Quantum mechanics of layers with a finite number of point perturbations

TL;DR

This paper investigates the spectral and scattering behavior of a quantum particle in a layered system with finite point perturbations, including effects of magnetic fields, using explicit resolvent formulas.

Contribution

It provides an explicit resolvent formula for the model and analyzes bound states, scattering, and magnetic field effects in layered quantum systems with point interactions.

Findings

Existence of bound states due to point perturbations.

Explicit form of the scattering operator.

Magnetic field induces eigenvalues in spectral gaps.

Abstract

We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Krein's formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.