this paper is a proxy for what deserves a book or at least a very long review article. In attempting to overview such a vast area in a few pages, I have had to focus on the high points. No proofs are given and I have settled for usually quoting the initial or especially significant papers. I have no doubt that I have left out some important papers, and if so, I ask the forgiveness of the reader (and their authors!).

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

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.

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