Abstract

We find supersymmetric partners of a family of self-adjoint operators which are self-adjoint extensions of the differential operator −d2/dx2 on L2[−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 ℓ-th order partner differs in one energy level from both the (ℓ−1)-th and the (ℓ+1)-th order partners. In general, the eigenvalues for each of the self-adjoint extensions of −d2/dx2 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 ℓ-th supersymmetric partners of each extension.

Highlights

  • The study of one dimensional models in quantum mechanics is useful in order to gain a better understanding of the properties of quantum systems

  • We intend to investigate the properties of the SUSY partners of the self-adjoint determinations of the operator −d2 /dx2 on L2 [− a, a], a > 0 and finite, with appropriate boundary conditions at the points − a and a

  • Let us consider the general solution of the time independent Scrödinger equation −d2 φ( x )/dx2 = Eφ( x ), with E = s2 /(2a)2 ≥ 0, where 2a is the infinite square well width

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Summary

Introduction

The study of one dimensional models in quantum mechanics is useful in order to gain a better understanding of the properties of quantum systems. We intend to investigate the properties of the SUSY partners of the self-adjoint determinations of the operator −d2 /dx on L2 [− a, a], a > 0 and finite, with appropriate boundary conditions at the points − a and a Note that this problem is closely related to the problem of the definition of the “free” Hamiltonian on the one dimensional infinite square well potential. The general formalism can be applied to obtain a sequence of Hamiltonians when the ground state of the original self-adjoint extension of −d2 /dx on L2 [− a, a] has zero or negative energy In this case, partner Hamiltonians may be very different from the original one in the sense that they may have a finite number of eigenvalues or no eigenvalues. Section and an Appendix in which we show what the correct form for the wave functions for the energy levels should be

Self-Adjoint Extensions
Spectrum of the Free Particle on a Finite Interval
The Angular Representation of the Self-Adjoint Extensions of H0
Some Simple Cases
About the Negative and Zero Energies
First Order SUSY Partners
Second Order SUSY Partners
Supersymmetric Self-Adjoint Extensions of the Infinite Well at-Order
Conclusions and Outlook
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