Compact submanifolds supporting singular interactions

TL;DR

This paper investigates quantum particles with singular interactions on compact embedded surfaces, revealing exact bound states related to the ambient manifold's heat kernel and providing methods to estimate ground state energies in multi-surface systems.

Contribution

It introduces a resolvent-based approach to analyze bound states on compact submanifolds and derives explicit solutions and energy bounds for complex configurations.

Findings

Exact bound state solutions expressed via heat kernel.

Bound state existence for a range of coupling strengths.

Estimation techniques for ground state energies in multi-surface systems.

Abstract

A quantum particle moving under the influence of singular interactions on embedded surfaces furnish an interesting example from the spectral point of view. In these problems, the possible occurrence of a bound state is perhaps the most important aspect. Such systems can be introduced as quadratic forms and generically they do not require renormalization. Yet an alternative path through the resolvent is also beneficial to study various properties. In the present work, we address these issues for compact surfaces embedded in a class of ambient manifolds. We discover that there is an exact bound state solution written in terms of the heat kernel of the ambient manifold for a range of coupling strengths. Moreover, we develop techniques to estimate bounds on the ground state energy when several surfaces, each of which admits a bound state solution, coexist.