Mathematical analysis and numerical methods for science and technology: 3 Spectral theory and applications
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English French |
| Published: |
Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London
Springer-Verlag
2000
|
| Edition: | [Nachdr.] |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 541 S. |
| ISBN: | 3540660992 |
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MARC
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| 100 | 1 | |a Dautray, Robert |d 1928- |0 (DE-588)133309347 |4 aut | |
| 240 | 1 | 0 | |a Analyse mathématique et calcul numérique pour les sciences et les techniques |
| 245 | 1 | 0 | |a Mathematical analysis and numerical methods for science and technology |n 3 |p Spectral theory and applications |c Robert Dautray ; Jacques-Louis Lions |
| 250 | |a [Nachdr.] | ||
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London |b Springer-Verlag |c 2000 | |
| 300 | |a X, 541 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 700 | 1 | |a Lions, Jacques-Louis |d 1928-2001 |0 (DE-588)124055397 |4 aut | |
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| _version_ | 1804127728416325633 |
|---|---|
| adam_text | Table
of Contents
Chapter
VIII.
Spectral Theory
Introduction
.......................... 1
§ 1.
Elements of Spectral Theory in a Banach Space. Dunford Integral
and Functional Calculus
................... 2
1.
Résolvant
Set and
Résolvant
Operator. Spectrum of A
.... 2
2.
Résolvant
Equation and Spectral Radius
.......... 7
3.
Dunford Integral and Operational Calculus
......... 10
4.
Isolated Singularities of the
Résolvant
............ 13
§2.
Spectral Decomposition of Self-Adjoint and Compact Normal
Operators in a Separable Hubert Space and Applications
..... 16
1.
Hubert Sums
...................... 16
2.
Spectral Decomposition of a Compact Self-Adjoint Operator
. . 20
3.
Spectral Decomposition of a Compact Normal Operator
.... 26
4.
Solution of the Equations
А и
=ƒ.
Fredholm
Alternative
.... 29
5.
Examples of Applications
................. 30
6.
Spectral Decomposition of an Unbounded Self-Adjoint Operator
with Compact Inverse
................... 38
7.
Sturm-Liouville Problems and Applications
......... 40
8.
Application to the Spectrum of the Laplacian in
Ω
c R
.... 82
9.
Determining the Eigenvalues of a Self-Adjoint Operator with
Compact Inverse.
Min-Max
and Courant-Fisher Formulas
... 98
§ 3.
Spectral Decomposition of a Bounded or Unbounded Self-Adjoint
Operator
.........................
Ill
Introduction
........................
Ill
1.
Spectral Family and Resolution of the Identity. Properties
... 113
2.
Spectral Family Associated with a Self-Adjoint Operator;
Spectral Theorem
.................... 120
3.
Properties of the Spectrum of a Self-Adjoint Operator.
Multiplicity. Examples
.................. 128
4.
Functions of a Self-Adjoint Operator
............ 138
5.
Operators which Commute with A and Functions of A
.... 145
6.
Fractional Powers of a Strictly Positive Self-Adjoint Operator
. 148
Table of Contents
§4.
Hubert Sum and Hubert Integral Associated with the Spectral
Decomposition of a Self-Adjoint Operator A in a Separable Hubert
Space
Я*
......................... 154
1.
Canonical Representation Associated with a Self-Adjoint
Operator Whose Spectrum is Simple
............ 154
2.
Hubert Sum Associated with the Spectral Decomposition of a
Self-Adjoint Operator
Л
in a Separable (and Complex) Hubert
Space
Я
........................ 159
3.
Hubert Integral. Diagonalisation Theorem of J.
von
Neumann
and J. Dixmier
......................
166
4.
An Application: The Intermediate Derivative and Trace
Theorems
........·................ 171
5.
Generalised Eigenvectors
................. 175
Appendix. Krein-Rutman Theorem *
.............. 187
Chapter IX. Examples in
Electromagnetism
and Quantum Physics
Introduction
.......................... 200
Part A. Examples in
Electromagnetism
............... 201
§ 1.
Basic Tools: Gradient, Divergence and Curl Operators
...... 201
1.
Introduction. Definitions (Gradient, Divergence, Curl)
.... 201
2.
The Spaces H(di ,
Ω)
and
Я(сиг1,
Ω).
Principal Properties
... 203
3.
Kernel and Image of the Gradient, Divergence and Curl
Operators. Introduction
.................. 213
4.
Some Results on Regularity
................ 234
§2.
Static
Electromagnetism
................... 239
1.
Magnetostatics of a Surface Current
............ 239
2.
Electrostatics of a Surface Charge
............. 251
Review of
§2........................ 262
§3.
The Spectral Problem in a Bounded Open Domain (Cavity) with
Perfect Conductor Boundary Conditions
............ 264
1.
Definition and Fundamental Properties of the Maxwell Operator
si in an Open Domain
fícR3
with Bounded Boundary
Γ=3Ω
265
2.
Spectral Properties of
si in a
Bounded Open Domain (Cavity)
. 268
Review of
§3........................ 270
§4.
Spectral Problems in a Wave Guide (Cylinder)
......... 271
1.
Introduction
....................... 271
2.
The Maxwell Operator si in a Cylinder. Definition of
D
(si) and
the Trace Theorem
.................... 279
3.
Study of the Kernel of the Operator si in the Space
ЈЃ
. . . . 283
Table
of Contents IX
4.
Spectral Decomposition of the Maxwell Operator
jé
in the Case
of a Cylinder ( Wave Guide )
Ω=ΩΓ
x R
with
Ωτ
a Connected
and Regular, Bounded Open Domain in R2
......... 285
Spaces Utilised
......................... 310
Part B. Examples in Quantum Physics
............... 315
introduction on the
Observables
of Quantum Physics
........ 315
§ 1.
Operators Corresponding to the Position, Momentum and Angular
Momentum
Observables
................... 316
1.
System Consisting of a Single
Non Relativistic
Particle
Without Spin, Located in the Space R3
........... 316
2.
System Consisting of a Single
Non
Relativistic Particle with
Spin
( Λ)
in R3
..................... 342
3.
System of a Single Particle Located in a Bounded Domain
ßcR3......................... 351
4.
System of
N
Distinguishable
non
Relativistic Particles in R3
. . 353
5.
System of
N
Indistinguishable
non
Relativistic Particles in R3
. 354
6.
System of a Single Free Relativistic Particle. Case of a Particle
with Spin V% Satisfying the Dirac Equation
......... 359
7.
Other Cases of Relativistic Particles
............. 376
§2.
Hamiltonian Operators in Quantum Physics
.......... 382
1.
Definition of Hamiltonian Operators as Self-Adjoint Operators
. 382
2.
Hamiltonian Operators and Essentially Self-Adjoint Operators
. 402
3.
Unbounded Below Hamiltonian Operators
.......... 418
4.
(Discrete) Point Spectrum, and Essential Spectrum of
(Hamiltonian) Self-Adjoint Operators
............ 428
5.
Continuous Spectrum of (Hamiltonian) Self-Adjoint Operators
. 451
Appendix. Some Spectral Notions
1.
General Definitions. Spectrum of a Commutative C*-Algebra
and Gelfand Transformation
................ 457
2.
Continuous Operational Calculus for a (Bounded) Normal
Operator
........................ 461
3.
Continuous Operational Calculus for an (Unbounded)
Self-Adjoint Operator
................... 463
4.
Simultaneous Spectrum of a Commutative Family of (Bounded)
Normal Operators
.................... 465
5.
Continuous Operational Calculus for a Finite Commutative
Family of Normal Bounded Operators in
Ж
......... 466
6.
Simultaneous Spectrum of a Finite Commutative Family of
(Unbounded) Self-Adjoint Operators in
Jť;
Continuous
Operational Calculus
.................. 467
X Table of Contents
7.
Spectral Measure and Basic Measure of a Commutative
C-Algebra
....................... 469
8. von
Neumann Algebras
.................. 471
9.
Bounded Operational Calculus
.............. 473
10.
Maximal Commutative
von
Neumann Algebras
....... 476
11.
Maximal Spectral Decomposition. Complete Family of
Observables
which Commute
............... 476
Bibliography
.......................... 484
Table of Notations
....................... 490
Index
............................. 504
Contents of Volumes
1, 2, 4-6.................. 537
|
| any_adam_object | 1 |
| author | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 |
| author_GND | (DE-588)133309347 (DE-588)124055397 |
| author_facet | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 |
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| author_sort | Dautray, Robert 1928- |
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| building | Verbundindex |
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| edition | [Nachdr.] |
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| id | DE-604.BV013031484 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:37:58Z |
| institution | BVB |
| isbn | 3540660992 |
| language | English French |
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| physical | X, 541 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer-Verlag |
| record_format | marc |
| spelling | Dautray, Robert 1928- (DE-588)133309347 aut Analyse mathématique et calcul numérique pour les sciences et les techniques Mathematical analysis and numerical methods for science and technology 3 Spectral theory and applications Robert Dautray ; Jacques-Louis Lions [Nachdr.] Berlin ; Heidelberg ; New York ; Paris ; Barcelona ; Hong Kong ; London Springer-Verlag 2000 X, 541 S. txt rdacontent n rdamedia nc rdacarrier Lions, Jacques-Louis 1928-2001 (DE-588)124055397 aut (DE-604)BV013031479 3 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878227&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dautray, Robert 1928- Lions, Jacques-Louis 1928-2001 Mathematical analysis and numerical methods for science and technology |
| title | Mathematical analysis and numerical methods for science and technology |
| title_alt | Analyse mathématique et calcul numérique pour les sciences et les techniques |
| title_auth | Mathematical analysis and numerical methods for science and technology |
| title_exact_search | Mathematical analysis and numerical methods for science and technology |
| title_full | Mathematical analysis and numerical methods for science and technology 3 Spectral theory and applications Robert Dautray ; Jacques-Louis Lions |
| title_fullStr | Mathematical analysis and numerical methods for science and technology 3 Spectral theory and applications Robert Dautray ; Jacques-Louis Lions |
| title_full_unstemmed | Mathematical analysis and numerical methods for science and technology 3 Spectral theory and applications Robert Dautray ; Jacques-Louis Lions |
| title_short | Mathematical analysis and numerical methods for science and technology |
| title_sort | mathematical analysis and numerical methods for science and technology spectral theory and applications |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008878227&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013031479 |
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