The Exact Point Spectrum and Eigenvector of the Unique Continuous L$^2(\mathbb{R}^2)$ Bound State Solution to the Dirac Delta Schrodinger Potential in Two Dimensions

TL;DR

This paper rigorously determines the unique bound state energy and eigenvector for a two-dimensional Dirac delta potential in quantum mechanics without using regularization or renormalization, resolving longstanding issues with the point spectrum.

Contribution

It provides a mathematically rigorous solution for the point spectrum of the 2D Dirac delta potential, establishing the exact eigenvalue and eigenvector without regularization.

Findings

Exactly one bound state eigenvalue identified

Explicit eigenvector constructed

No regularization or renormalization needed

Abstract

Analyzing the point spectrum, i.e. bound state energy eigenvalue, of the Dirac delta function in two and three dimensions is notoriously difficult without recourse to regularization or renormalization, typically both. The reason for this in two dimensions is two fold; 1) the coupling constant, together with the mass and Planck's constant form an unitless quantity. This causes there to be a missing anomalous length scale. 2) The immediately obvious L$^2$ solution is divergent at the origin, where the Dirac Delta potential has its important point of support as a measure. Due to the uniqueness of the solution presented here, it is immediate that the linear operator (the two dimensional Laplace operator on all of $\mathbb{R}^2$), with the specialized domain constructed here, ensures that the point spectrum has exactly one element. This element is determined precisely, and a natural mathematically rigorous resolution to the anomalous length scale arises. In this work, there is no recourse to renormalization or regularization of any kind.