Infinitely many singular interactions on noncompact manifolds

TL;DR

This paper proves that the ground state energy remains bounded from below on two-dimensional non-compact manifolds with infinitely many attractive delta potentials, using heat kernel techniques and geometric comparison theorems.

Contribution

It extends the analysis of delta interactions to non-compact manifolds, generalizing previous flat space results with geometric and heat kernel methods.

Findings

Ground state energy is bounded from below with infinitely many delta potentials.

Results apply to arbitrary locations with minimum separation on non-compact manifolds.

Uses heat kernel techniques and Riemannian geometry comparison theorems.

Abstract

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan-Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et. al. (2004).