The Birman-Schwinger Operator for a Parabolic Quantum Well in a Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity

TL;DR

This paper analyzes the spectral properties of a quantum particle in a thin layer with a parabolic well and Gaussian impurity, establishing the Hilbert-Schmidt nature of the Birman-Schwinger operator and constructing the limit Hamiltonian for a degenerate potential.

Contribution

It proves the Hilbert-Schmidt property of the Birman-Schwinger operator and constructs the self-adjoint Hamiltonian as a limit of scaled Gaussian potentials.

Findings

The Birman-Schwinger operator is Hilbert-Schmidt.

The Hamiltonian with a degenerate Gaussian potential is the norm resolvent limit of scaled Gaussian Hamiltonians.

Bounds on the ground state energies are provided.

Abstract

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the $x$-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impurity inside the layer and prove that such an integral operator is Hilbert-Schmidt, which allows the use of the modified Fredholm determinant in order to compute the bound states created by the impurity. Furthermore, we consider the case where the Gaussian potential degenerates to a $\delta$-potential in the $x$-direction and a Gaussian potential in the $y$-direction. We construct the corresponding self-adjoint Hamiltonian and prove that it is the limit in the norm resolvent sense of a sequence of corresponding Hamiltonians with suitably scaled Gaussian potentials. Satisfactory bounds on the ground state energies of all Hamiltonians involved are exhibited.