Exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice

Abstract

We address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice. As Kay and Moses showed in 1956, reflectionless potentials in one dimensional quantum mechanics are exactly solvable. With an additional time dependence these potentials are identified as the soliton solutions of the KdV hierarchy. An $N$-soliton potential has the time $t$ and $2N$ positive parameters, $k_1<...<k_N$ and $c_j$, $j=1,...,N$, corresponding to $N$ discrete eigenvalues $-k_j^2$. The eigenfunctions are elementary functions expressed by the ratio of determinants. The Darboux-Crum-Krein-Adler transformation or the Abraham-Moses transformations based on eigenfunctions deletions produce lower soliton number potentials with modified parameters $c_j^{\prime }$. We explore various identities satisfied by the eigenfunctions of the soliton potentials, which reflect the uniqueness theorem of Gel'fand-Levitan-Marchenko equations for separable (degenerate) kernels.

LaTeX 17 pages, no figure

References in corpus (9)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics
• Exact solution in the Heisenberg picture and annihilation-creation operators
• Casoratian Identities for the Wilson and Askey-Wilson Polynomials
• Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials
• Non-polynomial extensions of solvable potentials a la Abraham-Moses
• A novel scheme for measuring the relative phase difference between S and P polarization in optically denser medium
• Multi-indexed Extensions of Soliton Potential and Extended Integer Solitons of KdV Equation

Cited by in corpus (4)

• Supersymmetric Bragg gratings
• One-dimensional Schrödinger equation with non-analytic potential $V(x)=-g^2\exp (-|x|)$ and its exact Bessel-function solvability
• Reflectionless Potentials for Difference Schrödinger Equations
• Exactly Solvable Quantum Mechanics