On Persson's Theorem in Local Dirichiet Spaces

Abstract. Given a strongly local, regular and irreducible Dirichlet form 1, we prove a version of Perssons theorem concerning the variational characterization of the bottom of the essential spectrum of the generator H of E. Such a result is then used to prove LP -exponential decay of the "small eigenfunctions" of H. Keywords: Local Dirichlet spaces, Perssons theorem, small eigenfunctions AMS subject classification: Primary 31 C 25, secondary 35 B 05, 47 A 11, 58 G 25 1. Introduction Given a positive second-order differential operator of elliptic type in divergence form H 0 = - i1 (aij) axi on an open and connected domain D C R, with Dirichiet boundary conditions and suitable assumptions on the matrix ( a t,,), one can consider the bilinear form a(u,v) = (H 0 1/2 u,H 0 1/2 v)L2(D) defined for u, v E D(a) = Dom(H 2 ). This bilinear form mirrors many of the features of the operator H and enjoices several interesting properties which can be considered in a more abstract setting as the defining properties of mathematical objects which are known as