Convergence of Resolvent Differences

Abstract

In this chapter we investigate the resolvent convergence of Feller operators additively perturbed by potentials on sets Г ⊂ E of height ß. Let $K_0 \dot + V$ be the Feller operator as introduced in Definition 2.8 (see Theorem 2.5.(a) as well). Here K₀is the free Feller operator (see Definition 1.3) and V is a Kato-Feller potential given in Definition 2.1. The operator $K_0 \dot + V$ is perturbed by a potential of the form ß1Г Here ß is a positive parameter and Г is a closed region inEas discussed in the beginning of section C of Chapter 2. This means that we introduce a new self-adjoint operatorKßgiven by.

$$ K_\beta = K_0 \dot + V + \beta 1_\Gamma $$

((7.1))

in L²(E, m) with dom (K_β) = dom ($K_0 \dot + V$) We also introduced the operator $(K_0 \dot + V)_\Sigma$ in Definition 2.25 as the generator of the Dirichlet semigroup on L²(∑, m):

$$ [\exp ( - t (K_0 \dot + V)_\Sigma )f] (x) = \mathbb{E}_x \left[ {\exp \left( { - \int_0^t {V(X(u))du} } \right)f(X(t)):S > t} \right]. $$