Large coupling convergence with negative perturbations

Let E and P be nonnegative quadratic forms in the Hilbert space H . Suppose that for every β ≥ 0 the forms E ± β P are densely defined, lower semi-bounded, and closed. Let H β ± be the self-adjoint operator associated with E ± β P and R ∞ : = lim β ⟶ ∞ ( H β + + 1 ) − 1 . We discuss convergence of ( H β − + 1 ) − 1 towards 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 − 1 . Various examples supporting our results are elaborated.