The Infinite Square Well with a Point Interaction: A Discussion on the Different Parametrizations

TL;DR

This paper explores the relationship between two different parametrizations of point interactions in the infinite square well, focusing on their compatibility and how they can be related through boundary conditions and perturbations.

Contribution

It compares the self-adjoint extension approach with a matching condition approach for delta and delta prime interactions in the infinite square well.

Findings

The two parametrizations can be related through boundary conditions.

Perturbations of the form a delta + b delta' can be characterized within both frameworks.

The study clarifies the connection between different mathematical descriptions of point interactions.

Abstract

The construction of Dirac delta type potentials has been achieved with the use of the theory of self adjoint extensions of non-self adjoint formally Hermitian (symmetric) operators. The application of this formalism to investigate the possible self adjoint extensions of the one dimensional kinematic operator $K=-d^2/dx^2$ on the infinite square well potential is quite illustrative and has been given elsewhere. This requires the definition and use of four independent real parameters, which relate the boundary values of the wave functions at the walls. By means of a different approach, that fixes matching conditions at the origin for the wave functions, it is possible to define a perturbation of the type $a\delta(x)+b\delta'(x)$, thus depending on two parameters, on the infinite square well. The objective of this paper is to investigate whether these two approaches are compatible in the sense that perturbations like $a\delta(x)+b\delta'(x)$ can be fixed and determined using the first approach.