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Direct and Inverse Problems : Potentials in Quantum Scattering :  Potentials in Quantum Scattering - Boris N. Zakhariev

Direct and Inverse Problems : Potentials in Quantum Scattering

Potentials in Quantum Scattering


Published: November 1990
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This textbook can be viewed as a "how-to" manual for solving quantum inverse problems, for deriving the potential from spectra or scattering data, and as a quantum "picture book" which aims to enhance the reader's quantum intuition. The formal exposition of inverse methods is paralleled by a discussion of the direct problem. Differential and finite-difference equations are presented side by side. The common features and advantages of a variety of solution methods are analyzed. To foster a better understanding, the physical meaning of the mathematical quantities are discussed. Wave confinement in continuum bound states, resonance and collective tunnelling, energy shifts and the spectral and phase equivalence of various interactions are some of the physical problems covered.

I: One-dimensional, One-channel Systems.- 1 The principal Equations of Scattering Theory.- 1.1 General Remarks.- 1.2 Elements of the Direct and Inverse Problems.- 1.2.1 The Simplest Difference Schrodinger Equation.- 1.2.2 Potential Wells of Infinite Depth.- 1.2.3 The Direct Problem.- 1.2.4 The Inverse Problem.- 1.2.5 Scattering by a Potential of Finite Range.- 1.2.6 The Finite-Difference Analogue of the R-matrix Scattering Theory.- 1.2.7 Conditions of Orthonormality and Completeness of the Eigenfunctions of the Finite-Difference Schrodinger Operator on [O, ?].- 1.2.8 Relations Between the Scattering Parameters {E? and ??}.- 1.2.9 Reconstruction of the Potential on the Semi-axis 0 ?x Upper k Walf-plane.- 2.2.3 Potentials with S (k) with Two Poles in the Upper k Half-Plane.- 2.3 More General Models.- 2.3 Multi-term Degenerate Kernels of the Inverseproblem Equations.- 2.3.2 Models of One-Dimensional Motion on the Whole Axis.- 2.3.3 The Finite-Difference-Approach.- 2.3.4 The Rational Reflection Coefficient (no Bound States).- 2.3.5 The Finite-difference Approach.- 2.4 Potentials of the Finite-Range and Infinitely Deep Wells. R-matrix Models.- 2.5 Potentials Allowing Exact Solutions for Variable Angular Momenta.- 2.5.1 Potentials from Spectral Data at Fixed Energy and Variable l.- 2.5.2 Newton-Sabatier Potentials.- 2.5.3 The Generalized Crum-Krein Transformations.- 2.5.4 Lipperheide-Fidelday Potentials.- 2.6 Notes on the Literature.- 2.7 Exercises.- 3 Approximate Solutions.- 3.1 Convergence of the Approximations, Stability and Regularization of Solutions.- 3.2 Solutions Using Bargmann Potentials.- 3.4 Approximation of Datentials by Steps, at Discrete Points, and by Splines. The Role of the Upper Part of the Spectrum.- 3.5 Method of Multiple Solutions of the Direct Problem.- 3.6 Notes on the Literature.- 4 The Levinson Theorem.- 4.1 General Remarks.- 4.2 Simple Examples.- 4.3 The Coulomb Potential and Other Singular Interactions.- 4.4 Other Types of Interactions.- 4.4.1 Potentials Depending on Velocity.- 4.4.2 Potentials Depending on Energy.- 4.4.3 The Finite-Difference Schrodinger Equation.- 4.4.4 Motion Along the Axis.- 4.4.5 Nonlocal Potentials.- 4.5 Notes on the Literature.- II. Multi-channel, Multi-dimensional, Multi-particle Problems.- 5 Multi-channel Equations.- 5.1 General Remarks.- 5.2 The Formalism of Multi-channel Coupling.- 5.3 Finite-Difference Equations of Motion.- 5.4 Exactly Solvable Models.- 5.5 Notes on the Literature.- 5.6 Exercises.- 6 Multi-dimensional Problems.- 6.1 General Remarks.- 6.2 The Finite-Difference Formalism.- 6.3 Reduction of Multi-dimensional problems to Multichannel problems.- 6.4 The Multi-dimensional Inverse Problem.- 6.5 Separation of Variables in Noncentral Field.- 6.6 Exactly Solvable Models.- 6.7 Notes on the Literature.- 7 Multi-particle Systems.- 7.1 General Remarks.- 7.2 Asymptotic Hamiltonians and Boundary Conditions.- 7.3 Tunnelling Through Potential Barriers by Several Particles.- 7.4 Excitation of Collective Degrees of Freedom of Multi-particle Systems..- 7.4.1 Transformation of Amplitudes in Transition to the Reference Frame of the Target.- 7.5 The Method of Hyperspherical Functions (K-harmonics).- 7.6 The Levinson Theorem.- 7.7 Three-Particle Potentials.- 7.8 Notes on the Literature.- References.

ISBN: 9783540524847
ISBN-10: 3540524843
Audience: General
Format: Paperback
Language: English
Number Of Pages: 223
Published: November 1990
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.3
Weight (kg): 0.35