Search of a general form of superpotential in hierarchy with discrete energy spectrum

TL;DR

This paper generalizes the concept of superpotential in supersymmetric quantum mechanics, enabling the connection of potentials with discrete spectra through arbitrary bound or unbound states, and introduces new deformation methods.

Contribution

It proposes a unified definition of superpotential based on Riccati equation solutions, extending SUSY QM techniques to inverse problems and new potential deformations.

Findings

Reconstruction of potential deformations using SUSY QM and inverse problem methods.

Identification of parameters affecting wave function behavior and potential zero-points.

Development of new deformation types connecting potentials with non-overlapping spectra.

Abstract

A generalized definition of superpotential has proposed, which connects two one-dimensional potentials $V_{1}$ and $V_{2}$ with discrete energy spectra completely and where: 1) energy of factorization equals to arbitrary level of spectrum of $V_{1}$ and function of factorization is defined concerning bound state at this level, 2) energy of factorization equals to arbitrary energy and function of factorization is defined concerning unbound (or non-normalizable) state at this energy. It has shown, that for unknown superpotential such its definition follows from solution of Riccati equation at given $V_{1}$. Using arbitrary bound state in construction of superpotential, SUSY QM methods in detailed calculations of spectral characteristics have been coming to level of methods of inverse problem. So, if as starting $V_{1}$ to choose rectangular well with finite width and infinitely high walls, then we reconstruct by SUSY QM approach all pictures of deformation of this potential and its wave functions of lowest bound states, which were obtained early by methods of inverse problem. Dependence between parameters of deformation for methods of SUSY QM and inverse problem has found, analysis of behavior of wave functions and the potential under deformation has fulfilled, a classification has proposed for zero-points of potential, nodes of the deformed wave functions, points, where wave functions are not deformed, an analysis of angles of wave functions leaving from such points has fulfilled. Using unbound states at arbitrary energy of factorization, we obtain new types of deformations. So, using only one superpotential, one can join two potentials, which have real energy spectra with own bound states and without coincident levels.