Finitely Many Dirac-Delta Interactions on Riemannian Manifolds

TL;DR

This paper investigates the non-perturbative renormalization of bound states for finitely many Dirac-delta interactions on various Riemannian manifolds, employing heat kernel and spectral regularization methods to analyze the spectrum.

Contribution

It introduces a characteristic matrix approach for bound state problems on Riemannian manifolds and compares regularization methods, providing explicit calculations and spectrum bounds.

Findings

Characteristic matrix formulation for bound states.

Equivalence of regularization methods on compact manifolds.

Explicit heat kernel calculations for H^2 and H^3.

Abstract

This work is intended as an attempt to study the non-perturbative renormalization of bound state problem of finitely many Dirac-delta interactions on Riemannian manifolds, S^2, H^2 and H^3. We formulate the problem in terms of a finite dimensional matrix, called the characteristic matrix. The bound state energies can be found from the characteristic equation. The characteristic matrix can be found after a regularization and renormalization by using a sharp cut-off in the eigenvalue spectrum of the Laplacian, as it is done in the flat space, or using the heat kernel method. These two approaches are equivalent in the case of compact manifolds. The heat kernel method has a general advantage to find lower bounds on the spectrum even for compact manifolds as shown in the case of S^2. The heat kernels for H^2 and H^3 are known explicitly, thus we can calculate the characteristic matrix. Using the result, we give lower bound estimates of the discrete spectrum.