Jost functions and Jost solutions for Jacobi matrices, II. Decay and analyticity

This chapter examines the properties of one-dimensional Jost solutions for S-matrix problems. It first considers how the left–right transmission and reflections coefficients can be expressed in terms of the elements of the S-matrix for one-dimensional scattering problems on, focusing on poles of the transmission coefficient. It then uses the radial equation to revisit the problem of the square-well potential from the perspective of the Jost solution, with Jost boundary conditions at r = 0 and as r approaches infinity. It also presents the notations for the Jost functions and the S-matrix before discussing the problem of scattering from a constant spherical inhomogeneity.

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Effective Interactions: Resonances and Off-Shell Characteristics

HNPS Proceedings ◽

10.12681/hnps.2177 ◽

2019 ◽

Vol 10 ◽

pp. 93

Author(s):

S. E. Massen ◽

S. A. Sofianos ◽

S. A. Rakityansky ◽

S. Oryu

Keyword(s):

Inverse Scattering ◽

Nuclear Reactions ◽

Bound State ◽

Inverse Scattering Method ◽

Wave Functions ◽

Scattering Method ◽

Effective Interactions ◽

Analytical Properties ◽

Jost Functions ◽

Jost Solutions

The influence of resonances on the analytical properties and off-shell characteristics of effective interactions has been investigated. This requires, among others, the knowledge of the Jost function in regions of physical interest on the complex kplane when the potentials are given in a tabular form. The latter are encountered in inverse scattering and supersymmetric transformations. To investigate the effects of resonances on the analytical properties of the potential, we employed the Marchenko inverse scattering method to construct, phase and bound state equivalent local potentials but with different resonance spectra. It is shown that the inclusion of resonances changes the shape, strength, and range of the potential which in turn would modify the bound and scattering wave functions in the interior region. This could have important consequences in calculations of transition amplitudes in nuclear reactions, which strongly depend on the behaviour of the wave functions at short distances. Finally, an exact method to obtain the Jost solutions and the Jost functions for a repulsive singular potential is presented. The effectiveness of the method is demonstrated using the Lennard-Jones (12,6) potential.

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The Jost Solutions: Technical Details

Rays, Waves, and Scattering ◽

10.23943/princeton/9780691148373.003.0025 ◽

2017 ◽

Author(s):

John A. Adam

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Body Problem ◽

Complete Information ◽

Two Body Problem ◽

Jost Function ◽

Radial Schrödinger Equation ◽

Jost Functions ◽

Technical Details ◽

Jost Solutions

This chapter discusses the technical details of the Jost solutions of the Schrödinger equation. The nonrelativistic quantum mechanical two-body problem can be described in terms of the Jost functions and Jost solutions of the Schrödinger equation. When defined for all complex values of the momentum, the Jost functions contain complete information about the underlying physical system. Compared to the S-function which may have redundant poles, the Jost function is a more fundamental quantity because it does not suffer from ambiguities caused by redundant zeros. The chapter first considers the time-independent radial Schrödinger equation before analyzing the regular solution for the Jost function, the poles of the S-matrix, and the wavepacket approach.

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On‐shell singularities of the off‐shell Jost functions and Jost solutions for long‐range potentials

Journal of Mathematical Physics ◽

10.1063/1.526215 ◽

1984 ◽

Vol 25 (4) ◽

pp. 982-986 ◽

Cited By ~ 1

Author(s):

Jan Dušek

Keyword(s):

Long Range ◽

Jost Functions ◽

Jost Solutions

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Inverse problems for finite Jacobi matrices and Krein–Stieltjes strings

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0112 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Alexander Mikhaylov ◽

Victor Mikhaylov

Keyword(s):

Dynamical System ◽

Inverse Problems ◽

Jacobi Matrix ◽

Jacobi Matrices ◽

Unknown Parameters ◽

Stieltjes String ◽

Propagation Of Waves ◽

Dynamic Inverse

Abstract We consider dynamic inverse problems for a dynamical system associated with a finite Jacobi matrix and for a system describing propagation of waves in a finite Krein–Stieltjes string. We offer three methods of recovering unknown parameters: entries of a Jacobi matrix in the first problem and point masses and distances between them in the second, from dynamic Dirichlet-to-Neumann operators. We also answer a question on a characterization of dynamic inverse data for these two problems.

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