Supersymmetric Partners of the One-Dimensional Infinite Square Well Hamiltonian

TL;DR

This paper constructs supersymmetric partner Hamiltonians for various self-adjoint extensions of the infinite square well, classifies these extensions, and explicitly determines their eigenvalues and eigenfunctions.

Contribution

It classifies all self-adjoint extensions of the infinite square well and explicitly constructs their supersymmetric partners, including eigenvalues and eigenfunctions.

Findings

Extensions with positive ground state energy have infinite supersymmetric partners.

Eigenvalues are solutions to transcendental equations and form an infinite discrete spectrum.

Eigenfunctions are explicitly determined for all supersymmetric partners.

Abstract

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator $-d^2/dx^2$ on $L^2[-a,a]$, $a>0$, that is, the one dimensional infinite square well. First of all, we classify these self-adjoint extensions in terms of several choices of the parameters determining each of the extensions. There are essentially two big groups of extensions. In one, the ground state has strictly positive energy. On the other, either the ground state has zero or negative energy. In the present paper, we show that each of the extensions belonging to the first group (energy of ground state strictly positive) has an infinite sequence of supersymmetric partners, such that the $\ell$-th order partner differs in one energy level from both the $(\ell-1)$-th and the $(\ell+1)$-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of $-d^2/dx^2$ come from a transcendental equation and are all infinite. For the case under our study, we determine the eigenvalues, which are also infinite, {all the extensions have a purely discrete spectrum,} and their respective eigenfunctions for all of its $\ell$-th supersymmetric partners of each extension.