{'fraction_non_alphanumeric': 0.04534005037783375, 'fraction_numerical': 0.013853904282115869, 'mean_word_length': 3.8181818181818183, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
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.