Methods of inverse problems in physics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton u.a.
CRC Press
1991
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 488 S. graph. Darst. |
| ISBN: | 084936258X |
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MARC
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| 245 | 1 | 0 | |a Methods of inverse problems in physics |c author: Dilip N. Ghosh Roy |
| 264 | 1 | |a Boca Raton u.a. |b CRC Press |c 1991 | |
| 300 | |a 488 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Imaging systems | |
| 650 | 4 | |a Inverse problems (Differential equations) | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 0 | 7 | |a Inverses Problem |0 (DE-588)4125161-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematische Physik |0 (DE-588)4037952-8 |2 gnd |9 rswk-swf |
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Record in the Search Index
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|---|---|
| adam_text | TABLE OF CONTENTS
Chapter 1
Introduction 1
I. Structure of an Inverse Problem I
II. Two Different Settings of Inverse Problems 3
III. Quantum Scattering 7
IV. Evolution Equations and Nonlinear Optics 9
V. Plasma Physics 11
VI. Electron Density Profiles 11
VII. Physical Optics 12
VIII. Integral Geometry 12
IX. Geophysics 16
X. Acoustics 17
XI. Potential Theory 17
XII. Analytical Mechanics 17
XIII. Discrete Systems 19
XIV. Miscellaneous 21
Note 21
References 23
Chapter 2
Some Examples of Inverse Problems 37
I. Introduction 37
II. Examples 37
A. Example 1: Heat Conduction 37
B. Example 2: Determination of the Diffusion Coefficient 42
C. Example 3: Plasma Profile Inversion 46
D. Example 4: Vectorcardiography 49
E. Example 5: Wave Propagation in an Elastic Medium 53
F. Example 6: Integral Geometry 59
G. Example 7: Inverse Scattering Transform 65
References 73
Chapter 3
The Spectral Function 75
I. Introduction 75
II. Spectral Function 75
III. Parseval s Equality on the Half Line R+ 81
IV. Expansion Theorem 83
V. Connection Between Spectral Function and Spectral Family 84
VI. Spectral Functions on the Entire A. Axis 85
A. The Spectrum on the Positive Half Axis: k Real and Positive 87
B. Spectral Function on the Negative Half Axis: k2 z R 91
VII. Completeness Relation 93
VIII. Examples 94
Appendix 1 98
References 100
Chapter 4
The Povzner Levitan Transform 101
I. Introduction 101
II. The Generalized Displacement Operator 102
III. The Boundary Value Problem for T(x,y) 104
IV. The Solution of the Boundary Value Problem 106
A. Povzner Levitan Transform: Entire Real Axis, h = 0 109
B. Povzner Levitan Transform: Entire Real Axis, h = °° 110
C. Povzner Levitan Transformation on the Semiaxis 111
D. Inverse Povzner Levitan Transform: Semiaxis Ill
V. Two Theorems from Function Theory78 113
VI. Wiener Boas Theorem and Povzner Levitan Transform 115
VII. Derivation Due to Marcenko5 119
VIII. Povzner Levitan Transform and the Spectral Function 123
Appendix: Riemann s Solution of Cauchy s Problem 128
References 133
Chapter 5
The Gel fand Levitan Equation 135
I. Derivation of the Gel fand Levitan Equation 135
II. Uniqueness of Solutions 138
III. Continuity Properties of the Povzner Levitan Transforms 140
IV. Behavior of M(x,x ) with x 140
V. Relation Between M(x,x ) and the Potential q(x) 142
VI. Determination of the Boundary Value Problem 144
VII. A Brief Summary 149
VIII. Examples 149
References 153
Chapter 6
The Gel fand Levitan Theory in Time Domain 155
I. Introduction 155
II. Theory of Propagation of Discontinuities 158
III. Causal Impulse Response 161
IV. The Noncausal Impulse Function 164
V. Povzner Levitan Transform: Direct and Inverse, [x c R+] 166
VI. The Riemann Function 167
VII. Derivation of the Linear Gel fand Levitan Equation 170
VIII. Relation Between the Riemann and the Spectral Function 170
IX. Relation Between f(t,T) and the Spectral Function c(X) 172
X. Uniqueness of the Solution 173
XI. The Spectral Function p and the Inequality (6.46) 177
References 179
Chapter 7
Jost Functions and Their Regularity Properties 181
I. Introduction 181
II. Integral Representations of the Jost Solutions 182
III. Regularity Properties of the Jost Functions 185
A. k c C+ {0}: Regularity in x 185
B. k = C+ {0}: Regularity in k 189
C. kc C+: Im k 0, including k = 0. Regularity in x 191
D. Analyticity of the Jost Functions for k c C+ 196
E. Analyticity for Im k 0: Bargmann Strip 200
F. Uniqueness, Linear Independence, and Reality Properties 200
Appendix: The Jost Solutions and the Jost Functions on R+ 203
A.I. The Jost Solutions on R+ 204
A.I.a. Integral Representations of the Jost Solutions 205
A. l.b. Regularity Properties of the Jost Solutions 206
A.2. The Jost Functions and Their Properties 207
A.3. Application of Wiener Levy Theorem to Functions of F(k) 213
References 214
Chapter 8
Fourier Transform of the Jost Functions: The Levin Representation 217
I. Introduction 217
II. Fourier Transforms and Hardy Functions 217
III. The Levin Transform B+A(x,y) 220
A. Levin Transform Via Paley Wiener Theorem 220
B. Derivation of the Levin Transforms and Their Structures 220
IV. Differentiations of Levin Transforms 223
V. Equations Governing Levin Transforms 227
A. Differential Equation for BA(x,y) 227
B. Integral Representation of B±A(x,y) 228
VI. Iterative Solution of the Integral Equation for B+A(x,y) 230
VII. The Goursat Problem for the Levin Transforms 236
Appendix: The Levin Transforms on the Half Line R+ 238
References 240
Chapter 9
Scattering Matrix and Scattering Data 243
I. Introduction 243
II. The Fundamental Solutions 243
III. The Scattering Equations 244
IV. Relations Between the Coefficients t, t, r, and r 246
V. Asymptotic Behaviors 248
A. |x| °°, k * 0 248
B. |k| * 249
C. |x| °o and |k| °° 250
D. Behavior as k 0 252
VI. Analyticity and the Bound States 253
A. The Zeros of t(k) 253
B. The Number of Zeros: The Upper and the Lower Bounds 255
C. Multiplicity of Zeros 259
VII. The Scattering Matrix S 262
VIII. Asymptotics 264
A. Limit k 0 265
IX. Virtual Levels of the Zero Energy Bound States 266
X. Analyticity of T+(k) and R+(k) 269
XI. Determination of S: Hilbert Relations 271
XII. Fourier Transform of R+(K) and T+(K) 275
XIII. Spectral Representation on R 278
A. The Resolvent Operator 279
XIV. The Spectral Representation 284
XV. Krein Functionals and M0ller Operators 288
A. The Krein Functionals 288
B. The M0ller Operators 291
XVI. Derivation of the Scattering Operator S 294
XVII. Orthogonality Relations for the Jost Solutions on R 297
XVIII. Scattering Data and Spectral Transform 298
A. Relation Between the Fourier and the Spectral Transform 299
B. Time Evolution of the Scattering Data 305
XIX. A List of Some Important Nonlinear Evolution Equations Solvable by
Inverse Scattering Transform 308
References 311
Chapter 10
The Marcenko Integral Equation 313
I. Introduction 313
II. Marcenko s Equation on the Entire Line 314
A. The Fourier Transform Derivation 314
B. Derivation Using the Completeness Relation 318
1. The Generalized Levin Representations 318
2. Derivation of the Marcenko Equation 321
III. Derivation in the Time Domain 324
A. Method of Characteristic Fields 324
B. Fourier Transform Derivation 329
IV. Some Properties of Q 334
V. Estimates of K(x,x ) and Its First Derivatives 340
VI. The Uniqueness and the Existence of Solution 343
VII. Illustrations 345
Appendix A 362
References 365
Chapter 11
Discrete Systems and Transmission Lines 367
I. Introduction 367
II. Jacobi Matrices and Their Inversions 368
A. Jacobi Matrices in Inverse Problems in Vibration 368
B. Inversion of a Jacobi Matrix 372
1. Proof of Inverse Problem 1 (Hochstadt3) 376
2. Proof of Inverse Problem 1 (Gray and Wilson4) 378
3. Construction Due to de Boor and Golub6 381
4. Inversion of a Persymmetric Jacobi Matrix (Hochstadt3) 383
5. Proof of the Third Inverse Problem6 386
6. Well Posedness of the Inverse Jacobi Problem 386
III. Inverse Problems for Nonuniform, Lossless Transmission Lines 388
A. Inverse Solution of Discrete Lines:
Bruckstein Kailath Levy Formalism 388
1. The Basic Transmission Line 388
2. Inverse Solution of the Transmission Line Problem 393
3. Linear Matrix Equations of Discrete Inverse Scattering 396
4. Special Forms of Linear Matrix Equations 400
5. Connection with Continuous Inverse Scattering 403
6. Inverse Solutions 405
B. Brown Wilcox Theory of Inverse Problem of
Nonuniform Transmission Lines 406
1. Eigenvalue Problem for the Transmission Line
and Formulation of the Inverse Problem 406
2. The Operator T for the Transmission Line 408
3. Expansion Theorem 410
4. The Transformation Operator 411
5. The Functional Form of the Transformation Operator 413
6. The Differential Equation for the Inverse Solution 414
7. Integral Equations for the Kernels 416
8. Properties of R(ffl) and W(co) 418
Appendix A 419
A. Some Background Materials on Matrices:2 36
Definitions and Theorems 419
B. Jacobi Matrices 420
Appendix B 421
A. The Monic Polynomials 421
B. The Weight Functions w. 423
Appendix C 425
A. The Z Transform 425
Appendix D 426
A. Introduction: Toeplitz Matrix and Transmission Line Inversion 426
B. Relation between Impulse Response and Toeplitz Matrix 428
References 431
Chapter 12
The Radon Transform 433
I. Introduction 433
II. The Direct Radon Transform: Definition 434
III. Connections with Other Transforms 434
IV. The Radon Transform: Theorems and Properties 437
V. The Inverse Radon Transform 446
A. Derivatives and Spherical Means of Certain Generalized Functions 446
B. Derivation of the Inverse Radon Transform: John32 452
C. Derivation Due to Gel fand et al.:31 Method of
Plane Wave Expansion of Delta Functions 455
1. Preliminaries 456
2. The Derivation 460
D. Ludwig s Derivation:34 Fourier Technique 462
VI. The Adjoint Operator and Parseval s Theorem for Radon Transform 465
VII. Ill Posedness and Regularization 467
VIII. Radon Transform of Generalized Functions 469
IX. Application to Partial Differential Equations 471
X. Applications 472
References 474
Index 477
|
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| indexdate | 2024-07-09T16:12:31Z |
| institution | BVB |
| isbn | 084936258X |
| language | English |
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| physical | 488 S. graph. Darst. |
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| spelling | Ghosh Roy, Dilip N. 1939- Verfasser (DE-588)124293301 aut Methods of inverse problems in physics author: Dilip N. Ghosh Roy Boca Raton u.a. CRC Press 1991 488 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik Imaging systems Inverse problems (Differential equations) Mathematical physics Inverses Problem (DE-588)4125161-1 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Inverses Problem (DE-588)4125161-1 s Mathematische Physik (DE-588)4037952-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002730579&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ghosh Roy, Dilip N. 1939- Methods of inverse problems in physics Mathematische Physik Imaging systems Inverse problems (Differential equations) Mathematical physics Inverses Problem (DE-588)4125161-1 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4125161-1 (DE-588)4037952-8 |
| title | Methods of inverse problems in physics |
| title_auth | Methods of inverse problems in physics |
| title_exact_search | Methods of inverse problems in physics |
| title_full | Methods of inverse problems in physics author: Dilip N. Ghosh Roy |
| title_fullStr | Methods of inverse problems in physics author: Dilip N. Ghosh Roy |
| title_full_unstemmed | Methods of inverse problems in physics author: Dilip N. Ghosh Roy |
| title_short | Methods of inverse problems in physics |
| title_sort | methods of inverse problems in physics |
| topic | Mathematische Physik Imaging systems Inverse problems (Differential equations) Mathematical physics Inverses Problem (DE-588)4125161-1 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematische Physik Imaging systems Inverse problems (Differential equations) Mathematical physics Inverses Problem |
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