166:
253:
67:
720:
768:"Generalized Trotter's formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems"
331:
309:
808:
555:
518:
500:
202:
537:
705:
330:
into alternating increments of kinetic and potential operators (the Suzuki–Trotter decomposition, after
Trotter and
366:
676:
Kato, Tosio (1978), "Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups",
312:, in that both are replacements, in the context of noncommuting operators, for the classical exponential law.
574:
564:
569:
316:
678:
Topics in functional analysis (essays dedicated to M. G. KreÄn on the occasion of his 70th birthday)
320:
343:
827:
297:
to commute for the law to still hold. However, the Lie product formula holds for all matrices
509:
Appelbaum, David (2019). "The
Feynman-Kac Formula via the Lie-Kato-Trotter Product Formula".
339:
335:
511:
Semigroups of Linear
Operators : With Applications to Analysis, Probability and Physics
759:
689:
282:
54:
8:
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286:
173:
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779:
729:
663:
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486:
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646:
630:
613:
767:
755:
685:
474:
473:, Lecture Notes in Mathematics, vol. 423 (1st ed.), Berlin, New York:
681:
51:
648:
583:
821:
743:
715:
350:
36:
550:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
548:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
161:{\displaystyle e^{A+B}=\lim _{n\rightarrow \infty }(e^{A/n}e^{B/n})^{n},}
47:
20:
783:
751:
639:
491:
482:
323:
28:
582:
Cohen, Joel E.; Friedland, Shmuel; Kato, Tosio; Kelly, F. P. (1982).
734:
647:
Joel E. Cohen; Shmuel
Friedland; Tosio Kato; F. P. Kelly (1982),
349:
The
Trotter–Kato theorem can be used for approximation of linear
342:. Moreover, the Lie product theorem is sufficient to prove the
471:
Mathematical Theory of
Feynman Path Integrals: An Introduction
649:"Eigenvalue inequalities for products of matrix exponentials"
584:"Eigenvalue inequalities for products of matrix exponentials"
199:
This formula is an analogue of the classical exponential law
532:, Graduate Texts in Mathematics, vol. 267, Springer,
680:, Adv. in Math. Suppl. Stud., vol. 3, Boston, MA:
581:
385:
308:
The Lie product formula is conceptually related to the
718:(1959), "On the product of semi-groups of operators",
469:
Albeverio, Sergio A.; Høegh-Krohn, Raphael J. (1976),
205:
70:
614:"The Trotter-Kato Theorem and Approximation of PDEs"
468:
319:
of quantum mechanics. It allows one to separate the
801:
Lie Groups, Lie
Algebras, and Their Representations
700:(in German). New York: American Mathematical Soc.
315:The formula has applications, for example, in the
247:
188:extend this to certain unbounded linear operators
160:
819:
721:Proceedings of the American Mathematical Society
513:. Cambridge University Press. pp. 123–125.
334:). The same idea is used in the construction of
91:
257:which holds for all real or complex numbers
798:
733:
667:
629:
611:
602:
508:
490:
453:
442:
695:
714:
408:
820:
765:
696:Lie, Sophus; Engel, Friedrich (1970).
612:Ito, Kazufumi; Kappel, Franz (1998).
675:
545:
527:
430:
419:
396:
656:Linear Algebra and Its Applications
591:Linear Algebra and Its Applications
379:
31:(1875), but also widely called the
13:
698:Theorie der Transformationsgruppen
305:, even ones which do not commute.
248:{\displaystyle e^{x+y}=e^{x}e^{y}}
101:
14:
844:
530:Quantum Theory for Mathematicians
310:Baker–Campbell–Hausdorff formula
447:
436:
424:
413:
402:
390:
367:Time-evolving block decimation
338:for the numerical solution of
289:, it is usually necessary for
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106:
98:
16:Formula of matrix exponentials
1:
631:10.1090/S0025-5718-98-00915-6
462:
669:10.1016/0024-3795(82)90211-7
604:10.1016/0024-3795(82)90211-7
39:, states that for arbitrary
7:
570:Encyclopedia of Mathematics
360:
273:are replaced with matrices
182:Lie–Trotter product formula
10:
849:
799:Varadarajan, V.S. (1984),
618:Mathematics of Computation
565:"Trotter product formula"
317:path integral formulation
372:
546:Hall, Brian C. (2015),
528:Hall, Brian C. (2013),
33:Trotter product formula
766:Suzuki, Masuo (1976).
340:differential equations
249:
162:
454:Ito & Kappel 1998
321:Schrödinger evolution
250:
163:
684:, pp. 185–195,
203:
186:Trotter–Kato theorem
68:
803:, Springer-Verlag,
344:Feynman–Kac formula
25:Lie product formula
784:10.1007/bf01609348
483:10.1007/BFb0079827
287:matrix exponential
245:
174:matrix exponential
158:
105:
810:978-0-387-90969-1
557:978-0-387-40122-5
520:978-1-108-71637-6
502:978-3-540-07785-5
386:Cohen et al. 1982
336:splitting methods
90:
840:
813:
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772:Comm. Math. Phys
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846:
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814:, pp. 99.
809:
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778:(2): 183–190.
763:
728:(4): 545–551,
716:Trotter, H. F.
712:
706:
693:
682:Academic Press
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624:(221): 21–44.
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539:978-1461471158
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35:, named after
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9:
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2:
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828:Matrix theory
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433:Theorem 20.1
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332:Masuo Suzuki
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172:denotes the
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44:
40:
37:Hale Trotter
32:
27:, named for
24:
18:
492:10852/44049
355:-semigroups
283:exponential
21:mathematics
833:Lie groups
822:Categories
463:References
326:propagator
324:operator (
281:, and the
29:Sophus Lie
792:121900332
744:0002-9939
662:: 55–95,
597:: 55–95.
575:EMS Press
431:Hall 2013
420:Kato 1978
397:Hall 2015
102:∞
99:→
361:See also
184:and the
55:matrices
760:0108732
752:2033649
690:0538020
640:2584971
577:, 2001
180:. The
52:complex
43:×
807:
790:
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704:
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554:
536:
517:
499:
168:where
23:, the
788:S2CID
748:JSTOR
652:(PDF)
636:JSTOR
587:(PDF)
373:Notes
265:. If
805:ISBN
740:ISSN
702:ISBN
552:ISBN
534:ISBN
515:ISBN
497:ISBN
301:and
293:and
277:and
269:and
261:and
192:and
60:and
48:real
780:doi
730:doi
664:doi
626:doi
599:doi
487:hdl
479:doi
176:of
92:lim
50:or
19:In
824::
786:.
776:51
774:.
770:.
756:MR
754:,
746:,
738:,
726:10
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660:45
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593:.
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495:,
485:,
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357:.
346:.
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732::
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628::
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601::
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489::
481::
353:0
351:C
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303:B
299:A
295:B
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279:B
275:A
271:y
267:x
263:y
259:x
241:y
237:e
231:x
227:e
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212:x
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194:B
190:A
178:A
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151:n
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133:B
129:e
123:n
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115:A
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80:+
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45:m
41:m
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