An Introduction to Inverse Problems in Physics

Contents
Preface
Introduction
	References
1 Inverse Problems in Classical Dynamics
	1.1 Inverse Problem for Trajectory
	1.2 Determination of the Shape of the Potential Energy from the Period of Oscillation
	1.3 Action Equivalent Hamiltonians
	1.4 Abel’s Original Inverse Problem
	1.5 Inverse Scattering Problem in Classical Mechanics
	1.6 Inverse Problem of a Linear Chain of Masses Coupled to Springs
	1.7 Direct Problem of Non-exponential and of Exponential Decays in a Linear Chain
	1.8 Inverse Problem of Dynamics for a Non-uniform Chain
	1.9 Direct and Inverse Problems of Analytical Dynamics
	1.10 From the Classical Equations of Motion to the Lagrangian and Hamiltonian Formulations
	1.11 Langevin and Fokker–Planck Equations
	References
2 Inverse Problems in Semiclassical Formulation of Quantum Mechanics
	2.1 Quantum Mechanical Bound States for Confining Potentials
	2.2 Semiclassical Formulation of the Inverse Scattering Problem
	References
3 Inverse Problems and the Heisenberg Equations of Motion
	3.1 Equations of Motion Derived from the Hamiltonian Operator
	3.2 Determination of the Commutation Relations From the Equations of Motion
	3.3 Construction of the Hamiltonian Operator as an Inverse Problem
	References
4 Inverse Scattering Problem for the Schrodinger Equation and the Gel’fand–Levitan Formulation
	4.1 The Jost Solution
	4.2 The Jost Function
	4.3 The Levinson Theorem
	4.4 The Gel’fand–Levitan Equation
	4.5 Inverse Problem for One-dimensionalSchrodinger Equation
	4.6 Bargmann Potentials
	4.7 The Jost and Kohn Method of Inversion
	References
5 Marchenko’s Formulation of the Inverse Scattering Problem
	5.1 Mathematical Preliminaries
	5.2 Bound States Embedded in Continuum
	5.3 More Solvable Potentials Found from Inverse Scattering
	5.4 The Inverse Problem for Reflection and Transmission from a Barrier
	5.5 A Special Problem in Electromagnetic Inverse Scattering
	5.6 Construction of Reflectionless Potentials
	5.7 Symmetric Reflectionless Potentials Supporting a Given Set of Bound States
	References
6 Newton–Sabatier Approach to the Inverse Problem at Fixed Energy
	6.1 Construction of the Potential at Fixed Energy
	6.2 Criticism of the Newton–Sabatier Method of Inversion at a Fixed Energy
	6.3 On the Results of the Numerical Solution of Inverse Problems
	6.4 Modified Form of the Gel’fand–Levitan for Fixed Energy Problems and the Langer Transform
	6.5 Lipperheide and Fiedeldey Approach to the Inverse Problem at Fixed Energy
	6.6 Completeness of the Set of Jost Solutions f(λ, k, r)
	6.7 Generalized Gel’fand–Levitan Approach to Inversion
	6.8 The Method of Schnizer and Leeb
	6.9 Analysis of Atom-Atom Scattering Using Complex Angular Momentum Formulation
	References
7 Discrete Forms of the Schrodinger Equation and the Inverse Problem
	7.1 Zakhariev’s Method
	7.2 TheMethod of Case and Kac for Discrete Form of Inverse Scattering Problem
	7.3 Discrete Form of the Spectral Density for Solving the Inverse Problem on Semi-axis 0 ≤ r < ∞
	References
8 R Matrix Theory and Inverse Problems
	8.1 Inverse Problem for R Matrix Formulation of Scattering
	8.2 The Finite-difference Analogue of the R Matrix Theory of Scattering
	8.3 Shell-model Hamiltonian in Tri-diagonal Form
	8.4 Continued Fraction Expansion of the R Matrix
	References
9 Solvable Models of Fokker–Planck Equation Obtained Using the Gel’fand–Levitan Method
	9.1 Solution of the Fokker–Planck Equation for Symmetric and Asymmetric Double-Well Potentials
	References
10 The Eikonal Approximation
	10.1 Finding the Impact Parameter Phase Shifts from the Cross Section
	References
11 Inverse Methods Applied to Study Symmetries and Conservation Laws
	11.1 Classical Degeneracy and Its Quantum Counterpart
	11.2 Inverse Problem for Angular Momentum Eigenvalues
	11.3 Quantum Potentials Proportional to
	References
12 Inverse Problems in Quantum Tunneling
	12.1 Nonlinear Equation for Variable Reflection Amplitude
	12.2 Inverse One-dimensional Tunneling Problem
	12.3 A Method for Finding the Potential from the Reflection Amplitude
	12.4 Finding the Shape of the Potential Barrier in One-Dimensional Tunneling
	12.5 Construction of a Symmetric Double-Well Potential from the Known Energy Eigenvalues
	12.6 The Inverse Problem of Molecular Spectra
	12.7 The Inverse Problem of Tunneling for Gamow States
	12.8 Inverse Problem of Survival Probability
	References
13 Inverse Problems Related to the Classical Wave Propagation
	13.1 Determination of the Wave Velocity in an Inhomogeneous Medium from the Reflection Coefficient
	13.2 Solvable Examples
	13.3 Extension of the Inverse Method to Reflection from a Layered Medium where the Asymptotic Values of c(t) at t→±∞ are Different
	13.4 Direct and Inverse Problems of Wave Propagation Using Travel Time Coordinate
	13.5 R Matrix and the Inverse Problems of Wave Propagation
	13.6 Inverse Problem for Acoustic Waves: Determination of the Wave Velocity and Density Profiles
	13.7 Inversion of Travel Time Data in the Geometrical Acoustic Limit
	13.8 Riccati Equation for Solving the Direct Problem for Variable Velocity and Density
	13.9 Finite Difference Equation for Acoustic Pressure in an Inhomogeneous Medium: Direct and Inverse Problems
	13.10 Determination of the Wave Velocity and the Density of the Medium
	13.11 Rational Representation of the Input Data
	13.12 Direct and Inverse Methods Based on Continued Fraction Expansion Applied to Two Simple Models
	13.13 Inverse Problem of Wave Propagation Using Schwinger’s Approximation
	References
14 The Inverse Problem of Torsional Vibration
	References
15 Local Nucleon-Nucleon Potentials Found from the Inverse Scattering Problem at Fixed Energy
	15.1 Constructing the S Matrix from Empirical Data
	15.2 A Method for the Numerical Calculation of the Local Potential Using the Gel’fand–Levitan Formulation
	15.3 Direct and Inverse Problems for Nucleon-Nucleon Scattering Using Continued Fraction Formulation
	15.4 Inverse Problem of Scattering in the Presence of the Tensor Force
	15.5 Potential Model for Generating the Input Data for Testing the Inversion Method
	15.6 Inverse Method of Nucleon-Nucleon Phase Shift and the Calculation of Nuclear Structure
	References
16 The Inverse Problem of Nucleon-Nucleus Scattering
	16.1 Solving the Inverse Nucleon-Nucleus Problem
	16.2 Inverse Scattering Theory Incorporating Both Coulomb and Nuclear Forces
	16.3 Inverse Scattering Method for Two Identical Nuclei at Fixed Energy
	References
17 Two Inverse Problems of Electrical Conductivity in Geophysics
	17.1 Inverse Problem of Electrical Conductivity in One-Dimension
	17.2 The Inverse Problem of Geomagnetic Induction at a Fixed Frequency
	References
18 Determination of the Mass Density Distribution Inside or on the Surface of a Body from the Measurement of the External Potential
	References
19 The Inverse Problem of Reflection from a Moving Object
	References
A Expansion Algorithm for Continued J-fractions
B Reciprocal Differences of a Quotient and Thiele’s Theorem
Index