Large coupling limit for delta sphere interactions

Abstract

Let H ± β := −Δ ± βδ(|x| = R), be the quantum mechanical Hamiltonian describing a delta interaction living on a sphere of radius R in three dimensions, with strength β> 0. Let D ± := (−Δ + 1) −1 − (H ± + 1) −1 and D∞ be the strong limit w.r.t. β of the operators D + . For suitable sequence (βn) we prove uniform convergence of D − βn towards D∞ with optimal rate, namely β −1 n . Convergence of the above mentioned operators against each other within Schatten-von Neumann ideals Sp, is established as well and the rate of convergence (depending on p) is determined. At the final step we discuss aspects of differences between negative and positive perturbations at large coupling.