Self - adjoint extension theory of singular interactions - Türköz, Mustafa. (Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2017., 2017)
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.
Keywords: Riemannian manifolds.; Field extensions (Mathematics)