Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

A Kontorovich–Lebedev representation for zero-range potential eigensolutions

G. Gasaneo, S. Yu. Ovchinnikov, J. H. Macek

Published Oct 15, 2001• 14 citations
Publisher

Abstract

A novel mathematically simple Kontorovich–Lebedev representation of solutions to the Schrödinger equation for a three-particle problem where two of them interact via a zero-range potential is developed. The asymptotic limits and regularity properties are studied. The connection between the representations for E > 0 and E < 0 is also discussed.

DOI
10.1088/0305-4470/34/42/315

Related papers

Integral representation of one-dimensional three particle scattering for δ function interactions

Alejandro Amaya-Tapia, G. Gasaneo, S. Yu. Ovchinnikov +2 more

The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.

arXiv:math-ph/0406019•Journal of Mathematical Physics•Aug 6, 2004•7 citations

Scattering Theory in Quantum Mechanical Problems

Dimitri R. Yafaev

The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.

arXiv:2205.13204•May 26, 2022•0 citations

Eigenchannel method for nonrelativistic scattering from zero-range potentials

Radosław Szmytkowski, Marek Gruchowski

An eigenchannel method for nonrelativistic quantum scattering from an arbitrary system of zero-range potentials is presented. Eigenchannel vectors are defined as characteristic vectors of an auxiliary Hermitian matrix spectral problem with a positive semidefinite weight; real eigenvalues to this problem are the negative cotangents of eigenphase shifts. The eigenchannel vectors and the eigenphase shifts are used to construct particular solutions, called eigenchannels, of the pertinent Schr\"odinger equation. Analysis of the asymptotics of the eigenchannels in the far zone leads to a definition of eigenchannel harmonics, which for the problem at hand play the role analogous to that played by spherical harmonics for scattering in a central field. Representations of a far-field amplitude (for scattering of particles emitted from a point source) as well as a scattering amplitude, a scattering kernel, total, and total averaged cross sections (for scattering of a parallel beam of projectiles) in terms of the eigenphase shifts and the eigenchannel harmonics are derived. As an illustrative example, scattering of a plane wave from a system of four identical zero-range potentials, located in vertices of a regular tetrahedron, is worked out in detail.

Physical Review A•Dec 28, 2004•11 citations

Solution of Schrödinger Equation for a Particle Bound to More Than One Spherical Potential

Leonard J. Eyges

We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.

Physical Review•Aug 1, 1958•28 citations

Model of the effective separable potential in the problem of three one-dimensional quantum particles

S. B. Levin, A.S. Bagmutov, Victor O. Toropov

The goal of this paper is to construct an effective model for studying the asymptotic solution of the scattering problem of three one-dimensional quantum particles with finite (short-range) attractive pair potentials. The asymptotic nature of the solution is defined by the rapid decrease in its discrepancy in the Schrödinger equation.

arXiv:2405.11443•May 19, 2024•1 citation

Solution to One-Dimensional Schrödinger Equation for an Arbitrary Potential; Application to Radial Equation in Three Dimensions

James L. Sigel

The work of Reading and Sigel on the solution of the one-dimensional Schr\"odinger equation for a particle moving in the presence of a collection of $\ensuremath{\delta}$-function potentials of arbitrary strength and position is generalized so that any arbitrary potential $V(x)$ can be handled. A close connection is shown to exist between the methods described in this note and those, involving Jost functions, which have been developed for handling the radial Schr\"odinger equation in three dimensions.

Physical review. B, Solid state•Nov 15, 1972•1 citation
A novel mathematically simple Kontorovich–Lebedev representation of solutions to the Schrödinger equation for a three-particle problem where two of them interact via a zero-range potential is developed. The asymptotic limits and regularity properties are studied. The connection between the representations for E > 0 and E < 0 is also discussed.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
An eigenchannel method for nonrelativistic quantum scattering from an arbitrary system of zero-range potentials is presented. Eigenchannel vectors are defined as characteristic vectors of an auxiliary Hermitian matrix spectral problem with a positive semidefinite weight; real eigenvalues to this problem are the negative cotangents of eigenphase shifts. The eigenchannel vectors and the eigenphase shifts are used to construct particular solutions, called eigenchannels, of the pertinent Schr\"odinger equation. Analysis of the asymptotics of the eigenchannels in the far zone leads to a definition of eigenchannel harmonics, which for the problem at hand play the role analogous to that played by spherical harmonics for scattering in a central field. Representations of a far-field amplitude (for scattering of particles emitted from a point source) as well as a scattering amplitude, a scattering kernel, total, and total averaged cross sections (for scattering of a parallel beam of projectiles) in terms of the eigenphase shifts and the eigenchannel harmonics are derived. As an illustrative example, scattering of a plane wave from a system of four identical zero-range potentials, located in vertices of a regular tetrahedron, is worked out in detail.
We consider the quantum-mechanical problem of a particle bound to a configuration of spherical potentials, each of finite range. If the Schr\"odinger equation can be solved for each potential by itself, then it is shown how to solve it for the configuration, provided the potentials do not overlap. The energy levels are the zeros of a determinant of formally infinite order, but in practice this is always well approximated by a finite determinant and often by one of small order. As illustrative examples we consider some states of a particle bound to three square wells with the configuration of an equilateral triangle, and to two truncated Coulomb potentials. The possibility of extending an approximate version of the method to overlapping potentials is pointed out.
The work of Reading and Sigel on the solution of the one-dimensional Schr\"odinger equation for a particle moving in the presence of a collection of $\ensuremath{\delta}$-function potentials of arbitrary strength and position is generalized so that any arbitrary potential V(x)V(x)V(x) can be handled. A close connection is shown to exist between the methods described in this note and those, involving Jost functions, which have been developed for handling the radial Schr\"odinger equation in three dimensions.