Springer,
15632:
if and only if they are supported on disjoint Borel sets.
12777:
is not a symmetric operator, which certainly implies that
9576:
It has been customary to introduce the following notation
8244:. Any multiplication operator is a self-adjoint operator.
3942:{\displaystyle x\in \operatorname {Dom} A\setminus \{0\}.}
3347:{\displaystyle \sigma (A)=\mathbb {C} \setminus \rho (A).}
17157:"A remark on bound states in potential-scattering theory"
14927:{\displaystyle C_{0}^{\infty }(M)\to C_{0}^{\infty }(M).}
13530:
A symmetric operator that is not essentially self-adjoint
10815:{\displaystyle f_{n}(x)=\sin(n\pi x)\qquad n=1,2,\ldots }
16500:
14153:
We next give the example of differential operators with
9828:
9523:{\displaystyle f({\hat {H}})e_{j}=f(\lambda _{j})e_{j}.}
4579:{\displaystyle \operatorname {Dom} R_{\lambda }^{-1}=H.}
17170:(4). Springer Science and Business Media LLC: 655–662.
16631:
16476:
16452:
16392:
16368:
16356:
9364:{\displaystyle f({\hat {H}}):=e^{-it{\hat {H}}/\hbar }}
8988:
One application of the spectral theorem is to define a
5416:{\displaystyle Ax=y+\lambda x\in \operatorname {Im} A,}
3670:{\displaystyle m,M\in \mathbb {R} \cup \{\pm \infty \}}
16037:
10992:
10546:
9159:
One example from quantum mechanics is the case where
7439:
6238:
Symmetric operator with real spectrum is self-adjoint
4498:
The goal is to prove the existence and boundedness of
3823:
3576:
3517:
2633:{\displaystyle A:H\to \operatorname {Im} A\subseteq H}
16801:
16434:
16344:
16320:
16268:
16177:
16149:
16055:
15985:
15931:
15861:
15738:
15668:
15521:
15318:
15101:
15074:
14959:
14869:
14756:
14606:
14442:
14334:
14257:
14166:
14044:
13973:
13917:
13894:
13874:
13854:
13720:
13638:
13585:
13540:
13499:
13479:
13444:
13418:
13333:
13277:
13247:
13220:
13175:
13139:
13101:
13065:
13027:
12988:
12940:
12904:
12884:
12848:
12812:
12783:
12747:
12519:
12487:
12455:
12421:
12404:{\displaystyle {\hat {H}}:={\frac {P^{2}}{2m}}-X^{4}}
12350:
12283:
12214:
12180:
12152:
12100:
12007:
11961:
11929:
11897:
11861:
11797:
11770:
11750:
11730:
11710:
11672:
11637:
11610:
11519:
11485:
11361:
11322:
11207:
11143:
11078:
11023:
10938:
10746:
10678:
10630:
10572:
10433:
10336:
10303:
10253:
10182:
10173:, the theorem is referred to as resolution of unity:
10137:
10109:
10044:
9957:
9862:
9763:
9736:. The family of projection operators E(λ) is called
9707:
9659:
9585:
9559:
9539:
9453:
9377:
9300:
9273:
9246:
9217:
9188:
9165:
9136:
9130:
is the operator of multiplication by the composition
9107:
9087:
9067:
9038:
9018:
8998:
8963:
8939:
8919:
8899:
8785:
8758:
8728:
8708:
8642:
8622:
8602:
8526:
8481:
8432:
8398:
8371:
8332:
8293:
8273:
8253:
8120:
8016:
7939:
7919:
7881:
7839:
7807:
7787:
7760:
7727:
7700:
7653:
7617:
7576:
7538:
7483:
7277:
7239:
7200:
7164:
7144:
7108:
7057:
6995:
6904:
6844:
6808:
6733:
6652:
6598:
6519:
6472:
6434:
6384:
6356:
6305:
6272:
6252:
6195:
6160:
6118:
6098:
6071:
6002:
5967:
5932:
5887:
5824:
5780:
5725:
5649:
5597:
5564:
5544:
5521:
5488:
5429:
5376:
5323:
5303:
5242:
5183:
5154:
5123:
5022:
4993:
4924:
4891:
4846:
4810:
4775:
4709:
4682:
4637:
4592:
4542:
4504:
4455:
4402:
4371:
4329:
4309:
4258:
4220:
4179:
3962:
3908:
3822:
3748:
3711:
3683:
3634:
3575:
3516:
3452:
3404:
3363:
3305:
3282:
3256:
3223:
3203:
3102:
3079:
3025:
3008:
2986:
2954:
2912:
2870:
2850:
2830:
2803:
2694:
2670:
2650:
2602:
2559:
2527:
2495:
2463:
2422:
2390:
2355:
2319:
2235:
2200:
2177:
2139:
2091:
2067:
2027:
2007:
1983:
1858:
1829:
1790:
1770:
1743:
1723:
1678:
1658:
1625:
1601:
1546:
1491:
1461:
1441:
1411:
1391:
1364:
1335:
1311:
1266:
1176:
1153:
1121:
1082:
1037:
1004:
974:
879:
856:
817:
790:
745:
714:
667:
647:
624:
547:
527:
482:
458:
423:
393:
369:
239:
207:
173:. In quantum mechanics their importance lies in the
54:
17226:
Topological Vector Spaces, Distributions and Kernels
16440:
14244:
coefficients, where α ranges over a (finite) set of
11987:
is not self-adjoint, which by definition means that
9081:
is represented as the operator of multiplication by
8358:{\displaystyle \operatorname {Dom} B\subseteq H_{2}}
8319:{\displaystyle \operatorname {Dom} A\subseteq H_{1}}
7829:
Multiplication operator form of the spectral theorem
6343:{\displaystyle A-\lambda I\subseteq A^{*}-\lambda I}
5636:{\displaystyle R_{\lambda }^{*}=R_{\bar {\lambda }}}
2586:{\displaystyle \operatorname {Dom} \left(A\right)=H}
843:{\displaystyle \operatorname {Dom} A^{*}\subseteq H}
16999:
16752:
16661:
16464:
16410:
13841:, which in this case happens to be the unit circle
13619:{\displaystyle D:\phi \mapsto {\frac {1}{i}}\phi '}
11891:. Thus, in this case, the domain of the adjoint of
10034:. The resolution of the identity (sometimes called
10016:where the integral runs over the whole spectrum of
7430:some continuous analog thereof. In the case of the
7158:is self-adjoint. Indeed, it suffices to prove that
5290:{\displaystyle y_{n}+\lambda x_{n}\to y+\lambda x.}
4491:{\displaystyle \lambda \in \mathbb {C} \setminus .}
17625:Spectral theory of ordinary differential equations
17003:Quantum mechanics in rigged Hilbert space language
16536:
16286:
16191:
16163:
16110:
16018:
15967:
15910:
15818:
15708:
15658:sequence of countably additive finite measures on
15609:
15474:
15213:
15087:
15042:
14926:
14838:
14704:the Fourier transform considered as a unitary map
14663:
14578:
14422:
14314:
14225:
14123:
14016:
13947:
13900:
13880:
13860:
13792:
13674:
13618:
13568:
13512:
13485:
13459:
13430:
13386:
13290:
13263:
13233:
13206:
13161:
13125:
13087:
13051:
13013:
12974:
12926:
12890:
12870:
12834:
12798:
12769:
12730:
12502:
12470:
12437:
12403:
12320:
12261:
12192:
12166:
12138:
12076:
11979:
11947:
11915:
11879:
11840:
11783:
11756:
11736:
11716:
11696:
11658:
11623:
11585:
11498:
11468:
11340:
11312:Specifically, with the above choice of domain for
11286:
11180:
11119:
11057:
10973:
10814:
10714:
10653:
10613:
10522:
10413:
10315:
10281:
10236:
10151:
10123:
10083:
10005:
9937:
9817:
9728:
9690:{\displaystyle \mathbf {1} _{(-\infty ,\lambda ]}}
9689:
9642:
9565:
9545:
9522:
9436:
9363:
9286:
9259:
9232:
9203:
9171:
9148:
9122:
9093:
9073:
9053:
9024:
9004:
8969:
8945:
8925:
8905:
8877:
8764:
8734:
8714:
8694:
8628:
8608:
8585:
8511:
8464:
8411:
8384:
8357:
8318:
8279:
8259:
8227:
8100:
7999:
7925:
7901:
7863:
7813:
7793:
7773:
7746:
7713:
7683:
7636:
7600:
7562:
7524:
7469:
7402:
7264:
7225:
7186:
7150:
7130:
7094:
7040:
6981:
6890:
6830:
6794:
6719:
6638:
6582:
6505:
6458:
6420:
6370:
6342:
6291:
6258:
6219:
6181:
6142:
6104:
6084:
6057:
5988:
5953:
5918:
5865:
5810:
5766:
5711:
5635:
5583:
5550:
5530:
5507:
5473:
5415:
5363:
5309:
5289:
5228:
5169:
5136:
5110:
5008:
4979:
4910:
4875:{\displaystyle \operatorname {Im} R_{\lambda }=H.}
4874:
4829:
4796:
4761:
4695:
4666:{\displaystyle \operatorname {Im} R_{\lambda }=H.}
4665:
4623:
4578:
4528:
4490:
4438:
4388:
4357:
4315:
4273:
4244:
4206:
4162:
3941:
3892:
3805:
3732:
3697:
3669:
3620:
3561:
3503:
3427:
3386:
3346:
3288:
3262:
3238:
3209:
3186:
3085:
3055:
2992:
2972:
2940:
2898:
2856:
2836:
2809:
2783:
2679:
2656:
2632:
2585:
2545:
2501:
2481:
2440:
2408:
2376:
2337:
2302:
2218:
2183:
2163:
2121:
2073:
2033:
2013:
1989:
1966:
1841:
1815:
1776:
1756:
1729:
1709:
1664:
1644:
1607:
1581:
1526:
1474:
1447:
1427:
1397:
1377:
1350:
1317:
1297:
1249:
1159:
1139:
1107:
1068:
1023:
980:
953:
862:
842:
803:
763:
729:
700:
653:
630:
610:
533:
500:
464:
444:
399:
375:
310:
222:
72:
17039:
15716:such that the measures are pairwise singular and
14148:
2056:
445:{\displaystyle \operatorname {Dom} A\subseteq H.}
17987:
15975:is determined almost everywhere with respect to
12269:. Thus, in this case finding a domain such that
3840:
3584:
3525:
3428:{\displaystyle \sigma (A)\subseteq \mathbb {R} }
3387:{\displaystyle \sigma (A)\subseteq \mathbb {C} }
2709:
1527:{\displaystyle A\subseteq A^{**}\subseteq A^{*}}
321:which as an observable corresponds to the total
16858:Proceedings of the National Academy of Sciences
16208:may be taken to be Lebesgue measure on [0, ∞).
15224:
13314:Stone's theorem on one-parameter unitary groups
13302:Non-self-adjoint operators in quantum mechanics
10539:for the context where such operators appear in
6891:{\displaystyle \ker(A^{*}-\lambda I)\neq \{0\}}
5961:exists and is everywhere defined. The graph of
5866:{\displaystyle \ker R_{\bar {\lambda }}=\{0\}.}
5508:{\displaystyle \operatorname {Im} R_{\lambda }}
4911:{\displaystyle \operatorname {Im} R_{\lambda }}
1816:{\displaystyle \left\langle x,Ax\right\rangle }
411:(i.e. not necessarily bounded) operator with a
107:, this is equivalent to the condition that the
17040:Narici, Lawrence; Beckenstein, Edward (2011).
13707:(defined below) are given respectively by the
13412:The failure of essential self-adjointness for
12332:Schrödinger operators with singular potentials
7226:{\displaystyle x\in \operatorname {Dom} A^{*}}
3056:{\displaystyle A:\operatorname {Dom} (A)\to H}
2122:{\displaystyle A:\operatorname {Dom} (A)\to H}
17765:
17281:
17120:
16743:
16625:
16601:
14017:{\displaystyle \phi (1)=e^{i\theta }\phi (0)}
9240:has a true orthonormal basis of eigenvectors
7532:, which are clearly not in the Hilbert space
764:{\displaystyle \langle \cdot ,\cdot \rangle }
73:{\displaystyle \langle \cdot ,\cdot \rangle }
16853:"An Invariant for Certain Operator Algebras"
15573:
15557:
15437:
15421:
14941:there is another differential operator, the
12068:
12026:
11887:is the same as the domain of the adjoint of
11697:{\displaystyle f\in \operatorname {Dom} (A)}
11653:
11638:
10661:consisting of all complex-valued infinitely
10146:
10118:
10038:) formally resembles the rank-1 projections
9917:
9902:
8676:
8669:
8650:
8643:
6885:
6879:
6639:{\displaystyle A-\lambda I=A^{*}-\lambda I.}
6559:
6520:
6049:
6003:
5857:
5851:
5102:
5076:
5049:
5023:
4753:
4747:
4726:
4710:
4618:
4612:
4154:
4148:
4127:
4121:
4096:
4090:
4075:
4069:
4041:
4035:
4025:
3998:
3984:
3963:
3933:
3927:
3797:
3791:
3770:
3749:
3664:
3655:
3614:
3599:
3555:
3540:
3495:
3486:
3480:
3459:
2961:
2955:
2935:
2929:
2893:
2887:
2767:
2761:
2737:
2722:
2272:
2257:
2251:
2236:
1928:
1913:
1898:
1883:
1874:
1859:
1717:. Equivalently, a closed symmetric operator
1213:
1198:
1192:
1177:
923:
901:
895:
880:
758:
746:
602:
563:
67:
55:
17216:, Providence: American Mathematical Society
16905:
16850:
16727:Theory of Linear Operators in Hilbert Space
13689:is a symmetric operator as can be shown by
13473:There is no self-adjoint momentum operator
13059:, but the combination of them occurring in
8695:{\displaystyle \|A\|_{H_{1}}=\|B\|_{H_{2}}}
7095:{\displaystyle A-\lambda I=A^{*}-\lambda I}
3733:{\displaystyle x\in \operatorname {Dom} A,}
3439:) operators, however, the spectrum is real
17772:
17758:
17288:
17274:
16908:"The structure of intertwining isometries"
15291:is unitarily equivalent to the operator M
10614:{\displaystyle A=-{\frac {d^{2}}{dx^{2}}}}
8182:
8176:
7821:is the variable of the Fourier transform.
6220:{\displaystyle \lambda \notin \sigma (A).}
5229:{\displaystyle y_{n}+\lambda x_{n}=Ax_{n}}
4918:is closed. To prove this, pick a sequence
3170:
1140:{\displaystyle x\in \operatorname {Dom} A}
17085:
17066:
16925:
16888:
16878:
16817:
16768:
16362:
16314:
15889:
15620:Non-negative countably additive measures
15582:
13116:
13042:
12806:is not essentially self-adjoint. Indeed,
12709:
12586:
12160:
12139:{\displaystyle f_{\beta }(x)=e^{\beta x}}
12034:
11820:
11236:
10669:on satisfying the boundary conditions
10282:{\displaystyle H_{\text{eff}}=H-i\Gamma }
7895:
7754:, even though these functions are not in
7591:
7553:
6364:
5811:{\displaystyle d({\bar {\lambda }})>0}
5742:
4463:
4389:{\displaystyle \lambda \in \mathbb {C} .}
4379:
3691:
3648:
3421:
3380:
3322:
3138:
3134:
3130:
2760:
2164:{\displaystyle \operatorname {Dom} (A)=H}
1941:
17578:Group algebra of a locally compact group
17006:(PhD thesis). Universidad de Valladolid.
16993:Perturbation Theory for Linear Operators
16851:Carey, R. W.; Pincus, J. D. (May 1974).
16724:
16494:
16482:
16171:, otherwise −Δ has uniform multiplicity
7684:{\displaystyle \delta \left(p-p'\right)}
6506:{\displaystyle {\bar {\lambda }}\notin }
6371:{\displaystyle \lambda \in \mathbb {C} }
5926:has now been proven to be bijective, so
4358:{\displaystyle R_{\lambda }=A-\lambda I}
4295:Self-adjoint operator has real spectrum
3698:{\displaystyle \lambda \in \mathbb {C} }
2041:is essentially self-adjoint if it has a
452:This condition holds automatically when
177:of quantum mechanics, in which physical
27:Linear operator equal to its own adjoint
17010:
16934:
16386:
16350:
16326:
16287:{\displaystyle \operatorname {Dom} (A)}
15252:
13387:{\displaystyle V(x)=-(1+|x|)^{\alpha }}
12277:is symmetric, but large enough so that
11704:using integration by parts, then since
7426:orthonormal basis in the classic sense
4624:{\displaystyle \ker R_{\lambda }=\{0\}}
1582:{\displaystyle A=A^{**}\subseteq A^{*}}
501:{\displaystyle \operatorname {Dom} A=H}
14:
17988:
17250:
17220:
17206:
17154:
16913:Indiana University Mathematics Journal
16744:Berezin, F. A.; Shubin, M. A. (1991),
16542:
16143:= 1, then −Δ has uniform multiplicity
12262:{\displaystyle f_{n}(x):=e^{2\pi inx}}
12167:{\displaystyle \beta \in \mathbb {C} }
11120:{\displaystyle Af=-i{\frac {df}{dx}}.}
11007:
10873:
8983:
3217:is bounded, the definition reduces to
17753:
17269:
17228:. Mineola, N.Y.: Dover Publications.
17101:
16446:
16374:
16192:{\displaystyle {\text{mult}}=\omega }
15662:(some of which may be identically 0)
13169:to be nonsymmetric, even though both
12481:In this case, if we initially define
10024:is diagonalized by the eigenvectors Ψ
9829:Formulation in the physics literature
9061:. The spectral theorem shows that if
6421:{\displaystyle \sigma (A)\subseteq .}
4439:{\displaystyle \sigma (A)\subseteq .}
2489:is self-adjoint if the Hilbert space
16987:
16961:
16906:Carey, R. W.; Pincus, J. D. (1973).
16708:
16696:
16684:
16672:
16649:
16637:
16613:
16589:
16577:
16565:
16553:
16530:
16518:
16506:
16470:
16458:
16422:
16398:
16338:
15504:consists of vector-valued functions
13675:{\displaystyle \phi (0)=\phi (1)=0.}
13534:We first consider the Hilbert space
12563:twice differentiable functions
11659:{\displaystyle \langle g,Af\rangle }
10733:is symmetric. The eigenfunctions of
9573:has no normalizable eigenvectors).
9553:has continuous spectrum (i.e. where
2899:{\displaystyle |\lambda |\leq \|A\|}
2448:are bounded self-adjoint operators.
16038:Example: structure of the Laplacian
15836:
14141:is the distributional extension of
13126:{\displaystyle L^{2}(\mathbb {R} )}
13052:{\displaystyle L^{2}(\mathbb {R} )}
12174:are eigenvectors, with eigenvalues
11562: with two derivatives in
11409: with two derivatives in
10993:Symmetric vs self-adjoint operators
10547:Formulation for symmetric operators
10424:and write the spectral theorem as:
9729:{\displaystyle (-\infty ,\lambda ]}
9012:is a function on the real line and
7902:{\displaystyle h:X\to \mathbb {R} }
7601:{\displaystyle L^{2}(\mathbb {R} )}
7563:{\displaystyle L^{2}(\mathbb {R} )}
7414:
7048:This contradicts the bijectiveness.
2129:a symmetric operator. According to
701:{\displaystyle G(A)\subseteq G(B).}
165:Self-adjoint operators are used in
150:. This article deals with applying
24:
17779:
16084:
16056:
15945:
15942:
15939:
15601:
15464:
15297:of multiplication by the function
15172:
15169:
15166:
15163:
14978:
14975:
14972:
14969:
14907:
14880:
14646:
14613:
14545:
14513:
14484:
11971:
11968:
11939:
11936:
11907:
11904:
11871:
11868:
11479:whereas the domain of the adjoint
11381:
11378:
11332:
11329:
10942:
10638:
10635:
10632:
10502:
10472:
10393:
10357:
10310:
10276:
10221:
10203:
10143:
10115:
10068:
10050:
9990:
9966:
9921:
9893:
9797:
9786:
9778:
9714:
9673:
9617:
9586:
8848:
8562:
8070:
7849:
6143:{\displaystyle R_{\lambda }^{-1}.}
4529:{\displaystyle R_{\lambda }^{-1},}
3661:
3394:consists exclusively of (complex)
3139:
3009:Spectrum of self-adjoint operators
2279:
1949:
1220:
930:
284:
25:
18017:
17896:Compact operator on Hilbert space
16964:Quantum Theory for Mathematicians
16662:Bonneau, Faraut & Valent 2001
16435:Bebiano & da Providência 2019
16219:Compact operator on Hilbert space
13948:{\displaystyle \phi (0)=\phi (1)}
12741:It is then possible to show that
11991:is not essentially self-adjoint.
11980:{\displaystyle A^{\mathrm {cl} }}
11948:{\displaystyle A^{\mathrm {cl} }}
11916:{\displaystyle A^{\mathrm {cl} }}
11880:{\displaystyle A^{\mathrm {cl} }}
11341:{\displaystyle A^{\mathrm {cl} }}
11130:We must now specify a domain for
10999:Extensions of symmetric operators
10974:{\displaystyle -\Delta +|x|^{2}.}
10856:. The compact symmetric operator
10654:{\displaystyle \mathrm {Dom} (A)}
9429:
9356:
7525:{\displaystyle f_{p}(x):=e^{ipx}}
7187:{\displaystyle A^{*}\subseteq A.}
6182:{\displaystyle R_{\lambda }^{-1}}
5989:{\displaystyle R_{\lambda }^{-1}}
5954:{\displaystyle R_{\lambda }^{-1}}
5746:
4830:{\displaystyle \ker R_{\lambda }}
4797:{\displaystyle d(\lambda )>0.}
4467:
4245:{\displaystyle d(\lambda )>0,}
3924:
3326:
1358:is the graph of an operator). If
1325:, symmetric operators are always
17734:
17733:
17660:Topological quantum field theory
17000:de la Madrid Modino, R. (2001).
15868:
15798:
15789:
15646:be a self-adjoint operator on a
15528:
15494:is a Hilbert space of dimension
15415:
15388:
15379:
15349:
15340:
11841:{\displaystyle A^{*}g=-i\,dg/dx}
10866:. The same can then be said for
10729:of the inner product shows that
9662:
9606:
8465:{\displaystyle U:H_{1}\to H_{2}}
7864:{\displaystyle (X,\Sigma ,\mu )}
7470:{\textstyle P=-i{\frac {d}{dx}}}
6292:{\displaystyle A\subseteq A^{*}}
4207:{\displaystyle \lambda \notin ,}
2941:{\displaystyle |\lambda |=\|A\|}
2521:A bounded self-adjoint operator
1592:for closed symmetric operators.
1405:, the smallest closed extension
1024:{\displaystyle A\subseteq A^{*}}
16805:Journal of Mathematical Physics
16702:
16690:
16678:
16666:
16643:
16607:
16595:
16583:
16571:
16559:
16547:
16512:
16416:
16256:
16164:{\displaystyle {\text{mult}}=2}
15247:theory of spectral multiplicity
14674:is essentially self-adjoint on
13823:respectively. This shows that
13308:Non-Hermitian quantum mechanics
10915:consisting of eigenvectors for
10880:Discrete spectrum (mathematics)
10790:
10131:, if the system is prepared in
8847:
8561:
8069:
6831:{\displaystyle A^{*}-\lambda I}
6459:{\displaystyle \lambda \notin }
2817:are real and the corresponding
2593:has the following properties:
2278:
2021:is self-adjoint. Equivalently,
1948:
1737:is self-adjoint if and only if
1219:
929:
512:on a finite-dimensional space.
16725:Akhiezer, Naum Ilʹich (1981).
16332:
16281:
16275:
16028:spectral multiplicity function
15989:
15962:
15949:
15935:
15902:
15896:
15595:
15589:
15569:
15563:
15544:
15535:
15458:
15452:
15433:
15427:
15383:
15200:
15194:
15148:
15142:
14918:
14912:
14894:
14891:
14885:
14833:
14827:
14801:
14795:
14769:
14763:
14616:
14610:
14475:
14467:
14309:
14264:
14180:
14149:Constant-coefficient operators
14085:
14079:
14011:
14005:
13983:
13977:
13942:
13936:
13927:
13921:
13663:
13657:
13648:
13642:
13595:
13576:and the differential operator
13563:
13551:
13454:
13448:
13375:
13370:
13362:
13352:
13343:
13337:
13162:{\displaystyle {\hat {H}}^{*}}
13147:
13120:
13112:
13088:{\displaystyle {\hat {H}}^{*}}
13073:
13046:
13038:
13008:
13002:
12927:{\displaystyle {\hat {H}}^{*}}
12912:
12871:{\displaystyle {\hat {H}}^{*}}
12856:
12835:{\displaystyle {\hat {H}}^{*}}
12820:
12790:
12770:{\displaystyle {\hat {H}}^{*}}
12755:
12713:
12705:
12684:
12678:
12590:
12582:
12537:
12494:
12462:
12357:
12315:
12309:
12300:
12287:
12231:
12225:
12117:
12111:
12065:
12059:
12050:
12044:
12020:
12014:
11691:
11685:
11449:
11443:
11434:
11428:
11267:
11261:
11252:
11246:
11220:
11214:
11156:
11150:
11052:
11049:
11037:
11034:
10958:
10949:
10787:
10775:
10763:
10757:
10703:
10697:
10688:
10682:
10648:
10642:
10494:
10488:
10152:{\displaystyle |\Psi \rangle }
10139:
10124:{\displaystyle |\Psi \rangle }
10111:
9914:
9908:
9872:
9866:
9809:
9803:
9723:
9708:
9682:
9667:
9637:
9631:
9626:
9611:
9598:
9592:
9504:
9491:
9472:
9466:
9457:
9394:
9381:
9345:
9319:
9313:
9304:
9224:
9195:
9117:
9111:
9048:
9042:
8872:
8866:
8841:
8835:
8829:
8823:
8814:
8808:
8449:
8214:
8202:
8178:
8173:
8161:
8063:
8057:
8051:
8045:
8036:
8030:
7994:
7982:
7969:
7891:
7858:
7840:
7595:
7587:
7557:
7549:
7500:
7494:
7382:
7361:
7346:
7343:
7322:
7300:
7297:
7026:
7017:
7002:
6976:
6954:
6936:
6926:
6911:
6873:
6851:
6789:
6767:
6755:
6740:
6714:
6692:
6680:
6665:
6574:
6550:
6500:
6488:
6479:
6453:
6441:
6412:
6400:
6394:
6388:
6211:
6205:
6028:
6006:
5910:
5841:
5799:
5793:
5784:
5761:
5749:
5732:
5702:
5626:
5269:
5148:. Hence, it converges to some
5070:
5064:
4785:
4779:
4741:
4735:
4482:
4470:
4430:
4418:
4412:
4406:
4230:
4224:
4198:
4186:
4142:
4136:
4029:
4013:
4001:
3994:
3978:
3966:
3882:
3868:
3862:
3850:
3833:
3827:
3785:
3779:
3764:
3752:
3414:
3408:
3373:
3367:
3338:
3332:
3315:
3309:
3158:
3142:
3112:
3106:
3063:be an unbounded operator. The
3047:
3044:
3038:
3015:Spectrum (functional analysis)
2922:
2914:
2880:
2872:
2741:
2718:
2702:
2696:
2612:
2537:
2473:
2432:
2400:
2329:
2210:
2152:
2146:
2113:
2110:
2104:
2057:Bounded self-adjoint operators
1345:
1339:
692:
686:
677:
671:
581:
566:
557:
551:
358:
246:
214:
13:
1:
17456:Uniform boundedness principle
16718:
13271:with the confining potential
12975:{\displaystyle d^{2}f/dx^{2}}
12321:{\displaystyle D(A^{*})=D(A)}
12091:is essentially self-adjoint.
11923:is bigger than the domain of
10020:. The notation suggests that
7637:{\displaystyle \delta _{i,j}}
6838:would not be injective (i.e.
3296:is defined as the complement
3002:compact self-adjoint operator
2516:
1595:The densely defined operator
730:{\displaystyle A\subseteq B.}
175:Dirac–von Neumann formulation
17124:; Wolff, Manfred P. (1999).
17067:Reed, M.; Simon, B. (1980).
15225:Spectral multiplicity theory
15027:
13431:{\displaystyle \alpha >2}
13207:{\displaystyle d^{2}/dx^{2}}
11316:, the domain of the closure
10715:{\displaystyle f(0)=f(1)=0.}
10030:. Such a notation is purely
9287:{\displaystyle \lambda _{j}}
8722:is self-adjoint, then so is
7608:, after replacing the usual
6085:{\displaystyle R_{\lambda }}
4696:{\displaystyle R_{\lambda }}
4323:be self-adjoint and denote
1932:
1902:
1537:for symmetric operators and
1167:is symmetric if and only if
142:relative to this basis is a
126:. By the finite-dimensional
7:
16937:Applied functional analysis
16756:American Journal of Physics
16533:, pp. 144–147, 206–207
16212:
13524:
10293:and a skew-Hermitian (see
10289:is the sum of an Hermitian
9746:. Moreover, the following
5719:. The subsequent inclusion
4274:{\displaystyle A-\lambda I}
3239:{\displaystyle A-\lambda I}
3173:bounded and densely defined
850:consisting of the elements
521:of an (arbitrary) operator
91:to itself) that is its own
10:
18022:
17865:Hilbert projection theorem
17599:Invariant subspace problem
16927:10.1512/iumj.1973.22.22056
16249:
15406: measurable and
15267:. A self-adjoint operator
13711:solutions to the equation
13305:
12799:{\displaystyle {\hat {H}}}
12503:{\displaystyle {\hat {H}}}
12471:{\displaystyle {\hat {H}}}
10996:
10901:has an orthonormal basis {
10877:
10825:with the real eigenvalues
10536:Feshbach–Fano partitioning
10036:projection-valued measures
9739:resolution of the identity
9233:{\displaystyle {\hat {H}}}
9204:{\displaystyle {\hat {H}}}
8752:Any self-adjoint operator
8423:if and only if there is a
7418:
4396:It suffices to prove that
3012:
2191:is necessarily bounded. A
2131:Hellinger–Toeplitz theorem
1652:, that is, if and only if
351:equivalent to real-valued
340:are an important class of
223:{\displaystyle {\hat {H}}}
17944:
17888:
17857:
17844:Cauchy–Schwarz inequality
17831:
17787:
17729:
17688:
17612:
17591:
17550:
17489:
17431:
17377:
17319:
17312:
17126:Topological Vector Spaces
17042:Topological Vector Spaces
16626:Berezin & Shubin 1991
16602:Berezin & Shubin 1991
16244:Helffer–Sjöstrand formula
16199:. Moreover, the measure
15271:has uniform multiplicity
14696:a polynomial function on
14433:We also use the notation
13014:{\displaystyle x^{4}f(x)}
10316:{\displaystyle -i\Gamma }
9839:Borel functional calculus
7265:{\displaystyle y=A^{*}x,}
4586:We begin by showing that
3951:Cauchy–Schwarz inequality
3446:As a preliminary, define
1385:is a closed extension of
1108:{\displaystyle Ax=A^{*}x}
17568:Spectrum of a C*-algebra
16746:The Schrödinger Equation
16411:de la Madrid Modino 2001
16301:
14700:with real coefficients,
12193:{\displaystyle -i\beta }
10885:A self-adjoint operator
10878:Not to be confused with
9149:{\displaystyle f\circ h}
8893:. We might note that if
8596:If unitarily equivalent
8247:Secondly, two operators
7131:{\displaystyle A=A^{*},}
6266:is symmetric; therefore
5538:The self-adjointness of
4840:It remains to show that
2837:{\displaystyle \lambda }
2546:{\displaystyle A:H\to H}
2482:{\displaystyle A:H\to H}
2441:{\displaystyle B:H\to H}
2409:{\displaystyle A:H\to H}
2338:{\displaystyle T:H\to H}
2219:{\displaystyle A:H\to H}
1999:essentially self-adjoint
353:multiplication operators
162:of arbitrary dimension.
138:such that the matrix of
17665:Noncommutative geometry
16939:. Mineola, N.Y: Dover.
16935:Griffel, D. H. (2002).
16389:, pp. 224–230, 241
11058:{\displaystyle L^{2}()}
8933:, then the spectrum of
8241:multiplication operator
7747:{\displaystyle e^{ipx}}
5584:{\displaystyle A^{*}=A}
5170:{\displaystyle x\in H.}
5009:{\displaystyle y\in H.}
4703:is bounded below, i.e.
2973:{\displaystyle \|x\|=1}
2456:bounded linear operator
2313:Every bounded operator
1645:{\displaystyle A=A^{*}}
17721:Tomita–Takesaki theory
17696:Approximation property
17640:Calculus of variations
17102:Rudin, Walter (1991).
16880:10.1073/pnas.71.5.1952
16687:Theorems 7.19 and 10.9
16288:
16193:
16165:
16112:
16082:
16020:
15969:
15912:
15820:
15710:
15611:
15476:
15215:
15089:
15044:
14928:
14840:
14665:
14580:
14424:
14316:
14227:
14125:
14018:
13949:
13902:
13882:
13862:
13794:
13676:
13620:
13570:
13514:
13487:
13461:
13432:
13388:
13292:
13265:
13264:{\displaystyle -x^{4}}
13235:
13208:
13163:
13127:
13089:
13053:
13015:
12976:
12928:
12892:
12878:: There are functions
12872:
12836:
12800:
12771:
12732:
12504:
12472:
12439:
12438:{\displaystyle -x^{4}}
12405:
12322:
12263:
12194:
12168:
12140:
12078:
11981:
11949:
11917:
11881:
11842:
11785:
11758:
11738:
11718:
11698:
11660:
11625:
11587:
11500:
11470:
11342:
11288:
11182:
11121:
11059:
10975:
10840:can be seen to have a
10816:
10716:
10655:
10615:
10524:
10415:
10317:
10283:
10238:
10153:
10125:
10085:
10007:
9939:
9833:In quantum mechanics,
9819:
9730:
9691:
9644:
9567:
9547:
9524:
9438:
9365:
9288:
9261:
9234:
9205:
9173:
9150:
9124:
9095:
9075:
9055:
9026:
9006:
8971:
8947:
8927:
8907:
8879:
8766:
8736:
8716:
8696:
8630:
8610:
8587:
8513:
8466:
8425:unitary transformation
8413:
8386:
8359:
8320:
8281:
8261:
8229:
8102:
8001:
7927:
7903:
7873:σ-finite measure space
7865:
7815:
7795:
7775:
7748:
7715:
7685:
7638:
7602:
7564:
7526:
7471:
7404:
7266:
7227:
7188:
7152:
7132:
7096:
7042:
6983:
6892:
6832:
6796:
6721:
6640:
6584:
6507:
6460:
6422:
6372:
6344:
6293:
6260:
6221:
6183:
6144:
6106:
6086:
6059:
5990:
5955:
5920:
5867:
5812:
5768:
5713:
5637:
5585:
5552:
5532:
5509:
5475:
5417:
5365:
5311:
5291:
5230:
5171:
5138:
5112:
5010:
4981:
4912:
4876:
4831:
4798:
4763:
4697:
4667:
4625:
4580:
4530:
4492:
4440:
4390:
4359:
4317:
4275:
4246:
4208:
4164:
3943:
3894:
3807:
3734:
3699:
3671:
3622:
3563:
3505:
3429:
3388:
3357:In finite dimensions,
3348:
3290:
3264:
3240:
3211:
3188:
3087:
3057:
2994:
2974:
2942:
2900:
2858:
2838:
2811:
2785:
2681:
2658:
2634:
2587:
2547:
2503:
2483:
2442:
2410:
2378:
2377:{\displaystyle T=A+iB}
2345:can be written in the
2339:
2304:
2220:
2185:
2165:
2123:
2075:
2035:
2015:
1991:
1968:
1843:
1842:{\displaystyle x\in H}
1817:
1784:is self-adjoint, then
1778:
1758:
1731:
1711:
1666:
1646:
1609:
1583:
1528:
1476:
1449:
1429:
1428:{\displaystyle A^{**}}
1399:
1379:
1352:
1319:
1299:
1251:
1161:
1141:
1109:
1070:
1025:
982:
955:
864:
844:
805:
765:
739:Let the inner product
731:
702:
655:
632:
612:
535:
502:
466:
446:
401:
377:
338:Differential operators
325:of a particle of mass
312:
224:
74:
17875:Polarization identity
17818:Orthogonal complement
17716:Banach–Mazur distance
17679:Generalized functions
16652:Chapter 2, Exercise 4
16363:Reed & Simon 1980
16315:Reed & Simon 1980
16289:
16194:
16166:
16113:
16062:
16021:
15970:
15913:
15821:
15711:
15612:
15477:
15216:
15090:
15088:{\displaystyle L^{2}}
15045:
14929:
14860:is a linear operator
14841:
14744:is an open subset of
14666:
14581:
14425:
14317:
14228:
14155:constant coefficients
14126:
14019:
13950:
13903:
13883:
13863:
13795:
13677:
13621:
13571:
13569:{\displaystyle L^{2}}
13515:
13513:{\displaystyle p^{2}}
13488:
13462:
13433:
13389:
13293:
13291:{\displaystyle x^{4}}
13266:
13236:
13234:{\displaystyle X^{4}}
13209:
13164:
13128:
13090:
13054:
13016:
12977:
12929:
12893:
12873:
12837:
12801:
12772:
12733:
12505:
12473:
12440:
12406:
12338:Schrödinger operators
12323:
12264:
12195:
12169:
12141:
12079:
11982:
11955:itself, showing that
11950:
11918:
11882:
11843:
11786:
11784:{\displaystyle A^{*}}
11759:
11739:
11719:
11699:
11661:
11626:
11624:{\displaystyle A^{*}}
11588:
11501:
11499:{\displaystyle A^{*}}
11471:
11343:
11289:
11183:
11122:
11060:
10976:
10817:
10717:
10656:
10616:
10562:differential operator
10525:
10416:
10318:
10295:skew-Hermitian matrix
10284:
10239:
10154:
10126:
10086:
10008:
9940:
9820:
9731:
9692:
9645:
9568:
9548:
9525:
9439:
9366:
9289:
9262:
9260:{\displaystyle e_{j}}
9235:
9206:
9174:
9151:
9125:
9096:
9076:
9056:
9027:
9007:
8972:
8948:
8928:
8913:is multiplication by
8908:
8880:
8767:
8737:
8717:
8697:
8631:
8611:
8588:
8514:
8467:
8414:
8412:{\displaystyle H_{2}}
8387:
8385:{\displaystyle H_{1}}
8360:
8321:
8282:
8262:
8230:
8103:
8002:
7928:
7904:
7866:
7816:
7796:
7776:
7774:{\displaystyle L^{2}}
7749:
7716:
7714:{\displaystyle L^{2}}
7686:
7639:
7603:
7565:
7527:
7472:
7405:
7267:
7228:
7189:
7153:
7133:
7097:
7043:
6984:
6893:
6833:
6797:
6722:
6641:
6585:
6508:
6461:
6423:
6373:
6345:
6294:
6261:
6222:
6184:
6145:
6107:
6087:
6060:
5991:
5956:
5921:
5868:
5813:
5769:
5714:
5638:
5586:
5553:
5533:
5510:
5476:
5418:
5366:
5312:
5292:
5231:
5172:
5139:
5137:{\displaystyle x_{n}}
5113:
5011:
4982:
4913:
4877:
4832:
4799:
4764:
4698:
4668:
4626:
4581:
4531:
4493:
4441:
4391:
4360:
4318:
4276:
4247:
4209:
4165:
3944:
3895:
3808:
3735:
3700:
3672:
3623:
3564:
3506:
3430:
3389:
3349:
3291:
3265:
3241:
3212:
3189:
3088:
3058:
2995:
2975:
2943:
2901:
2859:
2839:
2812:
2786:
2682:
2659:
2640:is invertible if the
2635:
2588:
2548:
2504:
2484:
2451:Alternatively, every
2443:
2411:
2379:
2340:
2305:
2221:
2186:
2166:
2124:
2076:
2049:In physics, the term
2036:
2016:
1992:
1977:A symmetric operator
1969:
1844:
1818:
1779:
1759:
1757:{\displaystyle A^{*}}
1732:
1712:
1667:
1647:
1610:
1584:
1529:
1477:
1475:{\displaystyle A^{*}}
1455:must be contained in
1450:
1430:
1400:
1380:
1378:{\displaystyle A^{*}}
1353:
1329:(i.e. the closure of
1320:
1300:
1252:
1162:
1142:
1110:
1071:
1026:
983:
956:
865:
845:
811:acts on the subspace
806:
804:{\displaystyle A^{*}}
766:
732:
703:
656:
633:
613:
536:
503:
467:
447:
402:
378:
313:
225:
119:, i.e., equal to its
75:
36:self-adjoint operator
18:Self adjoint operator
17849:Riesz representation
17804:L-semi-inner product
17461:Kakutani fixed-point
17446:Riesz representation
16995:, New York: Springer
16962:Hall, B. C. (2013),
16266:
16175:
16147:
16053:
15983:
15929:
15859:
15736:
15666:
15654:. Then there is an
15519:
15316:
15259:uniform multiplicity
15253:Uniform multiplicity
15099:
15072:
14957:
14867:
14754:
14604:
14440:
14332:
14255:
14164:
14042:
13971:
13915:
13892:
13872:
13852:
13718:
13691:integration by parts
13636:
13583:
13538:
13497:
13477:
13460:{\displaystyle V(x)}
13442:
13416:
13331:
13275:
13245:
13218:
13173:
13137:
13099:
13063:
13025:
12986:
12938:
12902:
12882:
12846:
12810:
12781:
12745:
12517:
12485:
12453:
12419:
12348:
12281:
12212:
12178:
12150:
12098:
12005:
11959:
11927:
11895:
11859:
11795:
11768:
11764:is in the domain of
11748:
11728:
11708:
11670:
11635:
11608:
11517:
11483:
11359:
11320:
11205:
11141:
11076:
11021:
10936:
10744:
10727:integration by parts
10676:
10628:
10570:
10431:
10334:
10301:
10251:
10180:
10161:rigged Hilbert space
10135:
10107:
10042:
9955:
9860:
9845:is self-adjoint and
9761:
9705:
9657:
9583:
9557:
9537:
9451:
9375:
9298:
9271:
9244:
9215:
9186:
9181:Hamiltonian operator
9163:
9134:
9123:{\displaystyle f(T)}
9105:
9085:
9065:
9054:{\displaystyle f(T)}
9036:
9016:
8996:
8961:
8937:
8917:
8897:
8783:
8756:
8726:
8706:
8640:
8620:
8600:
8524:
8479:
8430:
8421:unitarily equivalent
8419:, respectively, are
8396:
8369:
8330:
8291:
8271:
8251:
8118:
8014:
7937:
7933:. Then the operator
7917:
7879:
7837:
7805:
7785:
7758:
7725:
7698:
7651:
7646:Dirac delta function
7615:
7574:
7536:
7481:
7437:
7275:
7237:
7198:
7162:
7142:
7106:
7055:
6993:
6902:
6842:
6806:
6731:
6650:
6596:
6517:
6470:
6432:
6382:
6354:
6303:
6270:
6250:
6193:
6158:
6152:closed graph theorem
6116:
6096:
6069:
6000:
5965:
5930:
5885:
5822:
5778:
5723:
5647:
5595:
5562:
5542:
5519:
5486:
5427:
5374:
5321:
5301:
5240:
5181:
5152:
5121:
5020:
4991:
4922:
4889:
4844:
4808:
4773:
4707:
4680:
4635:
4590:
4540:
4502:
4453:
4400:
4369:
4327:
4307:
4256:
4218:
4177:
3960:
3906:
3820:
3746:
3709:
3681:
3632:
3573:
3514:
3450:
3402:
3361:
3303:
3280:
3254:
3221:
3201:
3100:
3077:
3023:
2984:
2952:
2910:
2868:
2848:
2844:is an eigenvalue of
2828:
2801:
2692:
2668:
2648:
2600:
2557:
2525:
2493:
2461:
2420:
2388:
2353:
2317:
2233:
2226:is self-adjoint if
2198:
2175:
2137:
2089:
2065:
2025:
2005:
1981:
1856:
1827:
1788:
1768:
1741:
1721:
1676:
1656:
1623:
1599:
1544:
1489:
1459:
1439:
1409:
1389:
1362:
1351:{\displaystyle G(A)}
1333:
1309:
1264:
1174:
1151:
1119:
1080:
1035:
1002:
972:
877:
854:
815:
788:
743:
712:
665:
645:
622:
545:
525:
480:
456:
421:
391:
367:
237:
205:
146:with entries in the
52:
40:complex vector space
17870:Parseval's identity
17839:Bessel's inequality
17645:Functional calculus
17604:Mahler's conjecture
17583:Von Neumann algebra
17297:Functional analysis
17256:Functional Analysis
17176:1969NCimA..61..655R
17155:Ruelle, D. (1969).
17122:Schaefer, Helmut H.
17104:Functional Analysis
17024:2017stqm.book.....M
17018:, Springer-Verlag,
16972:2013qtm..book.....H
16871:1974PNAS...71.1952C
16828:2019JMP....60a2104B
16779:2001AmJPh..69..322B
16509:, pp. 127, 207
16401:, pp. 133, 177
16137: —
16104:
15878:
15853: —
15845:of Hilbert spaces:
15782:
15640: —
15333:
15058: —
14911:
14884:
14690: —
14572:
14540:
14511:
14416:
14391:
14369:
14236:be a polynomial on
11015:boundary conditions
11008:Boundary conditions
10874:Pure point spectrum
10515:
10406:
10370:
10351:
9790:
9750:representation for
8990:functional calculus
8984:Functional calculus
8750: —
8287:with dense domains
7911:measurable function
6590:are both bijective.
6236: —
6178:
6136:
6092:is closed (because
5985:
5950:
5818:and, consequently,
5612:
4987:converging to some
4566:
4522:
4293: —
708:This is written as
342:unbounded operators
167:functional analysis
121:conjugate transpose
17670:Riemann hypothesis
17369:Topological vector
17184:10.1007/bf02819607
17164:Il Nuovo Cimento A
17071:. Academic Press.
16729:. Boston: Pitman.
16640:, pp. 193–196
16497:, pp. 115–116
16461:, pp. 123–130
16377:, pp. 326–327
16317:, pp. 255–256
16284:
16224:Unbounded operator
16189:
16161:
16135:
16108:
16083:
16016:
15965:
15908:
15862:
15851:
15816:
15761:
15760:
15706:
15638:
15607:
15498:. The domain of M
15472:
15319:
15211:
15085:
15056:
15040:
14999:
14924:
14897:
14870:
14836:
14784:
14688:
14661:
14634:
14589:Then the operator
14576:
14544:
14512:
14483:
14420:
14395:
14370:
14348:
14312:
14223:
14202:
14121:
14014:
13945:
13898:
13878:
13858:
13790:
13788:
13672:
13616:
13566:
13510:
13483:
13457:
13428:
13384:
13288:
13261:
13231:
13204:
13159:
13133:. This allows for
13123:
13085:
13049:
13011:
12972:
12934:for which neither
12924:
12888:
12868:
12832:
12796:
12767:
12728:
12500:
12468:
12435:
12401:
12318:
12259:
12190:
12164:
12136:
12087:With this domain,
12074:
11977:
11945:
11913:
11877:
11838:
11781:
11754:
11734:
11714:
11694:
11656:
11621:
11583:
11496:
11466:
11338:
11284:
11178:
11117:
11055:
10971:
10812:
10737:are the sinusoids
10712:
10651:
10611:
10520:
10501:
10411:
10392:
10356:
10337:
10323:, one defines the
10313:
10279:
10234:
10149:
10121:
10081:
10003:
9935:
9815:
9770:
9748:Stieltjes integral
9726:
9699:indicator function
9687:
9640:
9563:
9543:
9520:
9434:
9361:
9284:
9257:
9230:
9201:
9169:
9146:
9120:
9091:
9071:
9051:
9022:
9002:
8967:
8943:
8923:
8903:
8875:
8762:
8748:
8732:
8712:
8692:
8636:are bounded, then
8626:
8606:
8583:
8509:
8462:
8409:
8382:
8365:in Hilbert spaces
8355:
8316:
8277:
8257:
8225:
8098:
7997:
7923:
7899:
7861:
7811:
7791:
7771:
7744:
7711:
7681:
7634:
7598:
7560:
7522:
7467:
7400:
7262:
7223:
7184:
7148:
7128:
7092:
7038:
6979:
6888:
6828:
6792:
6717:
6636:
6580:
6513:and the operators
6503:
6456:
6418:
6368:
6340:
6289:
6256:
6244:
6234:
6217:
6179:
6161:
6140:
6119:
6102:
6082:
6055:
5986:
5968:
5951:
5933:
5916:
5863:
5808:
5764:
5709:
5633:
5598:
5581:
5548:
5531:{\displaystyle H.}
5528:
5505:
5471:
5413:
5361:
5307:
5287:
5226:
5167:
5134:
5108:
5006:
4977:
4908:
4872:
4827:
4804:The triviality of
4794:
4759:
4693:
4663:
4621:
4576:
4549:
4526:
4505:
4488:
4436:
4386:
4355:
4313:
4301:
4291:
4271:
4242:
4204:
4160:
3939:
3890:
3889:
3866:
3803:
3730:
3695:
3677:. Then, for every
3667:
3618:
3617:
3598:
3559:
3558:
3539:
3501:
3425:
3384:
3344:
3286:
3260:
3236:
3207:
3184:
3083:
3053:
2990:
2970:
2938:
2896:
2854:
2834:
2807:
2781:
2680:{\displaystyle H.}
2677:
2654:
2630:
2583:
2543:
2499:
2479:
2438:
2406:
2374:
2335:
2300:
2216:
2181:
2161:
2119:
2071:
2031:
2011:
2001:if the closure of
1987:
1964:
1839:
1813:
1774:
1754:
1727:
1707:
1662:
1642:
1605:
1579:
1524:
1472:
1445:
1425:
1395:
1375:
1348:
1315:
1295:
1247:
1157:
1137:
1105:
1066:
1021:
978:
951:
860:
840:
801:
761:
727:
698:
651:
628:
608:
531:
498:
474:finite-dimensional
462:
442:
397:
373:
308:
220:
101:finite-dimensional
70:
17983:
17982:
17926:Sesquilinear form
17879:Parallelogram law
17823:Orthonormal basis
17747:
17746:
17650:Integral operator
17427:
17426:
17235:978-0-486-45352-1
17139:978-1-4612-7155-0
17113:978-0-07-054236-5
17078:978-0-12-585050-6
17033:978-3-319-70706-8
16836:10.1063/1.5048577
16787:10.1119/1.1328351
16628:, pp. 55, 86
16239:Positive operator
16229:Hermitian adjoint
16181:
16153:
16133:
16127:elliptic operator
16042:The Laplacian on
16030:of the operator.
15849:
15739:
15636:
15630:mutually singular
15556:
15407:
15279:is such that 1 ≤
15095:. Specifically:
15054:
15030:
14990:
14937:Corresponding to
14775:
14686:
14625:
14481:
14193:
14183:
13901:{\displaystyle D}
13881:{\displaystyle D}
13861:{\displaystyle D}
13606:
13486:{\displaystyle p}
13150:
13076:
13021:is separately in
12915:
12898:in the domain of
12891:{\displaystyle f}
12859:
12823:
12793:
12758:
12660:
12626:
12564:
12540:
12497:
12465:
12386:
12360:
12032:
11757:{\displaystyle g}
11737:{\displaystyle g}
11717:{\displaystyle f}
11563:
11555:
11410:
11402:
11234:
11169:
11112:
10833:being symmetric.
10609:
10541:scattering theory
10448:
10344:
10261:
9566:{\displaystyle T}
9546:{\displaystyle T}
9469:
9348:
9316:
9267:with eigenvalues
9227:
9198:
9172:{\displaystyle T}
9094:{\displaystyle h}
9074:{\displaystyle T}
9025:{\displaystyle T}
9005:{\displaystyle f}
8970:{\displaystyle h}
8946:{\displaystyle T}
8926:{\displaystyle h}
8906:{\displaystyle T}
8891:unitary operators
8765:{\displaystyle A}
8746:
8735:{\displaystyle B}
8715:{\displaystyle A}
8629:{\displaystyle B}
8609:{\displaystyle A}
8280:{\displaystyle B}
8260:{\displaystyle A}
7926:{\displaystyle X}
7814:{\displaystyle p}
7794:{\displaystyle p}
7465:
7432:momentum operator
7151:{\displaystyle A}
7020:
6929:
6553:
6482:
6259:{\displaystyle A}
6242:
6232:
6105:{\displaystyle A}
5844:
5796:
5735:
5705:
5629:
5551:{\displaystyle A}
5423:and consequently
5310:{\displaystyle A}
5074:
4316:{\displaystyle A}
4299:
4289:
4100:
4079:
4045:
3839:
3583:
3524:
3289:{\displaystyle A}
3263:{\displaystyle H}
3210:{\displaystyle A}
3174:
3086:{\displaystyle A}
2993:{\displaystyle A}
2857:{\displaystyle A}
2810:{\displaystyle A}
2657:{\displaystyle A}
2502:{\displaystyle H}
2184:{\displaystyle A}
2074:{\displaystyle H}
2034:{\displaystyle A}
2014:{\displaystyle A}
1990:{\displaystyle A}
1935:
1905:
1777:{\displaystyle A}
1764:is symmetric. If
1730:{\displaystyle A}
1672:is symmetric and
1665:{\displaystyle A}
1608:{\displaystyle A}
1448:{\displaystyle A}
1398:{\displaystyle A}
1318:{\displaystyle H}
1160:{\displaystyle A}
981:{\displaystyle A}
863:{\displaystyle y}
654:{\displaystyle A}
631:{\displaystyle B}
534:{\displaystyle A}
465:{\displaystyle H}
400:{\displaystyle A}
376:{\displaystyle H}
281:
249:
217:
171:quantum mechanics
136:orthonormal basis
105:orthonormal basis
16:(Redirected from
18013:
18006:Linear operators
17813:Prehilbert space
17774:
17767:
17760:
17751:
17750:
17737:
17736:
17655:Jones polynomial
17573:Operator algebra
17317:
17316:
17290:
17283:
17276:
17267:
17266:
17259:
17258:, Academic Press
17247:
17222:Trèves, François
17217:
17203:
17161:
17151:
17117:
17098:
17097:, Academic Press
17082:
17063:
17036:
17007:
16996:
16984:
16958:
16931:
16929:
16902:
16892:
16882:
16865:(5): 1952–1956.
16847:
16821:
16798:
16772:
16770:quant-ph/0103153
16749:
16740:
16712:
16711:Proposition 7.24
16706:
16700:
16699:Proposition 7.22
16694:
16688:
16682:
16676:
16670:
16664:
16659:
16653:
16647:
16641:
16635:
16629:
16623:
16617:
16611:
16605:
16599:
16593:
16587:
16581:
16575:
16569:
16568:Proposition 9.28
16563:
16557:
16556:Proposition 9.27
16551:
16545:
16540:
16534:
16528:
16522:
16516:
16510:
16504:
16498:
16492:
16486:
16480:
16474:
16468:
16462:
16456:
16450:
16444:
16438:
16432:
16426:
16420:
16414:
16413:, pp. 95–97
16408:
16402:
16396:
16390:
16384:
16378:
16372:
16366:
16360:
16354:
16348:
16342:
16336:
16330:
16324:
16318:
16312:
16295:
16293:
16291:
16290:
16285:
16260:
16198:
16196:
16195:
16190:
16182:
16179:
16170:
16168:
16167:
16162:
16154:
16151:
16138:
16117:
16115:
16114:
16109:
16103:
16098:
16097:
16096:
16081:
16076:
16046:is the operator
16025:
16023:
16022:
16017:
16015:
16011:
16010:
15974:
15972:
15971:
15966:
15961:
15960:
15948:
15917:
15915:
15914:
15909:
15888:
15887:
15877:
15872:
15871:
15854:
15843:direct integrals
15837:Direct integrals
15825:
15823:
15822:
15817:
15812:
15808:
15807:
15806:
15801:
15792:
15781:
15776:
15775:
15774:
15759:
15715:
15713:
15712:
15707:
15705:
15704:
15687:
15683:
15682:
15641:
15616:
15614:
15613:
15608:
15581:
15580:
15554:
15553:
15552:
15547:
15538:
15533:
15532:
15531:
15481:
15479:
15478:
15473:
15471:
15467:
15445:
15444:
15420:
15419:
15418:
15408:
15405:
15397:
15396:
15391:
15382:
15363:
15359:
15358:
15357:
15352:
15343:
15332:
15327:
15257:We first define
15220:
15218:
15217:
15212:
15207:
15203:
15193:
15192:
15177:
15176:
15175:
15141:
15140:
15117:
15116:
15094:
15092:
15091:
15086:
15084:
15083:
15059:
15049:
15047:
15046:
15041:
15039:
15035:
15031:
15026:
15025:
15016:
15009:
15008:
14998:
14983:
14982:
14981:
14933:
14931:
14930:
14925:
14910:
14905:
14883:
14878:
14845:
14843:
14842:
14837:
14826:
14822:
14818:
14817:
14794:
14793:
14783:
14691:
14670:
14668:
14667:
14662:
14654:
14653:
14644:
14643:
14633:
14585:
14583:
14582:
14577:
14571:
14570:
14569:
14559:
14558:
14557:
14539:
14538:
14537:
14527:
14526:
14525:
14510:
14509:
14508:
14498:
14497:
14496:
14482:
14480:
14479:
14478:
14470:
14457:
14452:
14451:
14429:
14427:
14426:
14421:
14415:
14414:
14413:
14403:
14390:
14389:
14388:
14378:
14368:
14367:
14366:
14356:
14344:
14343:
14321:
14319:
14318:
14313:
14308:
14307:
14289:
14288:
14276:
14275:
14232:
14230:
14229:
14224:
14222:
14221:
14212:
14211:
14201:
14189:
14185:
14184:
14176:
14130:
14128:
14127:
14122:
14120:
14116:
14100:
14099:
14078:
14077:
14054:
14053:
14023:
14021:
14020:
14015:
14001:
14000:
13954:
13952:
13951:
13946:
13907:
13905:
13904:
13899:
13887:
13885:
13884:
13879:
13867:
13865:
13864:
13859:
13799:
13797:
13796:
13791:
13789:
13769:
13738:
13681:
13679:
13678:
13673:
13625:
13623:
13622:
13617:
13615:
13607:
13599:
13575:
13573:
13572:
13567:
13550:
13549:
13519:
13517:
13516:
13511:
13509:
13508:
13492:
13490:
13489:
13484:
13466:
13464:
13463:
13458:
13437:
13435:
13434:
13429:
13408:
13401:
13393:
13391:
13390:
13385:
13383:
13382:
13373:
13365:
13297:
13295:
13294:
13289:
13287:
13286:
13270:
13268:
13267:
13262:
13260:
13259:
13240:
13238:
13237:
13232:
13230:
13229:
13213:
13211:
13210:
13205:
13203:
13202:
13190:
13185:
13184:
13168:
13166:
13165:
13160:
13158:
13157:
13152:
13151:
13143:
13132:
13130:
13129:
13124:
13119:
13111:
13110:
13094:
13092:
13091:
13086:
13084:
13083:
13078:
13077:
13069:
13058:
13056:
13055:
13050:
13045:
13037:
13036:
13020:
13018:
13017:
13012:
12998:
12997:
12981:
12979:
12978:
12973:
12971:
12970:
12958:
12950:
12949:
12933:
12931:
12930:
12925:
12923:
12922:
12917:
12916:
12908:
12897:
12895:
12894:
12889:
12877:
12875:
12874:
12869:
12867:
12866:
12861:
12860:
12852:
12841:
12839:
12838:
12833:
12831:
12830:
12825:
12824:
12816:
12805:
12803:
12802:
12797:
12795:
12794:
12786:
12776:
12774:
12773:
12768:
12766:
12765:
12760:
12759:
12751:
12737:
12735:
12734:
12729:
12724:
12720:
12719:
12716:
12712:
12704:
12703:
12691:
12687:
12674:
12673:
12661:
12659:
12658:
12657:
12644:
12640:
12639:
12629:
12627:
12625:
12617:
12616:
12607:
12589:
12581:
12580:
12565:
12562:
12552:
12548:
12547:
12542:
12541:
12533:
12509:
12507:
12506:
12501:
12499:
12498:
12490:
12477:
12475:
12474:
12469:
12467:
12466:
12458:
12444:
12442:
12441:
12436:
12434:
12433:
12410:
12408:
12407:
12402:
12400:
12399:
12387:
12385:
12377:
12376:
12367:
12362:
12361:
12353:
12327:
12325:
12324:
12319:
12299:
12298:
12268:
12266:
12265:
12260:
12258:
12257:
12224:
12223:
12208:, the functions
12199:
12197:
12196:
12191:
12173:
12171:
12170:
12165:
12163:
12145:
12143:
12142:
12137:
12135:
12134:
12110:
12109:
12083:
12081:
12080:
12075:
12033:
12031:smooth functions
12030:
11986:
11984:
11983:
11978:
11976:
11975:
11974:
11954:
11952:
11951:
11946:
11944:
11943:
11942:
11922:
11920:
11919:
11914:
11912:
11911:
11910:
11886:
11884:
11883:
11878:
11876:
11875:
11874:
11847:
11845:
11844:
11839:
11831:
11807:
11806:
11790:
11788:
11787:
11782:
11780:
11779:
11763:
11761:
11760:
11755:
11743:
11741:
11740:
11735:
11723:
11721:
11720:
11715:
11703:
11701:
11700:
11695:
11665:
11663:
11662:
11657:
11631:. If we compute
11630:
11628:
11627:
11622:
11620:
11619:
11592:
11590:
11589:
11584:
11579:
11575:
11574:
11573:
11564:
11561:
11556:
11553:
11543:
11539:
11538:
11505:
11503:
11502:
11497:
11495:
11494:
11475:
11473:
11472:
11467:
11462:
11458:
11421:
11420:
11411:
11408:
11403:
11400:
11390:
11386:
11385:
11384:
11347:
11345:
11344:
11339:
11337:
11336:
11335:
11293:
11291:
11290:
11285:
11280:
11276:
11235:
11233:smooth functions
11232:
11187:
11185:
11184:
11179:
11174:
11170:
11168:smooth functions
11167:
11126:
11124:
11123:
11118:
11113:
11111:
11103:
11095:
11064:
11062:
11061:
11056:
11033:
11032:
10980:
10978:
10977:
10972:
10967:
10966:
10961:
10952:
10865:
10821:
10819:
10818:
10813:
10756:
10755:
10721:
10719:
10718:
10713:
10660:
10658:
10657:
10652:
10641:
10620:
10618:
10617:
10612:
10610:
10608:
10607:
10606:
10593:
10592:
10583:
10553:spectral theorem
10529:
10527:
10526:
10521:
10519:
10514:
10509:
10484:
10480:
10479:
10454:
10450:
10449:
10446:
10420:
10418:
10417:
10412:
10410:
10405:
10400:
10387:
10386:
10374:
10369:
10364:
10350:
10345:
10342:
10322:
10320:
10319:
10314:
10288:
10286:
10285:
10280:
10263:
10262:
10259:
10243:
10241:
10240:
10235:
10233:
10229:
10228:
10215:
10211:
10210:
10172:
10158:
10156:
10155:
10150:
10142:
10130:
10128:
10127:
10122:
10114:
10101:spectral measure
10090:
10088:
10087:
10082:
10080:
10076:
10075:
10062:
10058:
10057:
10012:
10010:
10009:
10004:
10002:
9998:
9997:
9978:
9974:
9973:
9944:
9942:
9941:
9936:
9934:
9930:
9929:
9928:
9901:
9900:
9824:
9822:
9821:
9816:
9789:
9781:
9735:
9733:
9732:
9727:
9701:of the interval
9696:
9694:
9693:
9688:
9686:
9685:
9665:
9649:
9647:
9646:
9641:
9630:
9629:
9609:
9572:
9570:
9569:
9564:
9552:
9550:
9549:
9544:
9529:
9527:
9526:
9521:
9516:
9515:
9503:
9502:
9484:
9483:
9471:
9470:
9462:
9443:
9441:
9440:
9435:
9433:
9432:
9428:
9423:
9422:
9393:
9392:
9370:
9368:
9367:
9362:
9360:
9359:
9355:
9350:
9349:
9341:
9318:
9317:
9309:
9293:
9291:
9290:
9285:
9283:
9282:
9266:
9264:
9263:
9258:
9256:
9255:
9239:
9237:
9236:
9231:
9229:
9228:
9220:
9210:
9208:
9207:
9202:
9200:
9199:
9191:
9178:
9176:
9175:
9170:
9155:
9153:
9152:
9147:
9129:
9127:
9126:
9121:
9100:
9098:
9097:
9092:
9080:
9078:
9077:
9072:
9060:
9058:
9057:
9052:
9031:
9029:
9028:
9023:
9011:
9009:
9008:
9003:
8976:
8974:
8973:
8968:
8952:
8950:
8949:
8944:
8932:
8930:
8929:
8924:
8912:
8910:
8909:
8904:
8884:
8882:
8881:
8876:
8804:
8803:
8771:
8769:
8768:
8763:
8751:
8741:
8739:
8738:
8733:
8721:
8719:
8718:
8713:
8701:
8699:
8698:
8693:
8691:
8690:
8689:
8688:
8665:
8664:
8663:
8662:
8635:
8633:
8632:
8627:
8615:
8613:
8612:
8607:
8592:
8590:
8589:
8584:
8545:
8544:
8518:
8516:
8515:
8510:
8471:
8469:
8468:
8463:
8461:
8460:
8448:
8447:
8418:
8416:
8415:
8410:
8408:
8407:
8391:
8389:
8388:
8383:
8381:
8380:
8364:
8362:
8361:
8356:
8354:
8353:
8325:
8323:
8322:
8317:
8315:
8314:
8286:
8284:
8283:
8278:
8266:
8264:
8263:
8258:
8234:
8232:
8231:
8226:
8221:
8217:
8201:
8200:
8181:
8160:
8159:
8136:
8135:
8107:
8105:
8104:
8099:
8094:
8093:
8026:
8025:
8006:
8004:
8003:
7998:
7981:
7980:
7968:
7967:
7949:
7948:
7932:
7930:
7929:
7924:
7908:
7906:
7905:
7900:
7898:
7870:
7868:
7867:
7862:
7820:
7818:
7817:
7812:
7800:
7798:
7797:
7792:
7780:
7778:
7777:
7772:
7770:
7769:
7753:
7751:
7750:
7745:
7743:
7742:
7720:
7718:
7717:
7712:
7710:
7709:
7690:
7688:
7687:
7682:
7680:
7676:
7675:
7643:
7641:
7640:
7635:
7633:
7632:
7607:
7605:
7604:
7599:
7594:
7586:
7585:
7569:
7567:
7566:
7561:
7556:
7548:
7547:
7531:
7529:
7528:
7523:
7521:
7520:
7493:
7492:
7476:
7474:
7473:
7468:
7466:
7464:
7453:
7421:Spectral theorem
7415:Spectral theorem
7409:
7407:
7406:
7401:
7312:
7311:
7287:
7286:
7271:
7269:
7268:
7263:
7255:
7254:
7232:
7230:
7229:
7224:
7222:
7221:
7193:
7191:
7190:
7185:
7174:
7173:
7157:
7155:
7154:
7149:
7137:
7135:
7134:
7129:
7124:
7123:
7101:
7099:
7098:
7093:
7082:
7081:
7047:
7045:
7044:
7039:
7022:
7021:
7013:
6988:
6986:
6985:
6980:
6966:
6965:
6944:
6943:
6931:
6930:
6922:
6897:
6895:
6894:
6889:
6863:
6862:
6837:
6835:
6834:
6829:
6818:
6817:
6801:
6799:
6798:
6793:
6779:
6778:
6726:
6724:
6723:
6718:
6704:
6703:
6645:
6643:
6642:
6637:
6623:
6622:
6589:
6587:
6586:
6581:
6555:
6554:
6546:
6512:
6510:
6509:
6504:
6484:
6483:
6475:
6465:
6463:
6462:
6457:
6427:
6425:
6424:
6419:
6377:
6375:
6374:
6369:
6367:
6349:
6347:
6346:
6341:
6330:
6329:
6298:
6296:
6295:
6290:
6288:
6287:
6265:
6263:
6262:
6257:
6237:
6226:
6224:
6223:
6218:
6188:
6186:
6185:
6180:
6177:
6169:
6149:
6147:
6146:
6141:
6135:
6127:
6111:
6109:
6108:
6103:
6091:
6089:
6088:
6083:
6081:
6080:
6064:
6062:
6061:
6056:
6018:
6017:
5995:
5993:
5992:
5987:
5984:
5976:
5960:
5958:
5957:
5952:
5949:
5941:
5925:
5923:
5922:
5917:
5897:
5896:
5872:
5870:
5869:
5864:
5847:
5846:
5845:
5837:
5817:
5815:
5814:
5809:
5798:
5797:
5789:
5773:
5771:
5770:
5765:
5745:
5737:
5736:
5728:
5718:
5716:
5715:
5710:
5708:
5707:
5706:
5698:
5682:
5681:
5676:
5672:
5671:
5670:
5642:
5640:
5639:
5634:
5632:
5631:
5630:
5622:
5611:
5606:
5590:
5588:
5587:
5582:
5574:
5573:
5557:
5555:
5554:
5549:
5537:
5535:
5534:
5529:
5514:
5512:
5511:
5506:
5504:
5503:
5480:
5478:
5477:
5472:
5467:
5466:
5445:
5444:
5422:
5420:
5419:
5414:
5370:
5368:
5367:
5362:
5357:
5356:
5316:
5314:
5313:
5308:
5296:
5294:
5293:
5288:
5268:
5267:
5252:
5251:
5235:
5233:
5232:
5227:
5225:
5224:
5209:
5208:
5193:
5192:
5176:
5174:
5173:
5168:
5143:
5141:
5140:
5135:
5133:
5132:
5117:
5115:
5114:
5109:
5101:
5100:
5088:
5087:
5075:
5073:
5056:
5048:
5047:
5035:
5034:
5015:
5013:
5012:
5007:
4986:
4984:
4983:
4978:
4976:
4975:
4957:
4956:
4947:
4946:
4934:
4933:
4917:
4915:
4914:
4909:
4907:
4906:
4881:
4879:
4878:
4873:
4862:
4861:
4836:
4834:
4833:
4828:
4826:
4825:
4803:
4801:
4800:
4795:
4768:
4766:
4765:
4760:
4722:
4721:
4702:
4700:
4699:
4694:
4692:
4691:
4676:As shown above,
4672:
4670:
4669:
4664:
4653:
4652:
4630:
4628:
4627:
4622:
4608:
4607:
4585:
4583:
4582:
4577:
4565:
4557:
4535:
4533:
4532:
4527:
4521:
4513:
4497:
4495:
4494:
4489:
4466:
4445:
4443:
4442:
4437:
4395:
4393:
4392:
4387:
4382:
4364:
4362:
4361:
4356:
4339:
4338:
4322:
4320:
4319:
4314:
4294:
4280:
4278:
4277:
4272:
4251:
4249:
4248:
4243:
4213:
4211:
4210:
4205:
4169:
4167:
4166:
4161:
4117:
4113:
4106:
4102:
4101:
4099:
4085:
4080:
4078:
4064:
4046:
4044:
4033:
4032:
3997:
3991:
3948:
3946:
3945:
3940:
3899:
3897:
3896:
3891:
3885:
3871:
3865:
3812:
3810:
3809:
3804:
3739:
3737:
3736:
3731:
3704:
3702:
3701:
3696:
3694:
3676:
3674:
3673:
3668:
3651:
3627:
3625:
3624:
3619:
3597:
3568:
3566:
3565:
3560:
3538:
3510:
3508:
3507:
3502:
3434:
3432:
3431:
3426:
3424:
3393:
3391:
3390:
3385:
3383:
3353:
3351:
3350:
3345:
3325:
3295:
3293:
3292:
3287:
3269:
3267:
3266:
3261:
3245:
3243:
3242:
3237:
3216:
3214:
3213:
3208:
3193:
3191:
3190:
3185:
3180:
3176:
3175:
3172:
3169:
3168:
3133:
3092:
3090:
3089:
3084:
3062:
3060:
3059:
3054:
2999:
2997:
2996:
2991:
2979:
2977:
2976:
2971:
2947:
2945:
2944:
2939:
2925:
2917:
2905:
2903:
2902:
2897:
2883:
2875:
2863:
2861:
2860:
2855:
2843:
2841:
2840:
2835:
2816:
2814:
2813:
2808:
2790:
2788:
2787:
2782:
2780:
2776:
2744:
2721:
2705:
2686:
2684:
2683:
2678:
2663:
2661:
2660:
2655:
2639:
2637:
2636:
2631:
2592:
2590:
2589:
2584:
2576:
2552:
2550:
2549:
2544:
2508:
2506:
2505:
2500:
2488:
2486:
2485:
2480:
2447:
2445:
2444:
2439:
2415:
2413:
2412:
2407:
2383:
2381:
2380:
2375:
2344:
2342:
2341:
2336:
2309:
2307:
2306:
2301:
2225:
2223:
2222:
2217:
2193:bounded operator
2190:
2188:
2187:
2182:
2170:
2168:
2167:
2162:
2128:
2126:
2125:
2120:
2080:
2078:
2077:
2072:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
1996:
1994:
1993:
1988:
1973:
1971:
1970:
1965:
1944:
1936:
1931:
1911:
1906:
1901:
1881:
1848:
1846:
1845:
1840:
1823:is real for all
1822:
1820:
1819:
1814:
1812:
1808:
1783:
1781:
1780:
1775:
1763:
1761:
1760:
1755:
1753:
1752:
1736:
1734:
1733:
1728:
1716:
1714:
1713:
1708:
1706:
1705:
1671:
1669:
1668:
1663:
1651:
1649:
1648:
1643:
1641:
1640:
1614:
1612:
1611:
1606:
1588:
1586:
1585:
1580:
1578:
1577:
1565:
1564:
1533:
1531:
1530:
1525:
1523:
1522:
1510:
1509:
1481:
1479:
1478:
1473:
1471:
1470:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1424:
1423:
1404:
1402:
1401:
1396:
1384:
1382:
1381:
1376:
1374:
1373:
1357:
1355:
1354:
1349:
1324:
1322:
1321:
1316:
1304:
1302:
1301:
1296:
1282:
1281:
1256:
1254:
1253:
1248:
1166:
1164:
1163:
1158:
1147:. Equivalently,
1146:
1144:
1143:
1138:
1114:
1112:
1111:
1106:
1101:
1100:
1075:
1073:
1072:
1067:
1065:
1064:
1030:
1028:
1027:
1022:
1020:
1019:
987:
985:
984:
979:
960:
958:
957:
952:
919:
918:
869:
867:
866:
861:
849:
847:
846:
841:
833:
832:
810:
808:
807:
802:
800:
799:
782:adjoint operator
773:conjugate linear
770:
768:
767:
762:
736:
734:
733:
728:
707:
705:
704:
699:
660:
658:
657:
652:
637:
635:
634:
629:
617:
615:
614:
609:
540:
538:
537:
532:
507:
505:
504:
499:
471:
469:
468:
463:
451:
449:
448:
443:
406:
404:
403:
398:
382:
380:
379:
374:
317:
315:
314:
309:
292:
291:
282:
280:
272:
271:
262:
251:
250:
242:
229:
227:
226:
221:
219:
218:
210:
191:angular momentum
158:to operators on
128:spectral theorem
117:Hermitian matrix
79:
77:
76:
71:
21:
18021:
18020:
18016:
18015:
18014:
18012:
18011:
18010:
18001:Operator theory
17986:
17985:
17984:
17979:
17972:Segal–Bargmann
17940:
17911:Hilbert–Schmidt
17901:Densely defined
17884:
17853:
17827:
17783:
17778:
17748:
17743:
17725:
17689:Advanced topics
17684:
17608:
17587:
17546:
17512:Hilbert–Schmidt
17485:
17476:Gelfand–Naimark
17423:
17373:
17308:
17294:
17263:
17236:
17159:
17140:
17114:
17079:
17052:
17034:
16982:
16947:
16920:(22): 679–703.
16737:
16721:
16716:
16715:
16707:
16703:
16695:
16691:
16683:
16679:
16671:
16667:
16660:
16656:
16648:
16644:
16636:
16632:
16624:
16620:
16612:
16608:
16600:
16596:
16588:
16584:
16576:
16572:
16564:
16560:
16552:
16548:
16541:
16537:
16529:
16525:
16517:
16513:
16505:
16501:
16493:
16489:
16481:
16477:
16469:
16465:
16457:
16453:
16445:
16441:
16433:
16429:
16421:
16417:
16409:
16405:
16397:
16393:
16385:
16381:
16373:
16369:
16361:
16357:
16349:
16345:
16337:
16333:
16325:
16321:
16313:
16309:
16304:
16299:
16298:
16267:
16264:
16263:
16261:
16257:
16252:
16234:Normal operator
16215:
16210:
16207:
16178:
16176:
16173:
16172:
16150:
16148:
16145:
16144:
16136:
16099:
16092:
16088:
16087:
16077:
16066:
16054:
16051:
16050:
16040:
16006:
16002:
15998:
15984:
15981:
15980:
15979:. The function
15956:
15952:
15938:
15930:
15927:
15926:
15919:
15883:
15879:
15873:
15867:
15866:
15860:
15857:
15856:
15852:
15839:
15827:
15802:
15797:
15796:
15788:
15787:
15783:
15777:
15770:
15766:
15765:
15743:
15737:
15734:
15733:
15688:
15678:
15674:
15670:
15669:
15667:
15664:
15663:
15639:
15576:
15572:
15548:
15543:
15542:
15534:
15527:
15526:
15522:
15520:
15517:
15516:
15503:
15493:
15440:
15436:
15414:
15413:
15409:
15404:
15392:
15387:
15386:
15378:
15371:
15367:
15353:
15348:
15347:
15339:
15338:
15334:
15328:
15323:
15317:
15314:
15313:
15296:
15287:if and only if
15255:
15227:
15222:
15188:
15184:
15159:
15158:
15154:
15136:
15132:
15125:
15121:
15112:
15108:
15100:
15097:
15096:
15079:
15075:
15073:
15070:
15069:
15057:
15021:
15017:
15015:
15014:
15010:
15004:
15000:
14994:
14965:
14964:
14960:
14958:
14955:
14954:
14906:
14901:
14879:
14874:
14868:
14865:
14864:
14855:
14813:
14809:
14808:
14804:
14789:
14785:
14779:
14755:
14752:
14751:
14738:
14689:
14649:
14645:
14639:
14635:
14629:
14605:
14602:
14601:
14565:
14561:
14560:
14553:
14549:
14548:
14533:
14529:
14528:
14521:
14517:
14516:
14504:
14500:
14499:
14492:
14488:
14487:
14474:
14466:
14465:
14461:
14456:
14447:
14443:
14441:
14438:
14437:
14409:
14405:
14404:
14399:
14384:
14380:
14379:
14374:
14362:
14358:
14357:
14352:
14339:
14335:
14333:
14330:
14329:
14303:
14299:
14284:
14280:
14271:
14267:
14256:
14253:
14252:
14217:
14213:
14207:
14203:
14197:
14175:
14174:
14170:
14165:
14162:
14161:
14151:
14140:
14095:
14091:
14073:
14069:
14062:
14058:
14049:
14045:
14043:
14040:
14039:
14031:on an open set
13993:
13989:
13972:
13969:
13968:
13916:
13913:
13912:
13893:
13890:
13889:
13873:
13870:
13869:
13853:
13850:
13849:
13840:
13833:
13787:
13786:
13770:
13762:
13753:
13752:
13739:
13731:
13721:
13719:
13716:
13715:
13706:
13699:
13637:
13634:
13633:
13608:
13598:
13584:
13581:
13580:
13545:
13541:
13539:
13536:
13535:
13532:
13527:
13504:
13500:
13498:
13495:
13494:
13478:
13475:
13474:
13443:
13440:
13439:
13417:
13414:
13413:
13403:
13395:
13378:
13374:
13369:
13361:
13332:
13329:
13328:
13310:
13304:
13282:
13278:
13276:
13273:
13272:
13255:
13251:
13246:
13243:
13242:
13225:
13221:
13219:
13216:
13215:
13198:
13194:
13186:
13180:
13176:
13174:
13171:
13170:
13153:
13142:
13141:
13140:
13138:
13135:
13134:
13115:
13106:
13102:
13100:
13097:
13096:
13079:
13068:
13067:
13066:
13064:
13061:
13060:
13041:
13032:
13028:
13026:
13023:
13022:
12993:
12989:
12987:
12984:
12983:
12966:
12962:
12954:
12945:
12941:
12939:
12936:
12935:
12918:
12907:
12906:
12905:
12903:
12900:
12899:
12883:
12880:
12879:
12862:
12851:
12850:
12849:
12847:
12844:
12843:
12826:
12815:
12814:
12813:
12811:
12808:
12807:
12785:
12784:
12782:
12779:
12778:
12761:
12750:
12749:
12748:
12746:
12743:
12742:
12708:
12699:
12695:
12669:
12665:
12653:
12649:
12645:
12635:
12631:
12630:
12628:
12618:
12612:
12608:
12606:
12602:
12598:
12597:
12593:
12585:
12576:
12572:
12561:
12560:
12556:
12543:
12532:
12531:
12530:
12526:
12518:
12515:
12514:
12489:
12488:
12486:
12483:
12482:
12457:
12456:
12454:
12451:
12450:
12429:
12425:
12420:
12417:
12416:
12395:
12391:
12378:
12372:
12368:
12366:
12352:
12351:
12349:
12346:
12345:
12334:
12294:
12290:
12282:
12279:
12278:
12241:
12237:
12219:
12215:
12213:
12210:
12209:
12179:
12176:
12175:
12159:
12151:
12148:
12147:
12127:
12123:
12105:
12101:
12099:
12096:
12095:
12029:
12006:
12003:
12002:
11967:
11966:
11962:
11960:
11957:
11956:
11935:
11934:
11930:
11928:
11925:
11924:
11903:
11902:
11898:
11896:
11893:
11892:
11867:
11866:
11862:
11860:
11857:
11856:
11827:
11802:
11798:
11796:
11793:
11792:
11775:
11771:
11769:
11766:
11765:
11749:
11746:
11745:
11729:
11726:
11725:
11709:
11706:
11705:
11671:
11668:
11667:
11636:
11633:
11632:
11615:
11611:
11609:
11606:
11605:
11569:
11565:
11560:
11554:functions
11552:
11551:
11547:
11534:
11530:
11526:
11518:
11515:
11514:
11490:
11486:
11484:
11481:
11480:
11416:
11412:
11407:
11401:functions
11399:
11398:
11394:
11377:
11376:
11372:
11368:
11360:
11357:
11356:
11328:
11327:
11323:
11321:
11318:
11317:
11231:
11230:
11226:
11206:
11203:
11202:
11166:
11162:
11142:
11139:
11138:
11104:
11096:
11094:
11077:
11074:
11073:
11028:
11024:
11022:
11019:
11018:
11010:
11001:
10995:
10962:
10957:
10956:
10948:
10937:
10934:
10933:
10914:
10906:
10897:if and only if
10883:
10876:
10861:
10751:
10747:
10745:
10742:
10741:
10677:
10674:
10673:
10631:
10629:
10626:
10625:
10602:
10598:
10594:
10588:
10584:
10582:
10571:
10568:
10567:
10549:
10510:
10505:
10497:
10475:
10471:
10467:
10445:
10441:
10437:
10432:
10429:
10428:
10401:
10396:
10388:
10382:
10378:
10365:
10360:
10352:
10346:
10341:
10335:
10332:
10331:
10302:
10299:
10298:
10258:
10254:
10252:
10249:
10248:
10224:
10220:
10216:
10206:
10202:
10198:
10181:
10178:
10177:
10167:
10138:
10136:
10133:
10132:
10110:
10108:
10105:
10104:
10071:
10067:
10063:
10053:
10049:
10045:
10043:
10040:
10039:
10029:
9993:
9989:
9985:
9969:
9965:
9961:
9956:
9953:
9952:
9924:
9920:
9896:
9892:
9891:
9887:
9861:
9858:
9857:
9831:
9782:
9774:
9762:
9759:
9758:
9754:can be proved:
9706:
9703:
9702:
9666:
9661:
9660:
9658:
9655:
9654:
9610:
9605:
9604:
9584:
9581:
9580:
9558:
9555:
9554:
9538:
9535:
9534:
9511:
9507:
9498:
9494:
9479:
9475:
9461:
9460:
9452:
9449:
9448:
9424:
9418:
9414:
9404:
9400:
9388:
9384:
9376:
9373:
9372:
9351:
9340:
9339:
9329:
9325:
9308:
9307:
9299:
9296:
9295:
9278:
9274:
9272:
9269:
9268:
9251:
9247:
9245:
9242:
9241:
9219:
9218:
9216:
9213:
9212:
9190:
9189:
9187:
9184:
9183:
9164:
9161:
9160:
9135:
9132:
9131:
9106:
9103:
9102:
9086:
9083:
9082:
9066:
9063:
9062:
9037:
9034:
9033:
9017:
9014:
9013:
8997:
8994:
8993:
8986:
8962:
8959:
8958:
8955:essential range
8938:
8935:
8934:
8918:
8915:
8914:
8898:
8895:
8894:
8887:
8796:
8792:
8784:
8781:
8780:
8757:
8754:
8753:
8749:
8727:
8724:
8723:
8707:
8704:
8703:
8684:
8680:
8679:
8675:
8658:
8654:
8653:
8649:
8641:
8638:
8637:
8621:
8618:
8617:
8601:
8598:
8597:
8537:
8533:
8525:
8522:
8521:
8480:
8477:
8476:
8456:
8452:
8443:
8439:
8431:
8428:
8427:
8403:
8399:
8397:
8394:
8393:
8376:
8372:
8370:
8367:
8366:
8349:
8345:
8331:
8328:
8327:
8310:
8306:
8292:
8289:
8288:
8272:
8269:
8268:
8252:
8249:
8248:
8196:
8192:
8177:
8155:
8151:
8144:
8140:
8131:
8127:
8119:
8116:
8115:
8089:
8085:
8021:
8017:
8015:
8012:
8011:
7976:
7972:
7963:
7959:
7944:
7940:
7938:
7935:
7934:
7918:
7915:
7914:
7894:
7880:
7877:
7876:
7838:
7835:
7834:
7831:
7806:
7803:
7802:
7786:
7783:
7782:
7765:
7761:
7759:
7756:
7755:
7732:
7728:
7726:
7723:
7722:
7705:
7701:
7699:
7696:
7695:
7668:
7661:
7657:
7652:
7649:
7648:
7622:
7618:
7616:
7613:
7612:
7610:Kronecker delta
7590:
7581:
7577:
7575:
7572:
7571:
7552:
7543:
7539:
7537:
7534:
7533:
7510:
7506:
7488:
7484:
7482:
7479:
7478:
7457:
7452:
7438:
7435:
7434:
7423:
7417:
7412:
7307:
7303:
7282:
7278:
7276:
7273:
7272:
7250:
7246:
7238:
7235:
7234:
7217:
7213:
7199:
7196:
7195:
7169:
7165:
7163:
7160:
7159:
7143:
7140:
7139:
7119:
7115:
7107:
7104:
7103:
7077:
7073:
7056:
7053:
7052:
7012:
7011:
6994:
6991:
6990:
6961:
6957:
6939:
6935:
6921:
6920:
6903:
6900:
6899:
6858:
6854:
6843:
6840:
6839:
6813:
6809:
6807:
6804:
6803:
6774:
6770:
6732:
6729:
6728:
6699:
6695:
6651:
6648:
6647:
6618:
6614:
6597:
6594:
6593:
6545:
6544:
6518:
6515:
6514:
6474:
6473:
6471:
6468:
6467:
6433:
6430:
6429:
6383:
6380:
6379:
6363:
6355:
6352:
6351:
6325:
6321:
6304:
6301:
6300:
6283:
6279:
6271:
6268:
6267:
6251:
6248:
6247:
6240:
6235:
6229:
6194:
6191:
6190:
6189:is bounded, so
6170:
6165:
6159:
6156:
6155:
6128:
6123:
6117:
6114:
6113:
6097:
6094:
6093:
6076:
6072:
6070:
6067:
6066:
6013:
6009:
6001:
5998:
5997:
5977:
5972:
5966:
5963:
5962:
5942:
5937:
5931:
5928:
5927:
5892:
5888:
5886:
5883:
5882:
5878:
5875:
5836:
5835:
5831:
5823:
5820:
5819:
5788:
5787:
5779:
5776:
5775:
5741:
5727:
5726:
5724:
5721:
5720:
5697:
5696:
5692:
5677:
5666:
5662:
5655:
5651:
5650:
5648:
5645:
5644:
5621:
5620:
5616:
5607:
5602:
5596:
5593:
5592:
5569:
5565:
5563:
5560:
5559:
5543:
5540:
5539:
5520:
5517:
5516:
5499:
5495:
5487:
5484:
5483:
5462:
5458:
5440:
5436:
5428:
5425:
5424:
5375:
5372:
5371:
5352:
5348:
5322:
5319:
5318:
5302:
5299:
5298:
5263:
5259:
5247:
5243:
5241:
5238:
5237:
5220:
5216:
5204:
5200:
5188:
5184:
5182:
5179:
5178:
5153:
5150:
5149:
5128:
5124:
5122:
5119:
5118:
5096:
5092:
5083:
5079:
5060:
5055:
5043:
5039:
5030:
5026:
5021:
5018:
5017:
4992:
4989:
4988:
4971:
4967:
4952:
4948:
4942:
4938:
4929:
4925:
4923:
4920:
4919:
4902:
4898:
4890:
4887:
4886:
4857:
4853:
4845:
4842:
4841:
4821:
4817:
4809:
4806:
4805:
4774:
4771:
4770:
4717:
4713:
4708:
4705:
4704:
4687:
4683:
4681:
4678:
4677:
4648:
4644:
4636:
4633:
4632:
4603:
4599:
4591:
4588:
4587:
4558:
4553:
4541:
4538:
4537:
4514:
4509:
4503:
4500:
4499:
4462:
4454:
4451:
4450:
4401:
4398:
4397:
4378:
4370:
4367:
4366:
4334:
4330:
4328:
4325:
4324:
4308:
4305:
4304:
4297:
4292:
4257:
4254:
4253:
4219:
4216:
4215:
4178:
4175:
4174:
4089:
4084:
4068:
4063:
4059:
4055:
4054:
4050:
4034:
4028:
3993:
3992:
3990:
3961:
3958:
3957:
3907:
3904:
3903:
3881:
3867:
3843:
3821:
3818:
3817:
3747:
3744:
3743:
3710:
3707:
3706:
3690:
3682:
3679:
3678:
3647:
3633:
3630:
3629:
3587:
3574:
3571:
3570:
3528:
3515:
3512:
3511:
3451:
3448:
3447:
3420:
3403:
3400:
3399:
3379:
3362:
3359:
3358:
3321:
3304:
3301:
3300:
3281:
3278:
3277:
3255:
3252:
3251:
3222:
3219:
3218:
3202:
3199:
3198:
3171:
3161:
3157:
3129:
3122:
3118:
3101:
3098:
3097:
3078:
3075:
3074:
3024:
3021:
3020:
3017:
3011:
2985:
2982:
2981:
2953:
2950:
2949:
2921:
2913:
2911:
2908:
2907:
2879:
2871:
2869:
2866:
2865:
2849:
2846:
2845:
2829:
2826:
2825:
2821:are orthogonal.
2802:
2799:
2798:
2740:
2717:
2716:
2712:
2695:
2693:
2690:
2689:
2669:
2666:
2665:
2649:
2646:
2645:
2601:
2598:
2597:
2566:
2558:
2555:
2554:
2526:
2523:
2522:
2519:
2494:
2491:
2490:
2462:
2459:
2458:
2421:
2418:
2417:
2389:
2386:
2385:
2354:
2351:
2350:
2318:
2315:
2314:
2234:
2231:
2230:
2199:
2196:
2195:
2176:
2173:
2172:
2138:
2135:
2134:
2090:
2087:
2086:
2066:
2063:
2062:
2059:
2026:
2023:
2022:
2006:
2003:
2002:
1982:
1979:
1978:
1940:
1912:
1910:
1882:
1880:
1857:
1854:
1853:
1828:
1825:
1824:
1795:
1791:
1789:
1786:
1785:
1769:
1766:
1765:
1748:
1744:
1742:
1739:
1738:
1722:
1719:
1718:
1701:
1697:
1677:
1674:
1673:
1657:
1654:
1653:
1636:
1632:
1624:
1621:
1620:
1600:
1597:
1596:
1573:
1569:
1557:
1553:
1545:
1542:
1541:
1518:
1514:
1502:
1498:
1490:
1487:
1486:
1466:
1462:
1460:
1457:
1456:
1440:
1437:
1436:
1416:
1412:
1410:
1407:
1406:
1390:
1387:
1386:
1369:
1365:
1363:
1360:
1359:
1334:
1331:
1330:
1310:
1307:
1306:
1277:
1273:
1265:
1262:
1261:
1175:
1172:
1171:
1152:
1149:
1148:
1120:
1117:
1116:
1096:
1092:
1081:
1078:
1077:
1060:
1056:
1036:
1033:
1032:
1015:
1011:
1003:
1000:
999:
973:
970:
969:
966:densely defined
914:
910:
878:
875:
874:
855:
852:
851:
828:
824:
816:
813:
812:
795:
791:
789:
786:
785:
744:
741:
740:
713:
710:
709:
666:
663:
662:
646:
643:
642:
623:
620:
619:
546:
543:
542:
526:
523:
522:
510:linear operator
481:
478:
477:
457:
454:
453:
422:
419:
418:
392:
389:
388:
368:
365:
364:
361:
331:potential field
287:
283:
273:
267:
263:
261:
241:
240:
238:
235:
234:
209:
208:
206:
203:
202:
152:generalizations
144:diagonal matrix
53:
50:
49:
28:
23:
22:
15:
12:
11:
5:
18019:
18009:
18008:
18003:
17998:
17996:Hilbert spaces
17981:
17980:
17978:
17977:
17969:
17963:compact &
17948:
17946:
17942:
17941:
17939:
17938:
17933:
17928:
17923:
17918:
17913:
17908:
17906:Hermitian form
17903:
17898:
17892:
17890:
17886:
17885:
17883:
17882:
17872:
17867:
17861:
17859:
17855:
17854:
17852:
17851:
17846:
17841:
17835:
17833:
17829:
17828:
17826:
17825:
17820:
17815:
17806:
17797:
17791:
17789:
17788:Basic concepts
17785:
17784:
17781:Hilbert spaces
17777:
17776:
17769:
17762:
17754:
17745:
17744:
17742:
17741:
17730:
17727:
17726:
17724:
17723:
17718:
17713:
17708:
17706:Choquet theory
17703:
17698:
17692:
17690:
17686:
17685:
17683:
17682:
17672:
17667:
17662:
17657:
17652:
17647:
17642:
17637:
17632:
17627:
17622:
17616:
17614:
17610:
17609:
17607:
17606:
17601:
17595:
17593:
17589:
17588:
17586:
17585:
17580:
17575:
17570:
17565:
17560:
17558:Banach algebra
17554:
17552:
17548:
17547:
17545:
17544:
17539:
17534:
17529:
17524:
17519:
17514:
17509:
17504:
17499:
17493:
17491:
17487:
17486:
17484:
17483:
17481:Banach–Alaoglu
17478:
17473:
17468:
17463:
17458:
17453:
17448:
17443:
17437:
17435:
17429:
17428:
17425:
17424:
17422:
17421:
17416:
17411:
17409:Locally convex
17406:
17392:
17387:
17381:
17379:
17375:
17374:
17372:
17371:
17366:
17361:
17356:
17351:
17346:
17341:
17336:
17331:
17326:
17320:
17314:
17310:
17309:
17293:
17292:
17285:
17278:
17270:
17261:
17260:
17248:
17234:
17218:
17204:
17152:
17138:
17118:
17112:
17099:
17083:
17077:
17064:
17051:978-1584888666
17050:
17037:
17032:
17008:
16997:
16985:
16981:978-1461471158
16980:
16959:
16945:
16932:
16903:
16848:
16799:
16763:(3): 322–331.
16750:
16741:
16735:
16720:
16717:
16714:
16713:
16701:
16689:
16677:
16665:
16654:
16642:
16630:
16618:
16606:
16594:
16582:
16570:
16558:
16546:
16535:
16523:
16511:
16499:
16487:
16475:
16463:
16451:
16449:, pp. 327
16439:
16427:
16415:
16403:
16391:
16379:
16367:
16355:
16343:
16331:
16329:, pp. 224
16319:
16306:
16305:
16303:
16300:
16297:
16296:
16283:
16280:
16277:
16274:
16271:
16254:
16253:
16251:
16248:
16247:
16246:
16241:
16236:
16231:
16226:
16221:
16214:
16211:
16203:
16188:
16185:
16160:
16157:
16131:
16119:
16118:
16107:
16102:
16095:
16091:
16086:
16080:
16075:
16072:
16069:
16065:
16061:
16058:
16039:
16036:
16014:
16009:
16005:
16001:
15997:
15994:
15991:
15988:
15964:
15959:
15955:
15951:
15947:
15944:
15941:
15937:
15934:
15907:
15904:
15901:
15898:
15895:
15892:
15886:
15882:
15876:
15870:
15865:
15847:
15838:
15835:
15815:
15811:
15805:
15800:
15795:
15791:
15786:
15780:
15773:
15769:
15764:
15758:
15755:
15752:
15749:
15746:
15742:
15703:
15700:
15697:
15694:
15691:
15686:
15681:
15677:
15673:
15650:Hilbert space
15634:
15618:
15617:
15606:
15603:
15600:
15597:
15594:
15591:
15588:
15585:
15579:
15575:
15571:
15568:
15565:
15562:
15559:
15551:
15546:
15541:
15537:
15530:
15525:
15499:
15489:
15483:
15482:
15470:
15466:
15463:
15460:
15457:
15454:
15451:
15448:
15443:
15439:
15435:
15432:
15429:
15426:
15423:
15417:
15412:
15403:
15400:
15395:
15390:
15385:
15381:
15377:
15374:
15370:
15366:
15362:
15356:
15351:
15346:
15342:
15337:
15331:
15326:
15322:
15292:
15254:
15251:
15226:
15223:
15210:
15206:
15202:
15199:
15196:
15191:
15187:
15183:
15180:
15174:
15171:
15168:
15165:
15162:
15157:
15153:
15150:
15147:
15144:
15139:
15135:
15131:
15128:
15124:
15120:
15115:
15111:
15107:
15104:
15082:
15078:
15052:
15051:
15050:
15038:
15034:
15029:
15024:
15020:
15013:
15007:
15003:
14997:
14993:
14989:
14986:
14980:
14977:
14974:
14971:
14968:
14963:
14944:formal adjoint
14935:
14934:
14923:
14920:
14917:
14914:
14909:
14904:
14900:
14896:
14893:
14890:
14887:
14882:
14877:
14873:
14853:
14847:
14846:
14835:
14832:
14829:
14825:
14821:
14816:
14812:
14807:
14803:
14800:
14797:
14792:
14788:
14782:
14778:
14774:
14771:
14768:
14765:
14762:
14759:
14684:
14672:
14671:
14660:
14657:
14652:
14648:
14642:
14638:
14632:
14628:
14624:
14621:
14618:
14615:
14612:
14609:
14587:
14586:
14575:
14568:
14564:
14556:
14552:
14547:
14543:
14536:
14532:
14524:
14520:
14515:
14507:
14503:
14495:
14491:
14486:
14477:
14473:
14469:
14464:
14460:
14455:
14450:
14446:
14431:
14430:
14419:
14412:
14408:
14402:
14398:
14394:
14387:
14383:
14377:
14373:
14365:
14361:
14355:
14351:
14347:
14342:
14338:
14323:
14322:
14311:
14306:
14302:
14298:
14295:
14292:
14287:
14283:
14279:
14274:
14270:
14266:
14263:
14260:
14234:
14233:
14220:
14216:
14210:
14206:
14200:
14196:
14192:
14188:
14182:
14179:
14173:
14169:
14150:
14147:
14138:
14132:
14131:
14119:
14115:
14112:
14109:
14106:
14103:
14098:
14094:
14090:
14087:
14084:
14081:
14076:
14072:
14068:
14065:
14061:
14057:
14052:
14048:
14013:
14010:
14007:
14004:
13999:
13996:
13992:
13988:
13985:
13982:
13979:
13976:
13957:
13956:
13944:
13941:
13938:
13935:
13932:
13929:
13926:
13923:
13920:
13897:
13877:
13857:
13838:
13831:
13801:
13800:
13785:
13782:
13779:
13776:
13773:
13771:
13768:
13765:
13761:
13758:
13755:
13754:
13751:
13748:
13745:
13742:
13740:
13737:
13734:
13730:
13727:
13724:
13723:
13709:distributional
13704:
13697:
13693:. The spaces
13683:
13682:
13671:
13668:
13665:
13662:
13659:
13656:
13653:
13650:
13647:
13644:
13641:
13627:
13626:
13614:
13611:
13605:
13602:
13597:
13594:
13591:
13588:
13565:
13562:
13559:
13556:
13553:
13548:
13544:
13531:
13528:
13526:
13523:
13507:
13503:
13482:
13456:
13453:
13450:
13447:
13427:
13424:
13421:
13381:
13377:
13372:
13368:
13364:
13360:
13357:
13354:
13351:
13348:
13345:
13342:
13339:
13336:
13318:time evolution
13303:
13300:
13285:
13281:
13258:
13254:
13250:
13228:
13224:
13201:
13197:
13193:
13189:
13183:
13179:
13156:
13149:
13146:
13122:
13118:
13114:
13109:
13105:
13082:
13075:
13072:
13048:
13044:
13040:
13035:
13031:
13010:
13007:
13004:
13001:
12996:
12992:
12969:
12965:
12961:
12957:
12953:
12948:
12944:
12921:
12914:
12911:
12887:
12865:
12858:
12855:
12829:
12822:
12819:
12792:
12789:
12764:
12757:
12754:
12739:
12738:
12727:
12723:
12718:
12715:
12711:
12707:
12702:
12698:
12694:
12690:
12686:
12683:
12680:
12677:
12672:
12668:
12664:
12656:
12652:
12648:
12643:
12638:
12634:
12624:
12621:
12615:
12611:
12605:
12601:
12596:
12592:
12588:
12584:
12579:
12575:
12571:
12568:
12559:
12555:
12551:
12546:
12539:
12536:
12529:
12525:
12522:
12496:
12493:
12464:
12461:
12432:
12428:
12424:
12412:
12411:
12398:
12394:
12390:
12384:
12381:
12375:
12371:
12365:
12359:
12356:
12333:
12330:
12317:
12314:
12311:
12308:
12305:
12302:
12297:
12293:
12289:
12286:
12256:
12253:
12250:
12247:
12244:
12240:
12236:
12233:
12230:
12227:
12222:
12218:
12189:
12186:
12183:
12162:
12158:
12155:
12133:
12130:
12126:
12122:
12119:
12116:
12113:
12108:
12104:
12085:
12084:
12073:
12070:
12067:
12064:
12061:
12058:
12055:
12052:
12049:
12046:
12043:
12040:
12037:
12028:
12025:
12022:
12019:
12016:
12013:
12010:
11973:
11970:
11965:
11941:
11938:
11933:
11909:
11906:
11901:
11873:
11870:
11865:
11837:
11834:
11830:
11826:
11823:
11819:
11816:
11813:
11810:
11805:
11801:
11778:
11774:
11753:
11733:
11713:
11693:
11690:
11687:
11684:
11681:
11678:
11675:
11655:
11652:
11649:
11646:
11643:
11640:
11618:
11614:
11594:
11593:
11582:
11578:
11572:
11568:
11559:
11550:
11546:
11542:
11537:
11533:
11529:
11525:
11522:
11493:
11489:
11477:
11476:
11465:
11461:
11457:
11454:
11451:
11448:
11445:
11442:
11439:
11436:
11433:
11430:
11427:
11424:
11419:
11415:
11406:
11397:
11393:
11389:
11383:
11380:
11375:
11371:
11367:
11364:
11334:
11331:
11326:
11295:
11294:
11283:
11279:
11275:
11272:
11269:
11266:
11263:
11260:
11257:
11254:
11251:
11248:
11245:
11242:
11239:
11229:
11225:
11222:
11219:
11216:
11213:
11210:
11198:If we choose
11189:
11188:
11177:
11173:
11165:
11161:
11158:
11155:
11152:
11149:
11146:
11128:
11127:
11116:
11110:
11107:
11102:
11099:
11093:
11090:
11087:
11084:
11081:
11054:
11051:
11048:
11045:
11042:
11039:
11036:
11031:
11027:
11009:
11006:
10994:
10991:
10982:
10981:
10970:
10965:
10960:
10955:
10951:
10947:
10944:
10941:
10909:
10904:
10895:point spectrum
10875:
10872:
10823:
10822:
10811:
10808:
10805:
10802:
10799:
10796:
10793:
10789:
10786:
10783:
10780:
10777:
10774:
10771:
10768:
10765:
10762:
10759:
10754:
10750:
10723:
10722:
10711:
10708:
10705:
10702:
10699:
10696:
10693:
10690:
10687:
10684:
10681:
10663:differentiable
10650:
10647:
10644:
10640:
10637:
10634:
10622:
10621:
10605:
10601:
10597:
10591:
10587:
10581:
10578:
10575:
10548:
10545:
10531:
10530:
10518:
10513:
10508:
10504:
10500:
10496:
10493:
10490:
10487:
10483:
10478:
10474:
10470:
10466:
10463:
10460:
10457:
10453:
10444:
10440:
10436:
10422:
10421:
10409:
10404:
10399:
10395:
10391:
10385:
10381:
10377:
10373:
10368:
10363:
10359:
10355:
10349:
10340:
10312:
10309:
10306:
10278:
10275:
10272:
10269:
10266:
10257:
10245:
10244:
10232:
10227:
10223:
10219:
10214:
10209:
10205:
10201:
10197:
10194:
10191:
10188:
10185:
10148:
10145:
10141:
10120:
10117:
10113:
10079:
10074:
10070:
10066:
10061:
10056:
10052:
10048:
10025:
10014:
10013:
10001:
9996:
9992:
9988:
9984:
9981:
9977:
9972:
9968:
9964:
9960:
9946:
9945:
9933:
9927:
9923:
9919:
9916:
9913:
9910:
9907:
9904:
9899:
9895:
9890:
9886:
9883:
9880:
9877:
9874:
9871:
9868:
9865:
9851:Borel function
9841:. That is, if
9835:Dirac notation
9830:
9827:
9826:
9825:
9814:
9811:
9808:
9805:
9802:
9799:
9796:
9793:
9788:
9785:
9780:
9777:
9773:
9769:
9766:
9725:
9722:
9719:
9716:
9713:
9710:
9684:
9681:
9678:
9675:
9672:
9669:
9664:
9651:
9650:
9639:
9636:
9633:
9628:
9625:
9622:
9619:
9616:
9613:
9608:
9603:
9600:
9597:
9594:
9591:
9588:
9562:
9542:
9531:
9530:
9519:
9514:
9510:
9506:
9501:
9497:
9493:
9490:
9487:
9482:
9478:
9474:
9468:
9465:
9459:
9456:
9431:
9427:
9421:
9417:
9413:
9410:
9407:
9403:
9399:
9396:
9391:
9387:
9383:
9380:
9358:
9354:
9347:
9344:
9338:
9335:
9332:
9328:
9324:
9321:
9315:
9312:
9306:
9303:
9281:
9277:
9254:
9250:
9226:
9223:
9197:
9194:
9168:
9145:
9142:
9139:
9119:
9116:
9113:
9110:
9090:
9070:
9050:
9047:
9044:
9041:
9021:
9001:
8992:. That is, if
8985:
8982:
8966:
8942:
8922:
8902:
8886:
8885:
8874:
8871:
8868:
8865:
8862:
8859:
8856:
8853:
8850:
8846:
8843:
8840:
8837:
8834:
8831:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8802:
8799:
8795:
8791:
8788:
8761:
8744:
8731:
8711:
8687:
8683:
8678:
8674:
8671:
8668:
8661:
8657:
8652:
8648:
8645:
8625:
8605:
8594:
8593:
8582:
8579:
8576:
8573:
8570:
8567:
8564:
8560:
8557:
8554:
8551:
8548:
8543:
8540:
8536:
8532:
8529:
8519:
8508:
8505:
8502:
8499:
8496:
8493:
8490:
8487:
8484:
8459:
8455:
8451:
8446:
8442:
8438:
8435:
8406:
8402:
8379:
8375:
8352:
8348:
8344:
8341:
8338:
8335:
8313:
8309:
8305:
8302:
8299:
8296:
8276:
8256:
8236:
8235:
8224:
8220:
8216:
8213:
8210:
8207:
8204:
8199:
8195:
8191:
8188:
8185:
8180:
8175:
8172:
8169:
8166:
8163:
8158:
8154:
8150:
8147:
8143:
8139:
8134:
8130:
8126:
8123:
8109:
8108:
8097:
8092:
8088:
8084:
8081:
8078:
8075:
8072:
8068:
8065:
8062:
8059:
8056:
8053:
8050:
8047:
8044:
8041:
8038:
8035:
8032:
8029:
8024:
8020:
7996:
7993:
7990:
7987:
7984:
7979:
7975:
7971:
7966:
7962:
7958:
7955:
7952:
7947:
7943:
7922:
7897:
7893:
7890:
7887:
7884:
7860:
7857:
7854:
7851:
7848:
7845:
7842:
7830:
7827:
7810:
7790:
7768:
7764:
7741:
7738:
7735:
7731:
7708:
7704:
7679:
7674:
7671:
7667:
7664:
7660:
7656:
7631:
7628:
7625:
7621:
7597:
7593:
7589:
7584:
7580:
7559:
7555:
7551:
7546:
7542:
7519:
7516:
7513:
7509:
7505:
7502:
7499:
7496:
7491:
7487:
7463:
7460:
7456:
7451:
7448:
7445:
7442:
7419:Main article:
7416:
7413:
7411:
7410:
7399:
7396:
7393:
7390:
7387:
7384:
7381:
7378:
7375:
7372:
7369:
7366:
7363:
7360:
7357:
7354:
7351:
7348:
7345:
7342:
7339:
7336:
7333:
7330:
7327:
7324:
7321:
7318:
7315:
7310:
7306:
7302:
7299:
7296:
7293:
7290:
7285:
7281:
7261:
7258:
7253:
7249:
7245:
7242:
7220:
7216:
7212:
7209:
7206:
7203:
7183:
7180:
7177:
7172:
7168:
7147:
7127:
7122:
7118:
7114:
7111:
7091:
7088:
7085:
7080:
7076:
7072:
7069:
7066:
7063:
7060:
7049:
7037:
7034:
7031:
7028:
7025:
7019:
7016:
7010:
7007:
7004:
7001:
6998:
6978:
6975:
6972:
6969:
6964:
6960:
6956:
6953:
6950:
6947:
6942:
6938:
6934:
6928:
6925:
6919:
6916:
6913:
6910:
6907:
6887:
6884:
6881:
6878:
6875:
6872:
6869:
6866:
6861:
6857:
6853:
6850:
6847:
6827:
6824:
6821:
6816:
6812:
6791:
6788:
6785:
6782:
6777:
6773:
6769:
6766:
6763:
6760:
6757:
6754:
6751:
6748:
6745:
6742:
6739:
6736:
6727:. That is, if
6716:
6713:
6710:
6707:
6702:
6698:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6661:
6658:
6655:
6635:
6632:
6629:
6626:
6621:
6617:
6613:
6610:
6607:
6604:
6601:
6591:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6552:
6549:
6543:
6540:
6537:
6534:
6531:
6528:
6525:
6522:
6502:
6499:
6496:
6493:
6490:
6487:
6481:
6478:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6417:
6414:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6390:
6387:
6366:
6362:
6359:
6339:
6336:
6333:
6328:
6324:
6320:
6317:
6314:
6311:
6308:
6286:
6282:
6278:
6275:
6255:
6241:
6230:
6228:
6227:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6176:
6173:
6168:
6164:
6139:
6134:
6131:
6126:
6122:
6101:
6079:
6075:
6054:
6051:
6048:
6045:
6042:
6039:
6036:
6033:
6030:
6027:
6024:
6021:
6016:
6012:
6008:
6005:
5983:
5980:
5975:
5971:
5948:
5945:
5940:
5936:
5915:
5912:
5909:
5906:
5903:
5900:
5895:
5891:
5879:
5877:
5876:
5874:
5873:
5862:
5859:
5856:
5853:
5850:
5843:
5840:
5834:
5830:
5827:
5807:
5804:
5801:
5795:
5792:
5786:
5783:
5763:
5760:
5757:
5754:
5751:
5748:
5744:
5740:
5734:
5731:
5704:
5701:
5695:
5691:
5688:
5685:
5680:
5675:
5669:
5665:
5661:
5658:
5654:
5628:
5625:
5619:
5615:
5610:
5605:
5601:
5580:
5577:
5572:
5568:
5547:
5527:
5524:
5502:
5498:
5494:
5491:
5481:
5470:
5465:
5461:
5457:
5454:
5451:
5448:
5443:
5439:
5435:
5432:
5412:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5388:
5385:
5382:
5379:
5360:
5355:
5351:
5347:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5317:is closed, so
5306:
5286:
5283:
5280:
5277:
5274:
5271:
5266:
5262:
5258:
5255:
5250:
5246:
5223:
5219:
5215:
5212:
5207:
5203:
5199:
5196:
5191:
5187:
5166:
5163:
5160:
5157:
5131:
5127:
5107:
5104:
5099:
5095:
5091:
5086:
5082:
5078:
5072:
5069:
5066:
5063:
5059:
5054:
5051:
5046:
5042:
5038:
5033:
5029:
5025:
5005:
5002:
4999:
4996:
4974:
4970:
4966:
4963:
4960:
4955:
4951:
4945:
4941:
4937:
4932:
4928:
4905:
4901:
4897:
4894:
4883:
4871:
4868:
4865:
4860:
4856:
4852:
4849:
4838:
4824:
4820:
4816:
4813:
4793:
4790:
4787:
4784:
4781:
4778:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4720:
4716:
4712:
4690:
4686:
4673:
4662:
4659:
4656:
4651:
4647:
4643:
4640:
4620:
4617:
4614:
4611:
4606:
4602:
4598:
4595:
4575:
4572:
4569:
4564:
4561:
4556:
4552:
4548:
4545:
4536:and show that
4525:
4520:
4517:
4512:
4508:
4487:
4484:
4481:
4478:
4475:
4472:
4469:
4465:
4461:
4458:
4435:
4432:
4429:
4426:
4423:
4420:
4417:
4414:
4411:
4408:
4405:
4385:
4381:
4377:
4374:
4354:
4351:
4348:
4345:
4342:
4337:
4333:
4312:
4298:
4287:
4270:
4267:
4264:
4261:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4171:
4170:
4159:
4156:
4153:
4150:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4116:
4112:
4109:
4105:
4098:
4095:
4092:
4088:
4083:
4077:
4074:
4071:
4067:
4062:
4058:
4053:
4049:
4043:
4040:
4037:
4031:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3996:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3888:
3884:
3880:
3877:
3874:
3870:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3842:
3838:
3835:
3832:
3829:
3826:
3814:
3813:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3729:
3726:
3723:
3720:
3717:
3714:
3693:
3689:
3686:
3666:
3663:
3660:
3657:
3654:
3650:
3646:
3643:
3640:
3637:
3616:
3613:
3610:
3607:
3604:
3601:
3596:
3593:
3590:
3586:
3582:
3579:
3557:
3554:
3551:
3548:
3545:
3542:
3537:
3534:
3531:
3527:
3523:
3520:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3441:if and only if
3423:
3419:
3416:
3413:
3410:
3407:
3382:
3378:
3375:
3372:
3369:
3366:
3355:
3354:
3343:
3340:
3337:
3334:
3331:
3328:
3324:
3320:
3317:
3314:
3311:
3308:
3285:
3259:
3235:
3232:
3229:
3226:
3206:
3195:
3194:
3183:
3179:
3167:
3164:
3160:
3156:
3153:
3150:
3147:
3144:
3141:
3137:
3132:
3128:
3125:
3121:
3117:
3114:
3111:
3108:
3105:
3093:is defined as
3082:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3010:
3007:
3006:
3005:
2989:
2969:
2966:
2963:
2960:
2957:
2937:
2934:
2931:
2928:
2924:
2920:
2916:
2895:
2892:
2889:
2886:
2882:
2878:
2874:
2853:
2833:
2822:
2806:
2791:
2779:
2775:
2772:
2769:
2766:
2763:
2759:
2756:
2753:
2750:
2747:
2743:
2739:
2736:
2733:
2730:
2727:
2724:
2720:
2715:
2711:
2708:
2704:
2701:
2698:
2687:
2676:
2673:
2653:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2582:
2579:
2575:
2572:
2569:
2565:
2562:
2542:
2539:
2536:
2533:
2530:
2518:
2515:
2498:
2478:
2475:
2472:
2469:
2466:
2437:
2434:
2431:
2428:
2425:
2405:
2402:
2399:
2396:
2393:
2373:
2370:
2367:
2364:
2361:
2358:
2334:
2331:
2328:
2325:
2322:
2311:
2310:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2215:
2212:
2209:
2206:
2203:
2180:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2070:
2058:
2055:
2030:
2010:
1997:is said to be
1986:
1975:
1974:
1963:
1960:
1957:
1954:
1951:
1947:
1943:
1939:
1934:
1930:
1927:
1924:
1921:
1918:
1915:
1909:
1904:
1900:
1897:
1894:
1891:
1888:
1885:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1838:
1835:
1832:
1811:
1807:
1804:
1801:
1798:
1794:
1773:
1751:
1747:
1726:
1704:
1700:
1696:
1693:
1690:
1687:
1684:
1681:
1661:
1639:
1635:
1631:
1628:
1604:
1590:
1589:
1576:
1572:
1568:
1563:
1560:
1556:
1552:
1549:
1535:
1534:
1521:
1517:
1513:
1508:
1505:
1501:
1497:
1494:
1469:
1465:
1444:
1422:
1419:
1415:
1394:
1372:
1368:
1347:
1344:
1341:
1338:
1314:
1294:
1291:
1288:
1285:
1280:
1276:
1272:
1269:
1258:
1257:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1156:
1136:
1133:
1130:
1127:
1124:
1104:
1099:
1095:
1091:
1088:
1085:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1040:
1018:
1014:
1010:
1007:
977:
962:
961:
950:
947:
944:
941:
938:
935:
932:
928:
925:
922:
917:
913:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
859:
839:
836:
831:
827:
823:
820:
798:
794:
779:argument. The
760:
757:
754:
751:
748:
726:
723:
720:
717:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
650:
627:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
530:
497:
494:
491:
488:
485:
461:
441:
438:
435:
432:
429:
426:
396:
372:
360:
357:
319:
318:
307:
304:
301:
298:
295:
290:
286:
279:
276:
270:
266:
260:
257:
254:
248:
245:
216:
213:
160:Hilbert spaces
69:
66:
63:
60:
57:
26:
9:
6:
4:
3:
2:
18018:
18007:
18004:
18002:
17999:
17997:
17994:
17993:
17991:
17976:
17975:
17970:
17968:
17966:
17962:
17958:
17954:
17950:
17949:
17947:
17943:
17937:
17934:
17932:
17929:
17927:
17924:
17922:
17919:
17917:
17914:
17912:
17909:
17907:
17904:
17902:
17899:
17897:
17894:
17893:
17891:
17887:
17880:
17876:
17873:
17871:
17868:
17866:
17863:
17862:
17860:
17858:Other results
17856:
17850:
17847:
17845:
17842:
17840:
17837:
17836:
17834:
17830:
17824:
17821:
17819:
17816:
17814:
17810:
17809:Hilbert space
17807:
17805:
17801:
17800:Inner product
17798:
17796:
17793:
17792:
17790:
17786:
17782:
17775:
17770:
17768:
17763:
17761:
17756:
17755:
17752:
17740:
17732:
17731:
17728:
17722:
17719:
17717:
17714:
17712:
17711:Weak topology
17709:
17707:
17704:
17702:
17699:
17697:
17694:
17693:
17691:
17687:
17680:
17676:
17673:
17671:
17668:
17666:
17663:
17661:
17658:
17656:
17653:
17651:
17648:
17646:
17643:
17641:
17638:
17636:
17635:Index theorem
17633:
17631:
17628:
17626:
17623:
17621:
17618:
17617:
17615:
17611:
17605:
17602:
17600:
17597:
17596:
17594:
17592:Open problems
17590:
17584:
17581:
17579:
17576:
17574:
17571:
17569:
17566:
17564:
17561:
17559:
17556:
17555:
17553:
17549:
17543:
17540:
17538:
17535:
17533:
17530:
17528:
17525:
17523:
17520:
17518:
17515:
17513:
17510:
17508:
17505:
17503:
17500:
17498:
17495:
17494:
17492:
17488:
17482:
17479:
17477:
17474:
17472:
17469:
17467:
17464:
17462:
17459:
17457:
17454:
17452:
17449:
17447:
17444:
17442:
17439:
17438:
17436:
17434:
17430:
17420:
17417:
17415:
17412:
17410:
17407:
17404:
17400:
17396:
17393:
17391:
17388:
17386:
17383:
17382:
17380:
17376:
17370:
17367:
17365:
17362:
17360:
17357:
17355:
17352:
17350:
17347:
17345:
17342:
17340:
17337:
17335:
17332:
17330:
17327:
17325:
17322:
17321:
17318:
17315:
17311:
17306:
17302:
17298:
17291:
17286:
17284:
17279:
17277:
17272:
17271:
17268:
17264:
17257:
17253:
17249:
17245:
17241:
17237:
17231:
17227:
17223:
17219:
17215:
17214:
17209:
17205:
17201:
17197:
17193:
17189:
17185:
17181:
17177:
17173:
17169:
17165:
17158:
17153:
17149:
17145:
17141:
17135:
17131:
17127:
17123:
17119:
17115:
17109:
17105:
17100:
17096:
17092:
17088:
17084:
17080:
17074:
17070:
17065:
17061:
17057:
17053:
17047:
17043:
17038:
17035:
17029:
17025:
17021:
17017:
17013:
17009:
17005:
17004:
16998:
16994:
16990:
16986:
16983:
16977:
16973:
16969:
16965:
16960:
16956:
16952:
16948:
16946:0-486-42258-5
16942:
16938:
16933:
16928:
16923:
16919:
16915:
16914:
16909:
16904:
16900:
16896:
16891:
16886:
16881:
16876:
16872:
16868:
16864:
16860:
16859:
16854:
16849:
16845:
16841:
16837:
16833:
16829:
16825:
16820:
16815:
16812:(1): 012104.
16811:
16807:
16806:
16800:
16796:
16792:
16788:
16784:
16780:
16776:
16771:
16766:
16762:
16758:
16757:
16751:
16747:
16742:
16738:
16736:0-273-08496-8
16732:
16728:
16723:
16722:
16710:
16705:
16698:
16693:
16686:
16681:
16674:
16669:
16663:
16658:
16651:
16646:
16639:
16634:
16627:
16622:
16615:
16610:
16603:
16598:
16591:
16586:
16579:
16574:
16567:
16562:
16555:
16550:
16544:
16539:
16532:
16527:
16520:
16515:
16508:
16503:
16496:
16495:Akhiezer 1981
16491:
16485:, p. 152
16484:
16483:Akhiezer 1981
16479:
16473:, p. 207
16472:
16467:
16460:
16455:
16448:
16443:
16436:
16431:
16424:
16419:
16412:
16407:
16400:
16395:
16388:
16383:
16376:
16371:
16365:, p. 195
16364:
16359:
16353:, p. 238
16352:
16347:
16341:Corollary 9.9
16340:
16335:
16328:
16323:
16316:
16311:
16307:
16278:
16272:
16269:
16259:
16255:
16245:
16242:
16240:
16237:
16235:
16232:
16230:
16227:
16225:
16222:
16220:
16217:
16216:
16209:
16206:
16202:
16186:
16183:
16158:
16155:
16142:
16130:
16128:
16124:
16105:
16100:
16093:
16089:
16078:
16073:
16070:
16067:
16063:
16059:
16049:
16048:
16047:
16045:
16035:
16031:
16029:
16012:
16007:
16003:
15999:
15995:
15992:
15986:
15978:
15957:
15953:
15932:
15924:
15918:
15905:
15899:
15893:
15890:
15884:
15880:
15874:
15863:
15846:
15844:
15834:
15832:
15826:
15813:
15809:
15803:
15793:
15784:
15778:
15771:
15767:
15762:
15756:
15753:
15750:
15747:
15744:
15740:
15731:
15727:
15723:
15719:
15701:
15698:
15695:
15692:
15689:
15684:
15679:
15675:
15671:
15661:
15657:
15653:
15649:
15645:
15633:
15631:
15627:
15623:
15604:
15598:
15592:
15586:
15583:
15577:
15566:
15560:
15549:
15539:
15523:
15515:
15514:
15513:
15511:
15507:
15502:
15497:
15492:
15488:
15468:
15461:
15455:
15449:
15446:
15441:
15430:
15424:
15410:
15401:
15398:
15393:
15375:
15372:
15368:
15364:
15360:
15354:
15344:
15335:
15329:
15324:
15320:
15312:
15311:
15310:
15308:
15304:
15300:
15295:
15290:
15286:
15282:
15278:
15274:
15270:
15266:
15262:
15260:
15250:
15248:
15246:
15242:
15237:
15233:
15221:
15208:
15204:
15197:
15189:
15185:
15181:
15178:
15160:
15155:
15151:
15145:
15137:
15133:
15129:
15126:
15122:
15118:
15113:
15109:
15105:
15102:
15080:
15076:
15067:
15063:
15036:
15032:
15022:
15018:
15011:
15005:
15001:
14995:
14991:
14987:
14984:
14966:
14961:
14953:
14952:
14951:
14950:
14946:
14945:
14940:
14921:
14915:
14902:
14898:
14888:
14875:
14871:
14863:
14862:
14861:
14859:
14852:
14830:
14823:
14819:
14814:
14810:
14805:
14798:
14790:
14786:
14780:
14776:
14772:
14766:
14760:
14757:
14750:
14749:
14748:
14747:
14743:
14737:
14735:
14731:
14727:
14723:
14719:
14715:
14711:
14707:
14703:
14699:
14695:
14683:
14681:
14677:
14658:
14655:
14650:
14640:
14636:
14630:
14626:
14622:
14619:
14607:
14600:
14599:
14598:
14596:
14592:
14573:
14566:
14562:
14554:
14550:
14541:
14534:
14530:
14522:
14518:
14505:
14501:
14493:
14489:
14471:
14462:
14458:
14453:
14448:
14444:
14436:
14435:
14434:
14417:
14410:
14406:
14400:
14396:
14392:
14385:
14381:
14375:
14371:
14363:
14359:
14353:
14349:
14345:
14340:
14336:
14328:
14327:
14326:
14304:
14300:
14296:
14293:
14290:
14285:
14281:
14277:
14272:
14268:
14261:
14258:
14251:
14250:
14249:
14247:
14246:multi-indices
14243:
14239:
14218:
14214:
14208:
14204:
14198:
14194:
14190:
14186:
14177:
14171:
14167:
14160:
14159:
14158:
14156:
14146:
14144:
14137:
14117:
14113:
14110:
14107:
14104:
14101:
14096:
14092:
14088:
14082:
14074:
14070:
14066:
14063:
14059:
14055:
14050:
14046:
14038:
14037:
14036:
14034:
14030:
14025:
14008:
14002:
13997:
13994:
13990:
13986:
13980:
13974:
13966:
13962:
13939:
13933:
13930:
13924:
13918:
13911:
13910:
13909:
13895:
13875:
13855:
13846:
13844:
13837:
13830:
13826:
13822:
13818:
13814:
13810:
13806:
13803:which are in
13783:
13780:
13777:
13774:
13772:
13766:
13763:
13759:
13756:
13749:
13746:
13743:
13741:
13735:
13732:
13728:
13725:
13714:
13713:
13712:
13710:
13703:
13696:
13692:
13688:
13669:
13666:
13660:
13654:
13651:
13645:
13639:
13632:
13631:
13630:
13612:
13609:
13603:
13600:
13592:
13589:
13586:
13579:
13578:
13577:
13560:
13557:
13554:
13546:
13542:
13522:
13505:
13501:
13480:
13472:
13468:
13451:
13445:
13425:
13422:
13419:
13410:
13406:
13399:
13379:
13366:
13358:
13355:
13349:
13346:
13340:
13334:
13326:
13322:
13319:
13315:
13309:
13299:
13283:
13279:
13256:
13252:
13248:
13226:
13222:
13199:
13195:
13191:
13187:
13181:
13177:
13154:
13144:
13107:
13103:
13080:
13070:
13033:
13029:
13005:
12999:
12994:
12990:
12967:
12963:
12959:
12955:
12951:
12946:
12942:
12919:
12909:
12885:
12863:
12853:
12827:
12817:
12787:
12762:
12752:
12725:
12721:
12700:
12696:
12692:
12688:
12681:
12675:
12670:
12666:
12662:
12654:
12650:
12646:
12641:
12636:
12632:
12622:
12619:
12613:
12609:
12603:
12599:
12594:
12577:
12573:
12569:
12566:
12557:
12553:
12549:
12544:
12534:
12527:
12523:
12520:
12513:
12512:
12511:
12491:
12479:
12459:
12448:
12430:
12426:
12422:
12396:
12392:
12388:
12382:
12379:
12373:
12369:
12363:
12354:
12344:
12343:
12342:
12339:
12329:
12312:
12306:
12303:
12295:
12291:
12284:
12276:
12272:
12254:
12251:
12248:
12245:
12242:
12238:
12234:
12228:
12220:
12216:
12207:
12203:
12187:
12184:
12181:
12156:
12153:
12131:
12128:
12124:
12120:
12114:
12106:
12102:
12092:
12090:
12071:
12062:
12056:
12053:
12047:
12041:
12038:
12035:
12023:
12017:
12011:
12008:
12001:
12000:
11999:
11997:
11992:
11990:
11963:
11931:
11899:
11890:
11863:
11854:
11849:
11835:
11832:
11828:
11824:
11821:
11817:
11814:
11811:
11808:
11803:
11799:
11776:
11772:
11751:
11731:
11711:
11688:
11682:
11679:
11676:
11673:
11650:
11647:
11644:
11641:
11616:
11612:
11603:
11599:
11580:
11576:
11570:
11566:
11557:
11548:
11544:
11540:
11535:
11531:
11527:
11523:
11520:
11513:
11512:
11511:
11509:
11491:
11487:
11463:
11459:
11455:
11452:
11446:
11440:
11437:
11431:
11425:
11422:
11417:
11413:
11404:
11395:
11391:
11387:
11373:
11369:
11365:
11362:
11355:
11354:
11353:
11351:
11324:
11315:
11310:
11308:
11304:
11300:
11281:
11277:
11273:
11270:
11264:
11258:
11255:
11249:
11243:
11240:
11237:
11227:
11223:
11217:
11211:
11208:
11201:
11200:
11199:
11196:
11194:
11175:
11171:
11163:
11159:
11153:
11147:
11144:
11137:
11136:
11135:
11133:
11114:
11108:
11105:
11100:
11097:
11091:
11088:
11085:
11082:
11079:
11072:
11071:
11070:
11068:
11046:
11043:
11040:
11029:
11025:
11016:
11005:
11000:
10990:
10987:
10968:
10963:
10953:
10945:
10939:
10932:
10931:
10930:
10928:
10924:
10920:
10918:
10912:
10907:
10900:
10896:
10892:
10888:
10881:
10871:
10869:
10864:
10859:
10855:
10851:
10847:
10843:
10839:
10836:The operator
10834:
10832:
10828:
10809:
10806:
10803:
10800:
10797:
10794:
10791:
10784:
10781:
10778:
10772:
10769:
10766:
10760:
10752:
10748:
10740:
10739:
10738:
10736:
10732:
10728:
10709:
10706:
10700:
10694:
10691:
10685:
10679:
10672:
10671:
10670:
10668:
10664:
10645:
10603:
10599:
10595:
10589:
10585:
10579:
10576:
10573:
10566:
10565:
10564:
10563:
10559:
10554:
10544:
10542:
10538:
10537:
10516:
10511:
10506:
10498:
10491:
10485:
10481:
10476:
10468:
10464:
10461:
10458:
10455:
10451:
10442:
10438:
10434:
10427:
10426:
10425:
10407:
10402:
10397:
10389:
10383:
10379:
10375:
10371:
10366:
10361:
10353:
10347:
10338:
10330:
10329:
10328:
10326:
10307:
10304:
10296:
10292:
10273:
10270:
10267:
10264:
10255:
10230:
10225:
10217:
10212:
10207:
10199:
10195:
10192:
10189:
10186:
10183:
10176:
10175:
10174:
10170:
10164:
10162:
10102:
10098:
10094:
10077:
10072:
10064:
10059:
10054:
10046:
10037:
10033:
10028:
10023:
10019:
9999:
9994:
9986:
9982:
9979:
9975:
9970:
9962:
9958:
9951:
9950:
9949:
9931:
9925:
9911:
9905:
9897:
9888:
9884:
9881:
9878:
9875:
9869:
9863:
9856:
9855:
9854:
9852:
9848:
9844:
9840:
9836:
9812:
9806:
9800:
9794:
9791:
9783:
9775:
9771:
9767:
9764:
9757:
9756:
9755:
9753:
9749:
9745:
9741:
9740:
9720:
9717:
9711:
9700:
9679:
9676:
9670:
9634:
9623:
9620:
9614:
9601:
9595:
9589:
9579:
9578:
9577:
9574:
9560:
9540:
9517:
9512:
9508:
9499:
9495:
9488:
9485:
9480:
9476:
9463:
9454:
9447:
9446:
9445:
9425:
9419:
9415:
9411:
9408:
9405:
9401:
9397:
9389:
9385:
9378:
9352:
9342:
9336:
9333:
9330:
9326:
9322:
9310:
9301:
9279:
9275:
9252:
9248:
9221:
9192:
9182:
9166:
9157:
9143:
9140:
9137:
9114:
9108:
9088:
9068:
9045:
9039:
9019:
8999:
8991:
8981:
8978:
8964:
8956:
8940:
8920:
8900:
8892:
8869:
8863:
8860:
8857:
8854:
8851:
8844:
8838:
8832:
8826:
8820:
8817:
8811:
8805:
8800:
8797:
8793:
8789:
8786:
8779:
8778:
8777:
8775:
8759:
8743:
8729:
8709:
8685:
8681:
8672:
8666:
8659:
8655:
8646:
8623:
8603:
8580:
8577:
8574:
8571:
8568:
8565:
8558:
8555:
8552:
8549:
8546:
8541:
8538:
8534:
8530:
8527:
8520:
8506:
8503:
8500:
8497:
8494:
8491:
8488:
8485:
8482:
8475:
8474:
8473:
8457:
8453:
8444:
8440:
8436:
8433:
8426:
8422:
8404:
8400:
8377:
8373:
8350:
8346:
8342:
8339:
8336:
8333:
8311:
8307:
8303:
8300:
8297:
8294:
8274:
8254:
8245:
8243:
8242:
8222:
8218:
8211:
8208:
8205:
8197:
8193:
8189:
8186:
8183:
8170:
8167:
8164:
8156:
8152:
8148:
8145:
8141:
8137:
8132:
8128:
8124:
8121:
8114:
8113:
8112:
8095:
8090:
8086:
8082:
8079:
8076:
8073:
8066:
8060:
8054:
8048:
8042:
8039:
8033:
8027:
8022:
8018:
8010:
8009:
8008:
8007:, defined by
7991:
7988:
7985:
7977:
7973:
7964:
7960:
7956:
7953:
7950:
7945:
7941:
7920:
7912:
7888:
7885:
7882:
7874:
7855:
7852:
7846:
7843:
7833:Firstly, let
7826:
7822:
7808:
7788:
7766:
7762:
7739:
7736:
7733:
7729:
7706:
7702:
7692:
7677:
7672:
7669:
7665:
7662:
7658:
7654:
7647:
7629:
7626:
7623:
7619:
7611:
7582:
7578:
7544:
7540:
7517:
7514:
7511:
7507:
7503:
7497:
7489:
7485:
7461:
7458:
7454:
7449:
7446:
7443:
7440:
7433:
7429:
7422:
7397:
7394:
7391:
7388:
7385:
7379:
7376:
7373:
7370:
7367:
7364:
7358:
7355:
7352:
7349:
7340:
7337:
7334:
7331:
7328:
7325:
7319:
7316:
7313:
7308:
7304:
7294:
7291:
7288:
7283:
7279:
7259:
7256:
7251:
7247:
7243:
7240:
7218:
7214:
7210:
7207:
7204:
7201:
7181:
7178:
7175:
7170:
7166:
7145:
7125:
7120:
7116:
7112:
7109:
7089:
7086:
7083:
7078:
7074:
7070:
7067:
7064:
7061:
7058:
7051:The equality
7050:
7035:
7032:
7029:
7023:
7014:
7008:
7005:
6999:
6996:
6973:
6970:
6967:
6962:
6958:
6951:
6948:
6945:
6940:
6932:
6923:
6917:
6914:
6908:
6905:
6882:
6876:
6870:
6867:
6864:
6859:
6855:
6848:
6845:
6825:
6822:
6819:
6814:
6810:
6786:
6783:
6780:
6775:
6771:
6764:
6761:
6758:
6752:
6749:
6746:
6743:
6737:
6734:
6711:
6708:
6705:
6700:
6696:
6689:
6686:
6683:
6677:
6674:
6671:
6668:
6662:
6659:
6656:
6653:
6633:
6630:
6627:
6624:
6619:
6615:
6611:
6608:
6605:
6602:
6599:
6592:
6577:
6571:
6568:
6565:
6562:
6556:
6547:
6541:
6538:
6535:
6532:
6529:
6526:
6523:
6497:
6494:
6491:
6485:
6476:
6450:
6447:
6444:
6438:
6435:
6415:
6409:
6406:
6403:
6397:
6391:
6385:
6360:
6357:
6337:
6334:
6331:
6326:
6322:
6318:
6315:
6312:
6309:
6306:
6284:
6280:
6276:
6273:
6253:
6246:
6245:
6239:
6214:
6208:
6202:
6199:
6196:
6174:
6171:
6166:
6162:
6153:
6137:
6132:
6129:
6124:
6120:
6099:
6077:
6073:
6052:
6046:
6043:
6040:
6037:
6034:
6031:
6025:
6022:
6019:
6014:
6010:
5981:
5978:
5973:
5969:
5946:
5943:
5938:
5934:
5913:
5907:
5904:
5901:
5898:
5893:
5889:
5881:The operator
5880:
5860:
5854:
5848:
5838:
5832:
5828:
5825:
5805:
5802:
5790:
5781:
5758:
5755:
5752:
5738:
5729:
5699:
5693:
5689:
5686:
5683:
5678:
5673:
5667:
5663:
5659:
5656:
5652:
5623:
5617:
5613:
5608:
5603:
5599:
5578:
5575:
5570:
5566:
5545:
5525:
5522:
5500:
5496:
5492:
5489:
5482:
5468:
5463:
5459:
5455:
5452:
5449:
5446:
5441:
5437:
5433:
5430:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5383:
5380:
5377:
5358:
5353:
5349:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5324:
5304:
5284:
5281:
5278:
5275:
5272:
5264:
5260:
5256:
5253:
5248:
5244:
5221:
5217:
5213:
5210:
5205:
5201:
5197:
5194:
5189:
5185:
5177:Furthermore,
5164:
5161:
5158:
5155:
5147:
5129:
5125:
5105:
5097:
5093:
5089:
5084:
5080:
5067:
5061:
5057:
5052:
5044:
5040:
5036:
5031:
5027:
5003:
5000:
4997:
4994:
4972:
4968:
4964:
4961:
4958:
4953:
4949:
4943:
4939:
4935:
4930:
4926:
4903:
4899:
4895:
4892:
4885:
4884:
4869:
4866:
4863:
4858:
4854:
4850:
4847:
4839:
4822:
4818:
4814:
4811:
4791:
4788:
4782:
4776:
4756:
4750:
4744:
4738:
4732:
4729:
4723:
4718:
4714:
4688:
4684:
4675:
4674:
4660:
4657:
4654:
4649:
4645:
4641:
4638:
4615:
4609:
4604:
4600:
4596:
4593:
4573:
4570:
4567:
4562:
4559:
4554:
4550:
4546:
4543:
4523:
4518:
4515:
4510:
4506:
4485:
4479:
4476:
4473:
4459:
4456:
4448:
4447:
4446:
4433:
4427:
4424:
4421:
4415:
4409:
4403:
4383:
4375:
4372:
4352:
4349:
4346:
4343:
4340:
4335:
4331:
4310:
4296:
4286:
4284:
4283:bounded below
4268:
4265:
4262:
4259:
4239:
4236:
4233:
4227:
4221:
4201:
4195:
4192:
4189:
4183:
4180:
4157:
4151:
4145:
4139:
4133:
4130:
4124:
4118:
4114:
4110:
4107:
4103:
4093:
4086:
4081:
4072:
4065:
4060:
4056:
4051:
4047:
4038:
4022:
4019:
4016:
4010:
4007:
4004:
3987:
3981:
3975:
3972:
3969:
3956:
3955:
3954:
3952:
3936:
3930:
3921:
3918:
3915:
3912:
3909:
3900:
3886:
3878:
3875:
3872:
3859:
3856:
3853:
3847:
3844:
3836:
3830:
3824:
3800:
3794:
3788:
3782:
3776:
3773:
3767:
3761:
3758:
3755:
3742:
3741:
3740:
3727:
3724:
3721:
3718:
3715:
3712:
3687:
3684:
3658:
3652:
3644:
3641:
3638:
3635:
3611:
3608:
3605:
3602:
3594:
3591:
3588:
3580:
3577:
3552:
3549:
3546:
3543:
3535:
3532:
3529:
3521:
3518:
3498:
3492:
3489:
3483:
3477:
3474:
3471:
3468:
3465:
3462:
3456:
3453:
3444:
3442:
3438:
3417:
3411:
3405:
3397:
3376:
3370:
3364:
3341:
3335:
3329:
3318:
3312:
3306:
3299:
3298:
3297:
3283:
3275:
3274:
3257:
3249:
3233:
3230:
3227:
3224:
3204:
3181:
3177:
3165:
3162:
3154:
3151:
3148:
3145:
3135:
3126:
3123:
3119:
3115:
3109:
3103:
3096:
3095:
3094:
3080:
3072:
3068:
3067:
3066:resolvent set
3050:
3041:
3035:
3032:
3029:
3026:
3016:
3003:
2987:
2967:
2964:
2958:
2932:
2926:
2918:
2890:
2884:
2876:
2851:
2831:
2823:
2820:
2804:
2796:
2792:
2777:
2773:
2770:
2764:
2757:
2754:
2751:
2748:
2745:
2734:
2731:
2728:
2725:
2713:
2706:
2699:
2688:
2674:
2671:
2651:
2643:
2627:
2624:
2621:
2618:
2615:
2609:
2606:
2603:
2596:
2595:
2594:
2580:
2577:
2573:
2570:
2567:
2563:
2560:
2540:
2534:
2531:
2528:
2514:
2512:
2496:
2476:
2470:
2467:
2464:
2457:
2454:
2449:
2435:
2429:
2426:
2423:
2403:
2397:
2394:
2391:
2371:
2368:
2365:
2362:
2359:
2356:
2348:
2332:
2326:
2323:
2320:
2297:
2294:
2291:
2288:
2285:
2282:
2275:
2269:
2266:
2263:
2260:
2254:
2248:
2245:
2242:
2239:
2229:
2228:
2227:
2213:
2207:
2204:
2201:
2194:
2178:
2158:
2155:
2149:
2143:
2140:
2132:
2116:
2107:
2101:
2098:
2095:
2092:
2084:
2083:Hilbert space
2068:
2054:
2052:
2047:
2044:
2028:
2008:
2000:
1984:
1961:
1958:
1955:
1952:
1945:
1937:
1925:
1922:
1919:
1916:
1907:
1895:
1892:
1889:
1886:
1877:
1871:
1868:
1865:
1862:
1852:
1851:
1850:
1836:
1833:
1830:
1809:
1805:
1802:
1799:
1796:
1792:
1771:
1749:
1745:
1724:
1702:
1698:
1694:
1691:
1688:
1685:
1682:
1679:
1659:
1637:
1633:
1629:
1626:
1618:
1602:
1593:
1574:
1570:
1566:
1561:
1558:
1554:
1550:
1547:
1540:
1539:
1538:
1519:
1515:
1511:
1506:
1503:
1499:
1495:
1492:
1485:
1484:
1483:
1467:
1463:
1442:
1420:
1417:
1413:
1392:
1370:
1366:
1342:
1336:
1328:
1312:
1292:
1289:
1286:
1283:
1278:
1274:
1270:
1267:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1216:
1210:
1207:
1204:
1201:
1195:
1189:
1186:
1183:
1180:
1170:
1169:
1168:
1154:
1134:
1131:
1128:
1125:
1122:
1102:
1097:
1093:
1089:
1086:
1083:
1061:
1057:
1053:
1050:
1047:
1044:
1041:
1038:
1016:
1012:
1008:
1005:
997:
993:
992:
975:
967:
948:
945:
942:
939:
936:
933:
926:
920:
915:
911:
907:
904:
898:
892:
889:
886:
883:
873:
872:
871:
857:
837:
834:
829:
825:
821:
818:
796:
792:
784:
783:
778:
774:
755:
752:
749:
737:
724:
721:
718:
715:
695:
689:
683:
680:
674:
668:
648:
641:
625:
605:
599:
596:
593:
590:
587:
584:
578:
575:
572:
569:
560:
554:
548:
528:
520:
519:
513:
511:
495:
492:
489:
486:
483:
475:
459:
439:
436:
433:
430:
427:
424:
417:
414:
410:
394:
386:
385:Hilbert space
370:
356:
354:
350:
345:
343:
339:
335:
332:
328:
324:
305:
302:
299:
296:
293:
288:
277:
274:
268:
264:
258:
255:
252:
243:
233:
232:
231:
211:
200:
196:
192:
188:
184:
180:
176:
172:
168:
163:
161:
157:
153:
149:
145:
141:
137:
133:
129:
125:
122:
118:
114:
110:
106:
103:with a given
102:
98:
94:
90:
86:
83:
64:
61:
58:
48:
47:inner product
44:
41:
37:
33:
19:
17973:
17964:
17960:
17956:
17952:
17921:Self-adjoint
17920:
17832:Main results
17701:Balanced set
17675:Distribution
17613:Applications
17466:Krein–Milman
17451:Closed graph
17262:
17255:
17225:
17212:
17167:
17163:
17125:
17103:
17094:
17068:
17041:
17015:
17002:
16992:
16963:
16936:
16917:
16911:
16862:
16856:
16809:
16803:
16760:
16754:
16745:
16726:
16704:
16692:
16680:
16668:
16657:
16645:
16633:
16621:
16616:Section 9.10
16609:
16597:
16592:Theorem 9.41
16585:
16580:Example 9.25
16573:
16561:
16549:
16538:
16526:
16521:Section 10.4
16514:
16502:
16490:
16478:
16466:
16454:
16442:
16430:
16418:
16406:
16394:
16387:Griffel 2002
16382:
16370:
16358:
16351:Griffel 2002
16346:
16334:
16327:Griffel 2002
16322:
16310:
16258:
16204:
16200:
16140:
16132:
16122:
16120:
16043:
16041:
16032:
16027:
15976:
15922:
15920:
15848:
15840:
15830:
15828:
15729:
15725:
15721:
15717:
15659:
15655:
15651:
15647:
15643:
15635:
15629:
15625:
15621:
15619:
15509:
15505:
15500:
15495:
15490:
15486:
15484:
15306:
15302:
15298:
15293:
15288:
15284:
15280:
15276:
15272:
15268:
15264:
15263:
15258:
15256:
15239:
15235:
15231:
15228:
15065:
15061:
15060:The adjoint
15053:
14948:
14942:
14938:
14936:
14857:
14850:
14848:
14745:
14741:
14739:
14733:
14729:
14725:
14721:
14717:
14713:
14709:
14705:
14701:
14697:
14693:
14685:
14679:
14675:
14673:
14594:
14590:
14588:
14432:
14324:
14241:
14237:
14235:
14152:
14142:
14135:
14133:
14032:
14028:
14026:
13964:
13960:
13958:
13847:
13842:
13835:
13828:
13824:
13820:
13816:
13812:
13808:
13804:
13802:
13701:
13694:
13686:
13684:
13628:
13533:
13470:
13469:
13411:
13404:
13402:but not for
13397:
13324:
13323:
13311:
12740:
12480:
12446:
12413:
12335:
12274:
12270:
12205:
12201:
12093:
12088:
12086:
11995:
11993:
11988:
11888:
11852:
11850:
11601:
11597:
11595:
11507:
11478:
11349:
11313:
11311:
11302:
11298:
11296:
11197:
11192:
11190:
11131:
11129:
11066:
11014:
11011:
11002:
10986:Hamiltonians
10983:
10926:
10922:
10921:
10916:
10910:
10902:
10898:
10890:
10886:
10884:
10867:
10862:
10857:
10853:
10849:
10845:
10837:
10835:
10830:
10826:
10824:
10734:
10730:
10724:
10666:
10623:
10557:
10550:
10534:
10532:
10423:
10325:biorthogonal
10290:
10247:In the case
10246:
10168:
10165:
10026:
10021:
10017:
10015:
9947:
9846:
9842:
9832:
9751:
9743:
9738:
9652:
9575:
9532:
9158:
8987:
8979:
8953:is just the
8888:
8745:
8595:
8420:
8246:
8239:
8238:is called a
8237:
8110:
7832:
7823:
7693:
7427:
7424:
6989:and, hence,
6231:
5515:is dense in
4302:
4288:
4282:
4172:
3902:Indeed, let
3901:
3815:
3445:
3440:
3356:
3271:
3196:
3070:
3064:
3018:
2819:eigenvectors
2664:is dense in
2520:
2510:
2450:
2312:
2060:
2050:
2048:
2042:
1998:
1976:
1617:self-adjoint
1616:
1594:
1591:
1536:
1305:is dense in
1259:
995:
990:
963:
780:
776:
738:
639:
618:An operator
516:
514:
362:
346:
333:
326:
320:
164:
148:real numbers
139:
131:
123:
112:
96:
88:
84:
42:
35:
29:
17931:Trace class
17630:Heat kernel
17620:Hardy space
17527:Trace class
17441:Hahn–Banach
17403:Topological
17012:Moretti, V.
16675:Section 9.6
16543:Ruelle 1969
16425:Section 9.4
10297:) operator
10097:eigenstates
10093:eigenvalues
9444:such that:
8472:such that:
7102:shows that
6112:is), so is
5996:is the set
5146:fundamental
3396:eigenvalues
3071:regular set
2795:eigenvalues
2553:defined on
1031:, i.e., if
638:is said to
541:is the set
359:Definitions
230:defined by
199:Hamiltonian
179:observables
32:mathematics
17990:Categories
17563:C*-algebra
17378:Properties
17252:Yosida, K.
17208:Teschl, G.
16819:1808.08863
16719:References
16447:Rudin 1991
16375:Rudin 1991
15512:such that
15265:Definition
13306:See also:
10997:See also:
10929:, that is
10665:functions
10327:basis set
7194:For every
6350:for every
5591:) implies
4281:is called
3705:and every
3013:See also:
2517:Properties
1615:is called
988:is called
870:such that
508:for every
329:in a real
82:linear map
17537:Unbounded
17532:Transpose
17490:Operators
17419:Separable
17414:Reflexive
17399:Algebraic
17385:Barrelled
17244:853623322
17224:(2006) .
17192:0369-3546
17148:840278135
17091:Simon, B.
17060:144216834
16844:0022-2488
16795:0002-9505
16709:Hall 2013
16697:Hall 2013
16685:Hall 2013
16673:Hall 2013
16650:Hall 2013
16638:Hall 2013
16614:Hall 2013
16590:Hall 2013
16578:Hall 2013
16566:Hall 2013
16554:Hall 2013
16531:Hall 2013
16519:Hall 2013
16507:Hall 2013
16471:Hall 2013
16459:Hall 2013
16423:Hall 2013
16399:Hall 2013
16339:Hall 2013
16273:
16187:ω
16085:∂
16064:∑
16057:Δ
16008:λ
15996:
15990:↦
15987:λ
15958:λ
15936:↦
15933:λ
15900:λ
15894:μ
15885:λ
15875:⊕
15864:∫
15804:ℓ
15772:ℓ
15768:μ
15757:ω
15754:≤
15751:ℓ
15748:≤
15741:⨁
15702:ω
15699:≤
15696:ℓ
15693:≤
15680:ℓ
15676:μ
15648:separable
15602:∞
15593:λ
15587:μ
15574:‖
15567:λ
15561:ψ
15558:‖
15540:λ
15524:∫
15465:∞
15450:μ
15438:‖
15425:ψ
15422:‖
15411:∫
15402:ψ
15384:→
15373:ψ
15325:μ
15245:Hellinger
15182:∈
15161:∗
15130:∈
15114:∗
15106:
15033:ϕ
15028:¯
15023:α
15006:α
14996:α
14992:∑
14985:ϕ
14967:∗
14908:∞
14895:→
14881:∞
14820:ϕ
14815:α
14791:α
14781:α
14777:∑
14761:ϕ
14720:). Then
14659:ϕ
14656:
14651:α
14641:α
14631:α
14627:∑
14620:ϕ
14563:α
14546:∂
14542:⋯
14531:α
14514:∂
14502:α
14485:∂
14472:α
14449:α
14407:α
14393:⋯
14382:α
14360:α
14341:α
14301:α
14294:…
14282:α
14269:α
14259:α
14219:α
14209:α
14199:α
14195:∑
14181:→
14108:±
14067:∈
14051:±
14003:ϕ
13998:θ
13975:ϕ
13934:ϕ
13919:ϕ
13778:−
13757:−
13726:−
13655:ϕ
13640:ϕ
13610:ϕ
13596:↦
13593:ϕ
13420:α
13380:α
13350:−
13249:−
13155:∗
13148:^
13081:∗
13074:^
12920:∗
12913:^
12864:∗
12857:^
12828:∗
12821:^
12791:^
12763:∗
12756:^
12693:∈
12663:−
12610:ℏ
12604:−
12570:∈
12545:∗
12538:^
12524:
12495:^
12463:^
12423:−
12389:−
12358:^
12296:∗
12246:π
12188:β
12182:−
12157:∈
12154:β
12129:β
12107:β
12039:∣
12012:
11815:−
11804:∗
11777:∗
11683:
11677:∈
11654:⟩
11639:⟨
11617:∗
11536:∗
11524:
11492:∗
11423:∣
11366:
11241:∣
11212:
11148:
11089:−
10943:Δ
10940:−
10893:has pure
10810:…
10782:π
10773:
10580:−
10512:∗
10503:Ψ
10473:Ψ
10459:∫
10403:∗
10394:Ψ
10384:∗
10367:∗
10358:Ψ
10348:∗
10311:Γ
10305:−
10277:Γ
10271:−
10222:Ψ
10204:Ψ
10190:∫
10147:⟩
10144:Ψ
10119:⟩
10116:Ψ
10069:Ψ
10051:Ψ
9991:Ψ
9967:Ψ
9922:Ψ
9918:⟨
9903:⟩
9894:Ψ
9879:∫
9807:λ
9801:
9792:λ
9787:∞
9779:∞
9776:−
9772:∫
9721:λ
9715:∞
9712:−
9680:λ
9674:∞
9671:−
9624:λ
9618:∞
9615:−
9596:λ
9590:
9496:λ
9467:^
9430:ℏ
9416:λ
9406:−
9386:λ
9357:ℏ
9346:^
9331:−
9314:^
9276:λ
9225:^
9196:^
9141:∘
8864:
8855:∈
8852:ψ
8849:∀
8833:ψ
8806:ψ
8798:−
8774:separable
8677:‖
8670:‖
8651:‖
8644:‖
8575:
8569:∈
8566:ξ
8563:∀
8556:ξ
8547:ξ
8539:−
8501:
8489:
8450:→
8343:⊆
8337:
8304:⊆
8298:
8212:μ
8190:∈
8187:ψ
8171:μ
8149:∈
8146:ψ
8125:
8083:
8077:∈
8074:ψ
8071:∀
8055:ψ
8028:ψ
7992:μ
7970:→
7957:
7892:→
7856:μ
7850:Σ
7666:−
7655:δ
7620:δ
7447:−
7383:⇔
7377:λ
7374:−
7356:λ
7353:−
7344:⇔
7338:λ
7335:−
7317:λ
7314:−
7309:∗
7298:⇔
7284:∗
7252:∗
7219:∗
7211:
7205:∈
7176:⊆
7171:∗
7121:∗
7087:λ
7084:−
7079:∗
7065:λ
7062:−
7030:≠
7018:¯
7015:λ
7009:−
7000:
6971:λ
6968:−
6963:∗
6952:
6941:⊥
6927:¯
6924:λ
6918:−
6909:
6877:≠
6868:λ
6865:−
6860:∗
6849:
6823:λ
6820:−
6815:∗
6784:λ
6781:−
6776:∗
6765:
6759:⊊
6750:λ
6747:−
6738:
6709:λ
6706:−
6701:∗
6690:
6684:⊆
6675:λ
6672:−
6663:
6628:λ
6625:−
6620:∗
6606:λ
6603:−
6575:→
6569:
6551:¯
6548:λ
6542:−
6530:λ
6527:−
6486:∉
6480:¯
6477:λ
6439:∉
6436:λ
6398:⊆
6386:σ
6361:∈
6358:λ
6335:λ
6332:−
6327:∗
6319:⊆
6313:λ
6310:−
6285:∗
6277:⊆
6203:σ
6200:∉
6197:λ
6172:−
6167:λ
6130:−
6125:λ
6078:λ
6044:
6038:∈
6032:∣
6015:λ
5979:−
5974:λ
5944:−
5939:λ
5911:→
5905:
5899::
5894:λ
5842:¯
5839:λ
5829:
5794:¯
5791:λ
5747:∖
5739:∈
5733:¯
5730:λ
5703:¯
5700:λ
5690:
5679:⊥
5668:λ
5660:
5643:and thus
5627:¯
5624:λ
5609:∗
5604:λ
5571:∗
5501:λ
5493:
5464:λ
5456:
5450:∈
5442:λ
5405:
5399:∈
5393:λ
5354:λ
5346:
5334:
5328:∈
5279:λ
5270:→
5257:λ
5198:λ
5159:∈
5103:‖
5090:−
5077:‖
5068:λ
5053:≤
5050:‖
5037:−
5024:‖
4998:∈
4973:λ
4965:
4959:∈
4944:λ
4904:λ
4896:
4859:λ
4851:
4823:λ
4815:
4783:λ
4754:‖
4748:‖
4745:⋅
4739:λ
4730:≥
4727:‖
4719:λ
4711:‖
4689:λ
4650:λ
4642:
4605:λ
4597:
4560:−
4555:λ
4547:
4516:−
4511:λ
4468:∖
4460:∈
4457:λ
4416:⊆
4404:σ
4376:∈
4373:λ
4350:λ
4347:−
4336:λ
4266:λ
4263:−
4228:λ
4184:∉
4181:λ
4155:‖
4149:‖
4146:⋅
4140:λ
4131:≥
4128:‖
4122:‖
4119:⋅
4111:λ
4108:−
4097:‖
4091:‖
4076:‖
4070:‖
4042:‖
4036:‖
4026:⟩
4011:λ
4008:−
3999:⟨
3988:≥
3985:‖
3976:λ
3973:−
3964:‖
3925:∖
3919:
3913:∈
3879:λ
3876:−
3848:∈
3831:λ
3798:‖
3792:‖
3789:⋅
3783:λ
3774:≥
3771:‖
3762:λ
3759:−
3750:‖
3722:
3716:∈
3688:∈
3685:λ
3662:∞
3659:±
3653:∪
3645:∈
3615:⟩
3600:⟨
3592:∈
3556:⟩
3541:⟨
3533:∈
3487:‖
3481:‖
3478:∣
3472:
3466:∈
3418:⊆
3406:σ
3377:⊆
3365:σ
3330:ρ
3327:∖
3307:σ
3248:bijective
3231:λ
3228:−
3163:−
3152:λ
3149:−
3140:∃
3127:∈
3124:λ
3104:ρ
3048:→
3036:
2962:‖
2956:‖
2936:‖
2930:‖
2919:λ
2894:‖
2888:‖
2885:≤
2877:λ
2832:λ
2771:≤
2768:‖
2762:‖
2752:∈
2738:⟩
2723:⟨
2625:⊆
2619:
2613:→
2564:
2538:→
2474:→
2433:→
2401:→
2330:→
2292:∈
2280:∀
2273:⟩
2258:⟨
2252:⟩
2237:⟨
2211:→
2144:
2114:→
2102:
2051:Hermitian
1956:∈
1950:∀
1938:∈
1933:¯
1929:⟩
1914:⟨
1903:¯
1899:⟩
1884:⟨
1875:⟩
1860:⟨
1849:, i.e.,
1834:∈
1750:∗
1703:∗
1695:
1683:
1638:∗
1575:∗
1567:⊆
1562:∗
1559:∗
1520:∗
1512:⊆
1507:∗
1504:∗
1496:⊆
1482:. Hence,
1468:∗
1421:∗
1418:∗
1371:∗
1290:
1284:⊇
1279:∗
1271:
1239:
1233:∈
1221:∀
1214:⟩
1199:⟨
1193:⟩
1178:⟨
1132:
1126:∈
1098:∗
1062:∗
1054:
1048:⊆
1042:
1017:∗
1009:⊆
996:Hermitian
991:symmetric
968:operator
943:
937:∈
931:∀
924:⟩
916:∗
902:⟨
896:⟩
881:⟨
835:⊆
830:∗
822:
797:∗
759:⟩
756:⋅
750:⋅
747:⟨
719:⊆
681:⊆
597:
591:∈
585:∣
487:
434:⊆
428:
409:unbounded
349:unitarily
303:ψ
294:ψ
285:∇
265:ℏ
259:−
253:ψ
247:^
215:^
201:operator
68:⟩
65:⋅
59:⋅
56:⟨
17945:Examples
17739:Category
17551:Algebras
17433:Theorems
17390:Complete
17359:Schwartz
17305:glossary
17254:(1965),
17210:(2009),
17200:56050354
17093:(1972),
17087:Reed, M.
17014:(2017),
16991:(1966),
16989:Kato, T.
16955:49250076
16899:16592156
16748:, Kluwer
16213:See also
16123:negative
14248:. Thus
13868:. Since
13767:′
13736:′
13613:′
13525:Examples
13471:Example.
13325:Example.
11309:below).
10560:and the
10499:⟨
10482:⟩
10408:⟩
10372:⟩
10218:⟨
10213:⟩
10065:⟨
10060:⟩
10000:⟩
9976:⟩
7801:, where
7673:′
6646:Indeed,
5774:implies
4882:Indeed,
4837:follows.
4104:⟩
4057:⟨
3273:spectrum
2906:, where
2703:‖
2697:‖
2453:positive
1810:⟩
1793:⟨
1327:closable
1115:for all
187:momentum
183:position
181:such as
154:of this
17959:) with
17936:Unitary
17795:Adjoint
17542:Unitary
17522:Nuclear
17507:Compact
17502:Bounded
17497:Adjoint
17471:Min–max
17364:Sobolev
17349:Nuclear
17339:Hilbert
17334:Fréchet
17299: (
17172:Bibcode
17020:Bibcode
16968:Bibcode
16867:Bibcode
16824:Bibcode
16775:Bibcode
16250:Remarks
16134:Theorem
16026:is the
15850:Theorem
15637:Theorem
15055:Theorem
14687:Theorem
13396:0 <
11791:, with
11307:example
10923:Example
10842:compact
9697:is the
9294:, then
9179:is the
9101:, then
8747:Theorem
6898:). But
6233:Theorem
4290:Theorem
3949:By the
2511:complex
2347:complex
775:on the
156:concept
134:has an
93:adjoint
17916:Normal
17517:Normal
17354:Orlicz
17344:Hölder
17324:Banach
17313:Spaces
17301:topics
17242:
17232:
17198:
17190:
17146:
17136:
17110:
17075:
17058:
17048:
17030:
16978:
16953:
16943:
16897:
16890:388361
16887:
16842:
16793:
16733:
15555:
15485:where
15275:where
14849:where
14157:. Let
14134:where
13407:> 2
13095:is in
12447:unique
10032:formal
9653:where
8111:where
6378:. Let
6065:Since
5558:(i.e.
5016:Since
3816:where
3437:normal
3270:. The
3246:being
2384:where
2133:, if
2043:unique
1260:Since
777:second
640:extend
476:since
416:domain
323:energy
109:matrix
87:(from
17967:<∞
17329:Besov
17196:S2CID
17160:(PDF)
16814:arXiv
16765:arXiv
16604:p. 85
16302:Notes
15064:* of
14240:with
13959:then
13685:Then
11191:then
10725:Then
10624:with
10533:(See
9948:with
9849:is a
9211:. If
8772:on a
8702:; if
7871:be a
7644:by a
7138:i.e.
6802:then
6466:then
6243:Proof
4769:with
4365:with
4300:Proof
4214:then
3628:with
3073:) of
3000:is a
2864:then
2642:image
2349:form
2171:then
2081:be a
1076:and
998:) if
518:graph
413:dense
383:be a
115:is a
95:. If
80:is a
45:with
38:on a
17889:Maps
17811:and
17802:and
17677:(or
17395:Dual
17240:OCLC
17230:ISBN
17188:ISSN
17144:OCLC
17134:ISBN
17108:ISBN
17073:ISBN
17056:OCLC
17046:ISBN
17028:ISBN
16976:ISBN
16951:OCLC
16941:ISBN
16895:PMID
16840:ISSN
16791:ISSN
16731:ISBN
16205:mult
16180:mult
16152:mult
15728:) =
15642:Let
15628:are
15599:<
15462:<
15305:) =
15241:Hahn
15234:and
14712:) →
14692:Let
14325:and
14242:real
14139:dist
14097:dist
13815:and
13423:>
13214:and
12982:nor
12146:for
11666:for
10551:The
10095:and
9742:for
8616:and
8392:and
8326:and
8267:and
7875:and
7233:and
6299:and
5803:>
5236:and
4789:>
4631:and
4449:Let
4303:Let
4252:and
4234:>
3569:and
3069:(or
3019:Let
2980:and
2793:The
2416:and
2085:and
2061:Let
994:(or
964:The
515:The
387:and
363:Let
195:spin
193:and
169:and
34:, a
17180:doi
17130:GTM
16922:doi
16885:PMC
16875:doi
16832:doi
16783:doi
16270:Dom
16139:If
16129:).
15993:dim
15732:on
15508:on
15309:on
15103:dom
14947:of
14728:(D)
14682:).
14597:by
13400:≤ 2
12521:Dom
12009:Dom
11680:Dom
11521:Dom
11510:is
11506:of
11363:Dom
11352:is
11348:of
11209:Dom
11145:Dom
10913:∈ I
10889:on
10770:sin
10543:).
10447:eff
10343:eff
10260:eff
10171:= 1
10166:If
10103:of
8957:of
8861:Dom
8742:.
8572:Dom
8498:Dom
8486:Dom
8334:Dom
8295:Dom
8122:Dom
8080:Dom
7954:Dom
7913:on
7691:.
7208:Dom
6949:ker
6846:ker
6762:Dom
6735:Dom
6566:Dom
6428:If
6150:By
6041:Dom
5902:Dom
5826:ker
5687:ker
5343:Dom
5331:Dom
5144:is
4812:ker
4594:ker
4544:Dom
4173:If
3916:Dom
3841:inf
3719:Dom
3585:sup
3526:inf
3469:Dom
3276:of
3250:on
3197:If
3033:Dom
2948:if
2824:If
2797:of
2710:sup
2644:of
2561:Dom
2509:is
2141:Dom
2099:Dom
1692:Dom
1680:Dom
1619:if
1435:of
1287:Dom
1268:Dom
1236:Dom
1129:Dom
1051:Dom
1039:Dom
940:Dom
819:Dom
771:be
661:if
594:Dom
484:Dom
472:is
425:Dom
407:an
111:of
99:is
30:In
17992::
17303:–
17238:.
17194:.
17186:.
17178:.
17168:61
17166:.
17162:.
17142:.
17128:.
17089:;
17054:.
17026:,
16974:,
16949:.
16916:.
16910:.
16893:.
16883:.
16873:.
16863:71
16861:.
16855:.
16838:.
16830:.
16822:.
16810:60
16808:.
16789:.
16781:.
16773:.
16761:69
16759:.
15624:,
15283:≤
15261::
15249:.
14736:.
14145:.
14024:.
13845:.
13834:→
13819:→
13811:→
13700:,
13670:0.
13409:.
13298:.
12364::=
12328:.
12235::=
11848:.
10919:.
10870:.
10848:=
10846:Af
10710:0.
10163:.
9853:,
9398::=
9323::=
9156:.
8977:.
8138::=
7909:a
7504::=
7428:or
6997:Im
6906:Im
6687:Im
6660:Im
6154:,
5657:Im
5490:Im
5453:Im
5402:Im
4962:Im
4893:Im
4848:Im
4792:0.
4639:Im
4285:.
3953:,
2616:Im
2513:.
344:.
336:.
189:,
185:,
130:,
17974:F
17965:n
17961:K
17957:K
17955:(
17953:C
17881:)
17877:(
17773:e
17766:t
17759:v
17681:)
17405:)
17401:/
17397:(
17307:)
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17246:.
17202:.
17182::
17174::
17150:.
17116:.
17081:.
17062:.
17022::
16970::
16957:.
16930:.
16924::
16918:7
16901:.
16877::
16869::
16846:.
16834::
16826::
16816::
16797:.
16785::
16777::
16767::
16739:.
16437:.
16282:)
16279:A
16276:(
16201:μ
16184:=
16159:2
16156:=
16141:n
16106:.
16101:2
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16090:x
16079:n
16074:1
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16068:i
16060:=
16044:R
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16004:H
16000:(
15977:μ
15963:)
15954:H
15950:(
15946:m
15943:i
15940:d
15923:μ
15906:.
15903:)
15897:(
15891:d
15881:H
15869:R
15831:A
15814:.
15810:)
15799:H
15794:,
15790:R
15785:(
15779:2
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15745:1
15730:λ
15726:λ
15724:(
15722:f
15718:A
15690:1
15685:}
15672:{
15660:R
15656:ω
15652:H
15644:A
15626:ν
15622:μ
15605:.
15596:)
15590:(
15584:d
15578:2
15570:)
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15550:2
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15536:|
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15510:R
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15491:n
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15469:}
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15416:R
15399::
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15380:R
15376::
15369:{
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15350:H
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15330:2
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15301:(
15299:f
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15281:n
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15243:–
15236:B
15232:A
15209:.
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15201:)
15198:M
15195:(
15190:2
15186:L
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15170:r
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15152::
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15138:2
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15127:u
15123:{
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15110:P
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15077:L
15066:P
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15037:)
15019:a
15012:(
15002:D
14988:=
14979:m
14976:r
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14970:f
14962:P
14949:P
14939:P
14922:.
14919:)
14916:M
14913:(
14903:0
14899:C
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14876:0
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14831:x
14828:(
14824:]
14811:D
14806:[
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14787:a
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14770:)
14767:x
14764:(
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14730:F
14726:P
14724:*
14722:F
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14716:(
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14710:R
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14680:R
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14637:c
14623:=
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14614:D
14611:(
14608:P
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14567:n
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14535:2
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14468:|
14463:i
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14445:D
14418:.
14411:n
14401:n
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14350:x
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14310:)
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14215:x
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14118:}
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14105:=
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14089::
14086:)
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14080:(
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14064:u
14060:{
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13987:=
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13981:1
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13965:D
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13925:0
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13784:u
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13775:=
13764:u
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13744:=
13733:u
13729:i
13705:−
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13695:N
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12726:.
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12701:2
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12301:)
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9768:=
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9724:]
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9627:]
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7368:=
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