Heat kernel methods in many body problems

Abstract

The main goal of this thesis is to construct a non-perturbative renormalization for several models in quantum mechanics and field theory by means of the heat kernel in two or three dimensional Riemannian manifolds and study their spectral properties with several approximation and heat kernel techniques. All models investigated here are formulated in terms of a finite well defined operator, including all the information about the interaction, and it is called the principal operator. As a first model, a parti- cle interacting with finitely many Dirac delta potentials in two and three dimensional manifolds is considered and the problem is renormalized with two different methods. The relation between the self-adjoint extensions and the renormalization approach to the same problem is emphasized via a kind of Krein's formula that we obtain. We then give a comparison theorem between the bound state energy of N and N +1 point interactions. The estimate of the bound state energies in the tunnelling regime is calculated by applying the perturbation theory. Moreover, the pointwise upper bounds on the wave function corresponding to the bound states are obtained and the existence of the Hamiltonian operator from the resolvent is established. Using Ger·sgorin theorem, the ground state energy is proven to be bounded from below for compact and Cartan-Hadamard manifolds. In addition to these, non-degeneracy and uniqueness of the ground state is found as a simple consequence of the Perron-Frobenius theorem. The renormalization group equations are also derived and the function is exactly calculated. In the second model, the renormalization of the non-relativistic Lee model in two and three dimensional manifolds is constructed and its ground state energy is proven to be bounded from below for compact and Cartan-Hadamard manifolds. Then, a kind of mean field approximation is applied to the model in two and three dimensions separately. Finally, the construction of the renormalization of the non-relativistic limit of ̧Á4 model in two dimensional manifolds is given.