On the Number of Bound States of Finitely Many Point Interactions on Riemannian Manifolds

Fatih Erman

Published 2015-11-24Version 1

We first review the construction of the problem of a quantum particle interacting with $N$ attractive point $\delta$-interactions in two and three dimensional Riemannian manifolds and improve it in a more rigorous way so that our somewhat heuristic arguments, which were used in our earlier works, are justified. We prove that the principal matrix is a matrix-valued holomorphic function on the region $\mathcal{R}=\{ z \in \mathbb{C}| \Re(z) <0 \}$ for two particular classes of Riemannian manifolds. We show that the essential spectrum of the Hamiltonian and that of the free Hamiltonian coincides. We finally give a sufficient condition for the Hamiltonian to have $N$ bound states and give an explicit criterion for it in hyperbolic manifolds.