Existence of Hamiltonians for Some Singular Interactions on Manifolds

TL;DR

This paper proves the existence of Hamiltonians for certain singular interactions on Riemannian manifolds, including relativistic and non-relativistic models, using semigroup theory and resolvent formulas.

Contribution

It establishes the existence of Hamiltonians for renormalized point interactions and the Lee model on manifolds, extending previous results to more complex settings.

Findings

Existence of Hamiltonians for renormalized point interactions in 2D and 3D manifolds

Extension to relativistic models in 2D

Hamiltonian existence for the non-relativistic Lee model

Abstract

The existence of the Hamiltonians of the renormalized point interactions in two and three dimensional Riemannian manifolds and that of a relativistic extension of this model in two dimensions are proven. Although it is much more difficult, the proof of existence of the Hamiltonian for the renormalized resolvent for the non-relativistic Lee model can still be given. To accomplish these results directly from the resolvent formula, we employ some basic tools from the semigroup theory.