Bound states in two spatial dimensions in the non-central case

TL;DR

This paper develops a new bound on the number of negative energy bound states in two-dimensional potentials by adapting the Birman-Schwinger method, involving symmetrization of integral kernels and zero-energy state analysis.

Contribution

It introduces a novel adaptation of the Birman-Schwinger bound for two dimensions, using symmetrization and zero-energy state counting techniques.

Findings

Derived a bound on the number of bound states in 2D potentials.

Extended the Schwinger method to two-dimensional cases.

Provided a mathematical framework for analyzing bound states in non-central potentials.

Abstract

We derive a bound on the total number of negative energy bound states in a potential in two spatial dimensions by using an adaptation of the Schwinger method to derive the Birman-Schwinger bound in three dimensions. Specifically, counting the number of bound states in a potential gV for g=1 is replaced by counting the number of g_i's for which zero energy bound states exist, and then the kernel of the integral equation for the zero-energy wave functon is symmetrized. One of the keys of the solution is the replacement of an inhomogeneous integral equation by a homogeneous integral equation.