Mathematik-Bericht 2009/10 Large coupling convergence: Overview and new results

Let E and P be nonnegative quadratic forms in a Hilbert space H and assume that E + bP is densely defined and closed for every b ≥ 0. For every b > 0 let Hb be the self-adjoint operator associated with E + bP in the sense of Kato’s representation theorem. By Kato’s monotone convergence theorem, the operators (Hb + 1) −1 converge strongly to an operator L, as b tends to infinity. Let k ∈ N. We give conditions which are sufficient for convergence of (Hb + 1) −k − L w.r.t. the operator norm and convergence w.r.t. to a Schatten class norm, respectively. Moreover we derive a variety of results on the rate of convergence. We discuss in detail the case when E is a regular Dirichlet form and P a killing term.