Estimates for the number of eigenvalues of two dimensional Schroedinger operators lying below the essential spectrum

TL;DR

This paper develops estimates for the number of eigenvalues below the essential spectrum of two-dimensional Schrödinger operators, extending known inequalities to more complex settings involving various supports and boundary conditions.

Contribution

It provides new Cwikel-Lieb-Rozenblum type estimates for 2D Schrödinger operators considering different potential supports and boundary conditions, using weighted and Orlicz norms.

Findings

Derived estimates involving weighted L^1 norms.

Extended bounds to potentials with supports of varying dimensions.

Applied results to operators on strips with boundary conditions.

Abstract

The celebrated Cwikel-Lieb_Rozenblum inequality gives an upper estimate for the number of negative eigenvalues of Schroedinger operators in dimension three and higher. The situation is much more difficult in the two dimensional case. There has been significant progress in obtaining upper estimates for the number of negative eigenvalues of two dimensional Schroedinger operators on the whole plane. In this thesis, we present estimates of the Cwikel-Lieb_Rozenblum type for the number of eigenvalues (counted with multiplicities) of two dimensional Schroedinger operators lying below the essential spectrum in terms of the norms of the potential. The problem is considered on the whole plane with different supports of the potential of dimension between 0 and 2, and on a strip with various boundary conditions. In both cases, the estimates involve weighted L^1 norms and Orlicz norms of the potential.