Applied analysis by the Hilbert space method: an introduction with applications to the wave, heat, and Schrödinger equations
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Dekker
1990
|
| Ausgabe: | 1. print. |
| Schriftenreihe: | Pure and applied mathematics
137. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 560 S. graph. Darst. |
| ISBN: | 0824782593 |
Internformat
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| 100 | 1 | |a Holland, Samuel S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Applied analysis by the Hilbert space method |b an introduction with applications to the wave, heat, and Schrödinger equations |c Samuel S. Holland, Jr. |
| 250 | |a 1. print. | ||
| 264 | 1 | |a New York u.a. |b Dekker |c 1990 | |
| 300 | |a XII, 560 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Pure and applied mathematics |v 137. | |
| 650 | 7 | |a Espace Hilbert |2 ram | |
| 650 | 7 | |a Équations différentielles |2 ram | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Differential operators | |
| 650 | 4 | |a Hilbert space | |
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| 830 | 0 | |a Pure and applied mathematics |v 137. |w (DE-604)BV000001885 |9 137 | |
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Datensatz im Suchindex
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| adam_text | Contents
Preface v
A Note on Method vii
Chapter 1. First Order Linear Differential Equations 1
1.1. The Equation a(x)y + b(x)y = h(x) 1
1.2. First Order Linear Differential Expressions; the Kernel 4
1.3. Finding a Particular Solution by Variation of Parameters 9
1.4. Power Series Review 10
1.5. The Initial Value Problem for a First Order Linear Differ¬
ential Equation 18
Chapter 2. Second Order Linear Differential Equations 22
2.1. Basic Concepts of Linear Algebra for Function Spaces 22
2.2. The Initial Value Problem for a Second Order Linear
Homogeneous Differential Equation 28
2.3. Dimension of the Kernel; General Solution; Abel s
Formula 35
2.4. Kernel of Constant Coefficient Expressions 39
2.5. The Classical Linear Oscillator 43
2.6. Guessing a Particular Solution to a Constant Coefficient
Equation 48
2.7. Particular Solution by Variation of Parameters 51
2.8. The Kernel of Legendre s Differential Expression 53
ix
x Contents
2.9. The Kernels of Other Classical Expressions 57
2.10. Dirac s Delta Function and Green s Functions 59
Appendix 2.A. Second Order Linear Differential Equa¬
tions in the Complex Domain 69
Chapter 3. Hilbert Space 78
3.1. The Vibrating Wire 78
3.2. Fourier Series 83
3.3. Fourier Sine and Cosine Series 94
3.4. Fourier Series over Other Intervals 99
3.5. The Vibrating Wire, Revisited 102
3.6. The Inner Product 108
3.7. Schwarz s Inequality 118
3.8. The Mean Square Metric; Orthogonal Bases 127
3.9. L2 Spaces 138
3.10. Hilbert Space 149
Chapter 4. Linear Second Order Differential Operators in L2 Spaces
and Their Eigenvalues and Eigenfunctions 157
4.1. Compatibility 157
4.2. Eigenvalues and Eigenfunctions 168
4.3. Hermitian Operators 175
4.4. Some General Operator Theory 184
4.5. The One Dimensional Laplacian 191
4.6. Legendre s Operator and Its Eigenfunctions, the Legendre
Polynomials 202
4.7. Solving Operator Equations with Legendre s Operator 217
4.8. More on Legendre Polynomials: Rodrigues Formula, the
Recursion Relation, and the Generating Function 224
4.9. Hermite s Operator and Its Eigenfunctions, the Hermite
Polynomials 233
4.10. Solving Operator Equations with Hermite s Operator 243
4.11. More on Hermite Polynomials: Rodrigues Formula, the
Recursion Relation, and the Generating Function 248
Appendix 4.A. Mathematical Aspects of Differential
Operators in L2 Spaces 253
Chapter 5. Schrodinger s Equations in One Dimension 267
5.1. The Wave Equation by the Hilbert Space Method 267
5.2. The Heat Equation by the Hilbert Space Method 273
5.3. Quanta as Eigenvalues—the Time Independent Schrodin
ger Equation in One Dimension 282
Contents xi
5.4. Interpretation of the 4* Function. The Time Dependent
Schrodinger Equation in One Dimension 289
5.5. The Quantum Linear Oscillator 298
5.6. Solution of the Time Dependent Schrodinger Equation 308
5.7. A Brief History of Matrix Mechanics 313
5.8. A General Formulation of Quantum Mechanics: States 316
5.9. A General Formulation of Quantum Mechanics:
Observables 326
Chapter 6. BessePs Operator and Bessel Functions 331
6.1. The Wave Equation and Other Equations in Higher
Dimensions; Polar Coordinates 331
6.2. Bessel s Equation and Bessel s Operator of Order Zero 339
6.3. J0(x): The Bessel Function of the First Kind of Order Zero 345
6.4. J0(x): Calculating Its Values and Finding Its Zeros 350
6.5. The Eigenvalues and Eigenfunctions of Bessel s Operator
of Order Zero 359
6.6. The Vibrating Drumhead, the Heated Disk, and the
Quantum Particle Confined to a Circular Region (the 6
Independent Case) 367
6.7. 0 Dependence: Bessel s Equation and Bessel s Operator of
Integral Order p 375
6.8. Jp(x): The Bessel Functions of the First Kind of Integral
Order p 380
6.9. The Eigenvalues and Eigenfunctions of Bessel s Operator
of Integral Order p 387
6.10. Project on Bessel Functions of Nonintegral Order 392
Appendix 6.A. Mathematical Theory of Bessel s
Operator 393
Chapter 7. Eigenvalues of the Laplacian, with Applications 402
7.1. The Laplacian as a Hilbert Space Operator 402
7.2. Differential Forms, the Stokes Theorem, and Integration
by Parts in Two Variables 410
7.3. The Laplacian Is Hermitian 418
7.4. General Facts About the Eigenvalues of the Laplacian 426
7.5. Eigenvalues of the Rectangle 440
7.6. Eigenvalues of the Disk 450
7.7. The Laplacian on the Sphere 452
7.8. Eigenvalues of the Sphere; Spherical Harmonics 466
7.9. The Hydrogen Atom 481
7.10. Project on Laguerre s Operator and Laguerre Poly¬
nomials 493
xii Contents
7.11. Laplace s Equation and Harmonic Polynomials 500
Appendix 7.A. The Legendre, Laguerre, and Schrodin
ger Operators 507
Chapter 8. The Fourier Transform 514
8.1. Complex Methods in Fourier Series; the Fourier
Transform 514
8.2. Plancherel s Theorem; Examples of Fourier Transforms 520
8.3. Fourier Sine and Cosine Transforms 526
8.4. The Fourier Transform Is a Unitary Operator on
L2( oo,oo). 529
8.5. The Fourier Transform Converts Differentiation into Mul¬
tiplication by the Independent Variable 537
8.6. The Eigenvalues and Eigenfunctions of the Fourier
Transform 544
Appendix 8.A. J«„ ((sin x)/x) dx = n 551
Index of Symbols 553
Index 555
|
| any_adam_object | 1 |
| author | Holland, Samuel S. |
| author_facet | Holland, Samuel S. |
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| author_sort | Holland, Samuel S. |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.35 |
| dewey-search | 515/.35 |
| dewey-sort | 3515 235 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. print. |
| format | Book |
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| id | DE-604.BV008548806 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:20:58Z |
| institution | BVB |
| isbn | 0824782593 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005619314 |
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| owner_facet | DE-824 |
| physical | XII, 560 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Dekker |
| record_format | marc |
| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Holland, Samuel S. Verfasser aut Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations Samuel S. Holland, Jr. 1. print. New York u.a. Dekker 1990 XII, 560 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 137. Espace Hilbert ram Équations différentielles ram Differential equations Differential operators Hilbert space Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Differentialoperator (DE-588)4012251-7 gnd rswk-swf Differentialoperator (DE-588)4012251-7 s Hilbert-Raum (DE-588)4159850-7 s DE-604 Pure and applied mathematics 137. (DE-604)BV000001885 137 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005619314&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Holland, Samuel S. Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations Pure and applied mathematics Espace Hilbert ram Équations différentielles ram Differential equations Differential operators Hilbert space Hilbert-Raum (DE-588)4159850-7 gnd Differentialoperator (DE-588)4012251-7 gnd |
| subject_GND | (DE-588)4159850-7 (DE-588)4012251-7 |
| title | Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations |
| title_auth | Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations |
| title_exact_search | Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations |
| title_full | Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations Samuel S. Holland, Jr. |
| title_fullStr | Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations Samuel S. Holland, Jr. |
| title_full_unstemmed | Applied analysis by the Hilbert space method an introduction with applications to the wave, heat, and Schrödinger equations Samuel S. Holland, Jr. |
| title_short | Applied analysis by the Hilbert space method |
| title_sort | applied analysis by the hilbert space method an introduction with applications to the wave heat and schrodinger equations |
| title_sub | an introduction with applications to the wave, heat, and Schrödinger equations |
| topic | Espace Hilbert ram Équations différentielles ram Differential equations Differential operators Hilbert space Hilbert-Raum (DE-588)4159850-7 gnd Differentialoperator (DE-588)4012251-7 gnd |
| topic_facet | Espace Hilbert Équations différentielles Differential equations Differential operators Hilbert space Hilbert-Raum Differentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005619314&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000001885 |
| work_keys_str_mv | AT hollandsamuels appliedanalysisbythehilbertspacemethodanintroductionwithapplicationstothewaveheatandschrodingerequations |