Compact submanifolds supporting singular interactions

• Schrödinger operator with singular interactions supported on compact submanifolds.
• Exact bound-state solution in terms of the heat kernel of the ambient manifold.
• Generalization of the variational approach to a collection of submanifolds.
• Existence of a lower bound for a unique ground state energy.

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.