Finitely many Dirac-delta interactions on Riemannian manifolds

This work is intended as an attempt to study the nonperturbative 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 Phi. The bound state energies can be found from the characteristic equation Phi(-nu(2))A=0. 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 Phi. Using the result, we give lower bound estimates of the discrete spectrum. (c) 2006 American Institute of Physics.