Self - adjoint extension theory of singular interactions

Abstract

In this thesis, we study two-dimensional finite singular point interactions and three-dimensional finite singular interactions supported by curves on a compact Riemannian manifold. Our aim is to show that finite rank perturbations of self-adjoint operators on compact manifold agree with renormalization technique to describe singu lar Dirac delta potential on the compact Riemannian manifold. To achieve our goal, we rely on heat kernel techniques and the proper upper bounds. We review the heuristic construction of the resolvents via renormalization method. Later, we present rudiments of finite rank perturbation and self-adjoint extensions, suitable for our purposes. In the end, we prove our main results that singular interactions on compact manifold could be understood from the self-adjoint extension perspective.