Methods of Inverse Problems in Physics

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CRC Press, Mar 14, 1991 - Science - 504 pages
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This interesting volume focuses on the second of the two broad categories into which problems of physical sciences fall-direct (or forward) and inverse (or backward) problems. It emphasizes one-dimensional problems because of their mathematical clarity. The unique feature of the monograph is its rigorous presentation of inverse problems (from quantum scattering to vibrational systems), transmission lines, and imaging sciences in a single volume. It includes exhaustive discussions on spectral function, inverse scattering integral equations of Gel'fand-Levitan and Marcenko, Povzner-Levitan and Levin transforms, Møller wave operators and Krein's functionals, S-matrix and scattering data, and inverse scattering transform for solving nonlinear evolution equations via inverse solving of a linear, isospectral Schrodinger equation and multisoliton solutions of the K-dV equation, which are of special interest to quantum physicists and mathematicians. The book also gives an exhaustive account of inverse problems in discrete systems, including inverting a Jacobi and a Toeplitz matrix, which can be applied to geophysics, electrical engineering, applied mechanics, and mathematics. A rigorous inverse problem for a continuous transmission line developed by Brown and Wilcox is included. The book concludes with inverse problems in integral geometry, specifically Radon's transform and its inversion, which is of particular interest to imaging scientists. This fascinating volume will interest anyone involved with quantum scattering, theoretical physics, linear and nonlinear optics, geosciences, mechanical, biomedical, and electrical engineering, and imaging research.
 

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Contents

Chapter
1
Chapter
3
Geophysics
16
Chapter 2
37
References
73
Completeness Relation
93
The PovznerLevitan Transform
101
The Generalized Displacement Operator
102
The Spectral Function p and the Inequality 6 46
177
References
179
Chapter 7
181
Integral Representations of the Jost Solutions
182
HI Regularity Properties of the Jost Functions
185
The Jost Solutions and the Jost Functions on R+
203
References
214
The Levin Transform B+Ax y
220

IH The BoundaryValue Problem for Tx y
104
The Solution of the BoundaryValue Problem
106
Entire Real Axis h 0
109
PovznerLevitan Transformation on the Semiaxis
111
Two Theorems from Function Theory2
113
WienerBoas Theorem and PovznerLevitan Transform
115
Derivation Due to Marcenko?
119
PovznerLevitan Transform and the Spectral Function
123
Riemanns Solution of Cauchys Problem
128
References
133
Chapter 5
135
Uniqueness of Solutions
138
Continuity Properties of the PovznerLevitan Transforms
140
Relation Between Mx x and the Potential qx
142
Determination of the BoundaryValue Problem
144
A Brief Summary
149
References
153
Chapter 6
155
Theory of Propagation of Discontinuities
158
Causal Impulse Response
161
The Noncausal Impulse Function
164
Direct and Inverse x C R
166
The Riemann Function
167
Derivation of the Linear GelfandLevitan Equation
170
Relation Between ft T and the Spectral Function o
172
Uniqueness of the Solution
173
Equations Governing Levin Transforms
227
The Goursat Problem for the Levin Transforms
236
Chapter 9
243
Analyticity and the Bound States
253
The Scattering Matrix S
262
Analyticity of Tk and R+k
269
Fourier Transform of RK and TK
275
The Spectral Representation
284
Derivation of the Scattering Operator S
294
A List of Some Important Nonlinear Evolution Equations Solvable
308
Marcenkos Equation on the Entire Line
314
Derivation of the Marcenko Equation
321
Some Properties of Q
334
Estimates of Kxx and Its First Derivatives
340
Appendix A
361
Chapter I
367
Inverse Problems for Nonuniform Lossless Transmission Lines
388
Appendix A
419
Appendix C
425
References
431
Theorems and Properties
437
The Inverse Radon Transform
446
The Adjoint Operator and Parsevals Theorem for Radon Transform
465
Application to Partial Differential Equations
471
Index
477
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