Perturbation of Dirichlet forms by measures

Abstract

Abstract. Perturbations ofa Dirichlet form 0 by measures/~ are studied. The perturbed form 0 -#-+ /z+ is defined for/~_ in a suitable Kato class and #+ absolutely continuous with respect to capacity. Lp-properties of the corresponding semigroups are derived by approximating #_ by functions. For treating #+, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has Lp -Lq-smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ~ the same is shown to be true for the perturbed semigroup, for a large class of measures.Mathematics Subject Classifications (1991): Primary 47D07; Secondary 31 C25,

Sch’nol’s theorem for strongly local forms

“…In fact, we can even include measures as potentials. Here, we follow the approach from [32,33]. Measure perturbations have been regarded by a number of authors in different contexts, see e.g.…”

Approximation by point potentials in a magnetic field

“…An example of such measure is the Dirac measure, supported by a curve or graph; one can easily check that the condition above holds. By [SV,theorem 3.1] the potential generated by m is (−∆)-form bounded with infinitesimally small relative bound. In order to define H γm properly, we need a similar form-boundedness with respect to H 0 .…”

Eigenfunction expansions for generators of Dirichlet forms

“…We want to discuss the question whether H can be replaced by H þ V where V is a perturbation with some mild regularity assumption. The answer will be yes and the necessary analysis is available from [37]. We refer to the literature quoted there and start to introduce the necessary notions.…”

“…For every n þ A M 0 (in contrast to [37] we denote measures by n rather than m in order to distinguish them from the spectral measure m from Section 2) we get an operator H þ n þ by form methods. This operator is defined on L 2 ðY Þ where Y H X might be smaller than X but the resolvent and the semigroup of H act on L 2 ðX Þ (they are 0 on L 2 ðX nY Þ).…”

“…This operator is defined on L 2 ðY Þ where Y H X might be smaller than X but the resolvent and the semigroup of H act on L 2 ðX Þ (they are 0 on L 2 ðX nY Þ). Moreover we can add a negative n À from the generalized Kato classŜ S K provided the Kato constant c K ðn À Þ < 1; see [37] for details. In that case H þ n þ À n À is bounded below and, due to the results from [37] (A1) is satisfied for n þ À n À whenever it is satisfied for H. To construct an eigenfunction expansion we work with the intrinsic metric of unperturbed operator H and get:…”