On the Spectrum of a Quantum Dot with Impurity in the Lobachevsky Plane

TL;DR

This paper investigates the spectral properties of a quantum dot with impurity on the Lobachevsky plane, analyzing how curvature influences eigenvalues and eigenfunctions, and providing asymptotic expansions as curvature diminishes.

Contribution

It offers explicit formulas for the Green function and Krein Q-function in this geometric setting, and performs numerical analysis of the spectrum considering curvature effects.

Findings

Curvature significantly affects eigenvalues and eigenfunctions.

As curvature radius increases, eigenvalues approach the flat case limit.

Explicit formulas enable detailed spectral analysis in hyperbolic geometry.

Abstract

A model of a quantum dot with impurity in the Lobachevsky plane is considered. Relying on explicit formulae for the Green function and the Krein $Q-$function which have been derived in a previous work we focus on the numerical analysis of the spectrum. The analysis is complicated by the fact that the basic formulae are expressed in terms of spheroidal functions with general characteristic exponents. The effect of the curvature on eigenvalues and eigenfunctions is investigated. Moreover, there is given an asymptotic expansion of eigenvalues as the curvature radius tends to infinity (the flat case limit).