HEAT KERNEL ESTIMATE OF THE SCHRODINGER OPERATOR IN UNIFORM DOMAINS

Abstract

In this thesis we study the properties of the Schrodinger operator L=−∆+q on a Harnack-type Dirichlet space for q belonging to Kato class K or Kato-infinity class K∞. To be specific, it consists of three parts as follows: The first part is a generalization of [27]. For any Harnack-type Dirichlet space we give conditions under which there exists a positive Dirichlet solution (the profile) in an unbounded uniform domain for the operator L. In this setting, we further give the two-sided heat kernel estimate using the famous h-transform technique. The idea of second part comes from [64]. In the exterior of a compact set in a non- parabolic Harnack-type space, we can prove some equivalent statements connect- ing subcrilicality, positiveness of the Green function, gaugeability and the bound- edness of the Dirichlet-type solution provided the potential q ∈ K∞. Particularly, we can apply the boundedness result of the profile to the first part and conclude a more precise heat kernel estimate. In the third part we provide some typical examples and explore some properties when the potential decays faster than the quadratic one. Some other examples are given in the domain outside an unbounded domain and we propose some hypothesis as an supplement to the second part.