On the Number of Bound States of Point Interactions on Hyperbolic Manifolds

TL;DR

This paper investigates the spectral properties of quantum particles with multiple point interactions on hyperbolic manifolds, providing conditions for the existence of bound states and extending the analysis to relativistic models.

Contribution

It offers explicit criteria for the number of bound states in hyperbolic manifolds and extends the spectral analysis to relativistic cases on hyperbolic and Euclidean spaces.

Findings

Derived sufficient conditions for N bound states on hyperbolic manifolds

Provided explicit criteria for bound state existence in hyperbolic geometries

Extended spectral analysis to relativistic quantum models

Abstract

We consider the problem of a quantum particle interacting with $N$ attractive point $\delta$-interactions in two and three dimensional Riemannian manifolds and discuss its some spectral properties. The main aim of this paper is to give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds $\mathbb{H}^2$ and $\mathbb{H}^3$. Furthermore, we study the same spectral problem for a relativistic extension of the model on $\mathbb{R}^2$ and $\mathbb{H}^2$.