. We consider the generalized Schrodinger operator \Gamma\Delta + where is a nonnegative Radon measure in R n , n 3. Assuming that satisfies certain scale-invariant Kato condition and doubling condition, we establish the following bounds for the fundamental solution of \Gamma\Delta + in R n : c e \Gamma" 2 d(x;y;) jx \Gamma yj n\Gamma2 \Gamma (x; y) C e \Gamma" 1 d(x;y;) jx \Gamma yj n\Gamma2 where d(x; y; ) is the distance function for the modified Agmon metric m(x; )dx 2 associated with . We also study the boundedness of the corresponding Riesz transforms r(\Gamma\Delta + ) \Gamma1=2 on L p (R n ; dx). Keywords. Schrodinger Operators, Fundamental Solutions, Riesz Transforms. Introduction Consider the generalized Schrodinger operator (0.1) \Gamma\Delta + in R n ; n 3 where is a nonnegative Radon measure on R n . The main purpose of this paper is to establish optimal upper and lower bounds for the fundamental solution of \Gamma\Delta + under suita...

The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.

Abstract. We prove a Harnack inequality for the positive solutions of a Schrödinger type equation L0 u + V u = 0, where L0 is an operator satisfying the Hörmander’s condition and V belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem. 1.

Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.

curvature flow of graphs

We prove the uniformintegrability of the approximate Green functions of some degenerate elliptic operators in divergence formwith lower order termcoefficients satisfying a Kato type condition. Some further properties of the approximate Green functions of such operators are also established. 1. Introduction. In this paper, we study the approximate Green functions of certain degenerate elliptic operators L on balls in Rn,n>2, when L has the divergence form

Communicated by Heinz Siedentop Abstract. The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. 2000 Mathematics Subject Classification: 35P05, 81Q10