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

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).