Teoman Turgut: Singular systems on manifolds

Abstract: We discuss a singular extension of Laplacian on a two or three dimensional manifold. Under certain mild assumptions, we show that the bottom of the spectrum becomes a non-degenerate eigenvalue.  We define an oversimplified version of the Lee model, in which a two level system is interacting with bosonic particles  on a three dimensional manifold. We show that this system can be understood by an associated operator,  after removing an infinity (renormalization). We indicate that this operator contains all the information about the dynamics and the spectrum is bounded from below. We explain that there is a well defined self-adjoint Hamiltonian  defined by this operator.  We sketch the proof that on a compact manifold the lowest eigenstate of this operator is unique, using the theory of positivity improving semigroups.