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.

this paper, we obtain analogues of Harnack's inequality for L-harmonic functions of operators in the class L(#, d, n). We use the technique introduced by Krylov for estimating the oscillation of a harmonic function on bounded sets [3]. The main results are given in Section 2. Section 3 is devoted to proofs and auxiliary results. 2. Main Results

Abstract. We prove a Harnack inequality for the positive solutions of ultraparabolic equations of the type L0 u + V u = 0, where L0 is a linear second order hypoelliptic operator 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.

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