## Methods of Inverse Problems in PhysicsThis 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 |

477 | |

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