Finitely many Dirac-delta interactions on Riemannian manifolds

Abstract

Articles you may be interested inThree lectures on global boundary conditions and the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary AIP Conf. Proc. 1460, 15 (2012); 10.1063/1.4733360 Nonrelativistic Lee model in three dimensional Riemannian manifoldsThis 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 ⌽. The bound state energies can be found from the characteristic equation ⌽͑− 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 ⌽. Using the result, we give lower bound estimates of the discrete spectrum.

A Perturbative Approach to the Tunneling Phenomena

“…In this paper, we have first reviewed the basic results about some singular interactions, such as the Dirac delta potentials supported by points on flat spaces and hyperbolic manifolds, and delta potentials supported by curves in flat spaces. Moreover, the results in the relativistic extensions of the above-mentioned potentials have been also reviewed which was essentially given in [13,14,15,17,18,19]. The main result of this paper is to develop some kind of perturbation theory applied to a class of singular potentials in order to find the splitting in the energy due to the tunneling.…”

“…Furthermore, point like Dirac delta interactions have also been extended to various more general cases. For our approach, to illustrate the main ideas, we are mainly concerned with the delta potentials supported by points on flat and hyperbolic manifolds [13,14,15], and delta potentials supported by curves in flat spaces, and its various relativistic extensions in flat spaces [16,17,18,19].…”

On the number of bound states of point interactions on hyperbolic manifolds

“…≥ c g(., .)) in two and three dimensions [8,9,10] (they were not stated as theorems in there) through Theorem 1 and Theorem 2. Their proofs are given in Appendices.…”

Dirac lattices, zero-range potentials, and self-adjoint extension

“…g. This was discussed in [7] and, more recently in [8], in application to delta functions on a manifold using a regularization in terms of separable potentials and application of renormaliation group techniques. A finite dimensional analog of this is known as Koster-Slater perturbation technique [9].…”