On large coupling convergence within trace ideals

Abstract

Let $\mathcal E$ and $\mathcal P$ be nonnegative quadratic forms such that $\mathcal E + b \mathcal P$ is closed and densely defined for every nonnegative real number $b$. Let $H_b$ be the selfadjoint operator associated with $\mathcal E + b\mathcal P.$ By Kato's monotone convergence theorem, there exists an operator $L$ such that $(H_b+1)^{-1}$ converges to $L$ strongly, as $b$ tends to infinity. We give a condition which is sufficient in order that the operators $(H_b+1)^{-1}$ converge w.r.t. the trace norm with convergence rate $O(1/b)$. As an application we discuss trace norm resolvent convergence of Schrodinger operators with point interactions.