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

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 ${\mathbb{H}}^2$.