917:
379:
212:
in quantum mechanics correspond to self-adjoint operators on some
Hilbert space, but some observables (like energy) are unbounded. By Hellinger–Toeplitz, such operators cannot be everywhere defined (but they may be defined on a
149:
74:
247:
806:
469:
205:
197:. One relies on the symmetric assumption, therefore the inner product structure, in proving the theorem. Also crucial is the fact that the given operator
632:
941:
759:
614:
590:
438:
482:
571:
462:
174:, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem is named after
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374:{\displaystyle (x)=-{\frac {1}{2}}{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}f(x)+{\frac {1}{2}}x^{2}f(x).}
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Mathematical
Methods in Quantum Mechanics; With Applications to Schrödinger Operators
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40:
36:
882:
535:
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214:
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24:
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The
Hellinger–Toeplitz theorem reveals certain technical difficulties in the
201:
is defined everywhere (and, in turn, the completeness of
Hilbert spaces).
222:
237:
is defined by (assuming the units are chosen such that ℏ =
388:
are 1/2, 3/2, 5/2, ...), so it cannot be defined on the whole of L(
451:
410:
Methods of
Mathematical Physics, Volume 1: Functional Analysis.
185:
This theorem can be viewed as an immediate corollary of the
144:{\displaystyle \langle Ax|y\rangle =\langle x|Ay\rangle }
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48:
870:
794:
773:
732:
671:
613:
559:
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807:Spectral theory of ordinary differential equations
373:
143:
68:
384:This operator is self-adjoint and unbounded (its
229:), the space of square integrable functions on
463:
206:mathematical formulation of quantum mechanics
69:{\displaystyle \langle \cdot |\cdot \rangle }
16:Theorem on boundedness of symmetric operators
8:
193:. Alternatively, it can be argued using the
138:
121:
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63:
49:
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412:Academic Press, 1980. See Section III.5.
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96:
55:
47:
760:Group algebra of a locally compact group
7:
304:
292:
31:states that an everywhere-defined
14:
916:
915:
842:Topological quantum field theory
189:, as self-adjoint operators are
942:Theorems in functional analysis
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359:
330:
324:
269:
263:
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251:
241: = ω = 1)
128:
108:
56:
1:
638:Uniform boundedness principle
431:American Mathematical Society
195:uniform boundedness principle
80:. By definition, an operator
221:. Here the Hilbert space is
219:quantum harmonic oscillator
963:
781:Invariant subspace problem
233:, and the energy operator
170:operators are necessarily
29:Hellinger–Toeplitz theorem
911:
501:
217:). Take for instance the
750:Spectrum of a C*-algebra
847:Noncommutative geometry
166:. Note that symmetric
903:Tomita–Takesaki theory
878:Approximation property
822:Calculus of variations
375:
145:
70:
898:Banach–Mazur distance
861:Generalized functions
376:
176:Ernst David Hellinger
146:
71:
643:Kakutani fixed-point
628:Riesz representation
248:
187:closed graph theorem
95:
46:
827:Functional calculus
786:Mahler's conjecture
765:Von Neumann algebra
479:Functional analysis
21:functional analysis
852:Riemann hypothesis
551:Topological vector
371:
168:everywhere-defined
141:
66:
33:symmetric operator
929:
928:
832:Integral operator
609:
608:
440:978-0-8218-4660-5
344:
319:
286:
162:in the domain of
954:
919:
918:
837:Jones polynomial
755:Operator algebra
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871:Advanced topics
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790:
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694:Hilbert–Schmidt
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658:Gelfand–Naimark
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947:Hilbert spaces
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888:Choquet theory
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740:Banach algebra
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663:Banach–Alaoglu
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591:Locally convex
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417:Teschl, Gerald
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23:, a branch of
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3:
2:
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893:Weak topology
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817:Index theorem
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774:Open problems
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402:Reed, Michael
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180:Otto Toeplitz
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52:
42:
41:inner product
38:
37:Hilbert space
34:
30:
26:
22:
883:Balanced set
857:Distribution
795:Applications
648:Krein–Milman
633:Closed graph
421:
409:
406:Simon, Barry
389:
383:
238:
234:
230:
226:
215:dense subset
203:
198:
184:
172:self-adjoint
167:
163:
159:
155:
153:
85:
81:
28:
18:
812:Heat kernel
802:Hardy space
709:Trace class
623:Hahn–Banach
585:Topological
386:eigenvalues
210:Observables
25:mathematics
936:Categories
745:C*-algebra
560:Properties
427:Providence
396:References
719:Unbounded
714:Transpose
672:Operators
601:Separable
596:Reflexive
581:Algebraic
567:Barrelled
276:−
139:⟩
122:⟨
116:⟩
99:⟨
86:symmetric
64:⟩
61:⋅
53:⋅
50:⟨
921:Category
733:Algebras
615:Theorems
572:Complete
541:Schwartz
487:glossary
419:(2009).
154:for all
724:Unitary
704:Nuclear
689:Compact
684:Bounded
679:Adjoint
653:Min–max
546:Sobolev
531:Nuclear
521:Hilbert
516:Fréchet
481: (
78:bounded
699:Normal
536:Orlicz
526:Hölder
506:Banach
495:Spaces
483:topics
437:
191:closed
27:, the
511:Besov
39:with
35:on a
859:(or
577:Dual
435:ISBN
404:and
178:and
88:if
392:).
182:.
84:is
76:is
19:In
938::
485:–
433:.
429::
425:.
408::
208:.
158:,
863:)
587:)
583:/
579:(
489:)
471:e
464:t
457:v
443:.
390:R
369:.
366:)
363:x
360:(
357:f
352:2
348:x
342:2
339:1
334:+
331:)
328:x
325:(
322:f
314:2
310:x
305:d
298:2
293:d
284:2
281:1
273:=
270:)
267:x
264:(
261:]
258:f
255:H
252:[
239:m
235:H
231:R
227:R
225:(
223:L
199:A
164:A
160:y
156:x
136:y
133:A
129:|
125:x
119:=
113:y
109:|
105:x
102:A
82:A
57:|
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