| Preface |
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vii | |
| Introduction |
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1 | (4) |
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1 Inverse Problems in Classical Dynamics |
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5 | (34) |
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1.1 Inverse Problem for Trajectory |
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5 | (1) |
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1.2 Determination of the Shape of the Potential Energy from the Period of Oscillation |
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6 | (2) |
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1.3 Action Equivalent Hamiltonians |
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8 | (2) |
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1.4 Abel's Original Inverse Problem |
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10 | (2) |
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1.5 Inverse Scattering Problem in Classical Mechanics |
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12 | (3) |
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1.6 Inverse Problem of a Linear Chain of Masses Coupled to Springs |
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15 | (7) |
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1.7 Direct Problem of Non-exponential and of Exponential Decays in a Linear Chain |
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22 | (2) |
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1.8 Inverse Problem of Dynamics for a Non-uniform Chain |
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24 | (3) |
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1.9 Direct and Inverse Problems of Analytical Dynamics |
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27 | (2) |
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1.10 From the Classical Equations of Motion to the Lagrangian and Hamiltonian Formulations |
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29 | (5) |
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1.11 Langevin and Fokker-Planck Equations |
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34 | (5) |
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2 Inverse Problems in Semiclassical Formulation of Quantum Mechanics |
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39 | (8) |
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2.1 Quantum Mechanical Bound States for Confining Potentials |
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39 | (2) |
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2.2 Semiclassical Formulation of the Inverse Scattering Problem |
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41 | (6) |
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3 Inverse Problems and the Heisenberg Equations of Motion |
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47 | (8) |
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3.1 Equations of Motion Derived from the Hamiltonian Operator |
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48 | (1) |
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3.2 Determination of the Commutation Relations From the Equations of Motion |
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49 | (3) |
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3.3 Construction of the Hamiltonian Operator as an Inverse Problem |
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52 | (3) |
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4 Inverse Scattering Problem for the Schrodinger Equation and the Gel'fand---Levitan Formulation |
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55 | (28) |
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56 | (2) |
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58 | (3) |
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61 | (2) |
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4.4 The Gel'fand-Levitan Equation |
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63 | (5) |
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4.5 Inverse Problem for One-dimensional Schrodinger Equation |
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68 | (5) |
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73 | (4) |
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4.7 The Jost and Kohn Method of Inversion |
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77 | (6) |
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5 Marchenko's Formulation of the Inverse Scattering Problem |
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83 | (32) |
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5.1 Mathematical Preliminaries |
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83 | (8) |
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5.2 Bound States Embedded in Continuum |
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91 | (1) |
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5.3 More Solvable Potentials Found from Inverse Scattering |
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92 | (4) |
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5.4 The Inverse Problem for Reflection and Transmission from a Barrier |
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96 | (2) |
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5.5 A Special Problem in Electromagnetic Inverse Scattering |
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98 | (6) |
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5.6 Construction of Reflectionless Potentials |
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104 | (4) |
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5.7 Symmetric Reflectionless Potentials Supporting a Given Set of Bound States |
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108 | (7) |
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6 Newton-Sabatier Approach to the Inverse Problem at Fixed Energy |
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115 | (38) |
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6.1 Construction of the Potential at Fixed Energy |
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115 | (6) |
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6.2 Criticism of the Newton-Sabatier Method of Inversion at a Fixed Energy |
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121 | (2) |
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6.3 On the Results of the Numerical Solution of Inverse Problems |
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123 | (1) |
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6.4 Modified Form of the Gel'fand-Levitan for Fixed Energy Problems and the Langer Transform |
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124 | (6) |
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6.5 Lipperheide and Fiedeldey Approach to the Inverse Problem at Fixed Energy |
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130 | (6) |
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6.6 Completeness of the Set of Jost Solutions /(A, k, r) |
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136 | (5) |
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6.7 Generalized Gel'fand-Levitan Approach to Inversion |
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141 | (4) |
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6.8 The Method of Schnizer and Leeb |
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145 | (2) |
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6.9 Analysis of Atom-Atom Scattering Using Complex Angular Momentum Formulation |
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147 | (6) |
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7 Discrete Forms of the Schrodinger Equation and the Inverse Problem |
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153 | (20) |
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154 | (1) |
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7.2 The Method of Case and Kac for Discrete Form of Inverse Scattering Problem |
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155 | (8) |
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7.3 Discrete Form of the Spectral Density for Solving the Inverse Problem on Semi-axis 0 ≥ r > ∞ |
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163 | (10) |
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8 R Matrix Theory and Inverse Problems |
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173 | (14) |
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8.1 Inverse Problem for R Matrix Formulation of Scattering |
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178 | (1) |
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8.2 The Finite-difference Analogue of the R Matrix Theory of Scattering |
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179 | (3) |
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8.3 Shell-model Hamiltonian in Tri-diagonal Form |
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182 | (1) |
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8.4 Continued Fraction Expansion of the R Matrix |
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183 | (4) |
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9 Solvable Models of Fokker-Planck Equation Obtained Using the Gel'fand-Levitan Method |
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187 | (8) |
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9.1 Solution of the Fokker-Planck Equation for Symmetric and Asymmetric Double-Well Potentials |
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191 | (4) |
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10 The Eikonal Approximation |
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195 | (12) |
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10.1 Finding the Impact Parameter Phase Shifts from the Cross Section |
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201 | (6) |
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11 Inverse Methods Applied to Study Symmetries and Conservation Laws |
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207 | (10) |
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11.1 Classical Degeneracy and Its Quantum Counterpart |
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208 | (1) |
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11.2 Inverse Problem for Angular Momentum Eigenvalues |
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209 | (4) |
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11.3 Quantum Potentials Proportional to h |
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213 | (4) |
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12 Inverse Problems in Quantum Tunneling |
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217 | (24) |
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12.1 Nonlinear Equation for Variable Reflection Amplitude |
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217 | (2) |
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12.2 Inverse One-dimensional Tunneling Problem |
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219 | (3) |
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12.3 A Method for Finding the Potential from the Reflection Amplitude |
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222 | (2) |
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12.4 Finding the Shape of the Potential Barrier in One-Dimensional Tunneling |
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224 | (4) |
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12.5 Construction of a Symmetric Double-Well Potential from the Known Energy Eigenvalues |
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228 | (2) |
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12.6 The Inverse Problem of Molecular Spectra |
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230 | (3) |
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12.7 The Inverse Problem of Tunneling for Gamow States |
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233 | (3) |
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12.8 Inverse Problem of Survival Probability |
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236 | (5) |
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13 Inverse Problems Related to the Classical Wave Propagation |
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241 | (44) |
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13.1 Determination of the Wave Velocity in an Inhomogeneous Medium from the Reflection Coefficient |
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241 | (5) |
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246 | (2) |
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13.3 Extension of the Inverse Method to Reflection from a Layered Medium where the Asymptotic Values of c(t) at t → ∞ are Different |
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248 | (6) |
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13.4 Direct and Inverse Problems of Wave Propagation Using Travel Time Coordinate |
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254 | (8) |
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13.5 R Matrix and the Inverse Problems of Wave Propagation |
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262 | (3) |
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13.6 Inverse Problem for Acoustic Waves: Determination of the Wave Velocity and Density Profiles |
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265 | (1) |
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13.7 Inversion of Travel Time Data in the Geometrical Acoustic Limit |
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265 | (2) |
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13.8 Riccati Equation for Solving the Direct Problem for Variable Velocity and Density |
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267 | (1) |
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13.9 Finite Difference Equation for Acoustic Pressure in an Inhomogeneous Medium: Direct and Inverse Problems |
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268 | (2) |
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13.10 Determination of the Wave Velocity and the Density of the Medium |
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270 | (1) |
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13.11 Rational Representation of the Input Data |
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271 | (1) |
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13.12 Direct and Inverse Methods Based on Continued Fraction Expansion Applied to Two Simple Models |
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271 | (3) |
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13.13 Inverse Problem of Wave Propagation Using Schwinger's Approximation |
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274 | (11) |
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14 The Inverse Problem of Torsional Vibration |
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285 | (8) |
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15 Local Nucleon-Nucleon Potentials Found from the Inverse Scattering Problem at Fixed Energy |
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293 | (24) |
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15.1 Constructing the S Matrix from Empirical Data |
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294 | (5) |
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15.2 A Method for the Numerical Calculation of the Local Potential Using the Gel'fand-Levitan Formulation |
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299 | (3) |
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15.3 Direct and Inverse Problems for Nucleon-Nucleon Scattering Using Continued Fraction Formulation |
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302 | (3) |
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15.4 Inverse Problem of Scattering in the Presence of the Tensor Force |
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305 | (4) |
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15.5 Potential Model for Generating the Input Data for Testing the Inversion Method |
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309 | (3) |
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15.6 Inverse Method of Nucleon-Nucleon Phase Shift and the Calculation of Nuclear Structure |
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312 | (5) |
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16 The Inverse Problem of Nucleon-Nucleus Scattering |
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317 | (16) |
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16.1 Solving the Inverse Nucleon-Nucleus Problem |
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320 | (4) |
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16.2 Inverse Scattering Theory Incorporating Both Coulomb and Nuclear Forces |
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324 | (3) |
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16.3 Inverse Scattering Method for Two Identical Nuclei at Fixed Energy |
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327 | (6) |
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17 Two Inverse Problems of Electrical Conductivity in Geophysics |
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333 | (16) |
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17.1 Inverse Problem of Electrical Conductivity in One-Dimension |
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333 | (6) |
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17.2 The Inverse Problem of Geomagnetic Induction at a Fixed Frequency |
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339 | (10) |
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18 Determination of the Mass Density Distribution Inside or on the Surface of a Body from the Measurement of the External Potential |
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349 | (6) |
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19 The Inverse Problem of Reflection from a Moving Object |
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355 | (16) |
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A Expansion Algorithm for Continued J-fractions |
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361 | (6) |
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B Reciprocal Differences of a Quotient and Thiele's Theorem |
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367 | (4) |
| Index |
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371 | |
Lisainfo e-raamatute kohta