Exact solutions of a particle in a box with a delta function potential: The factorization method

TL;DR

This paper applies the factorization method to exactly solve for the eigenvalues and eigenfunctions of a particle in a box with a delta function potential, revealing how the potential affects the ladder operators and spectrum.

Contribution

It introduces a novel application of the factorization method to a particle in a box with a delta potential, providing exact solutions and insights into the operator discontinuities.

Findings

Exact eigenvalues and eigenfunctions obtained

Discontinuity in ladder operators due to delta potential

Full spectrum derived even at weak coupling

Abstract

We use the factorization method to find the exact eigenvalues and eigenfunctions for a particle in a box with the delta function potential $V(x)=\lambda\delta(x-x_{0})$. We show that the presence of the potential results in the discontinuity of the corresponding ladder operators. The presence of the delta function potential allows us to obtain the full spectrum in the first step of the factorization procedure even in the weak coupling limit $\lambda\to 0$.