Large coupling convergence with negative perturbations

Let EE and PP be nonnegative quadratic forms in the Hilbert space HH. Suppose that for every β≥0β≥0 the forms E±βPE±βP are densely defined, lower semi-bounded, and closed. Let Hβ± be the self-adjoint operator associated with E±βPE±βP and R∞:=limβ⟶∞(Hβ++1)−1. We discuss convergence of (Hβ−+1)−1 towards R∞R∞ strongly, uniformly and with respect to the trace and Hilbert–Schmidt norms. We also give estimates of the speed of convergence for the indicated norms. Conditions ensuring trace and Hilbert–Schmidt norm convergence are also given as well as a condition ensuring trace norm convergence with rate proportional to β−1β−1. Various examples supporting our results are elaborated.