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_β^±:=(−Δ+1)⁻¹−(H_β^±+1)⁻¹ and D_∞ be the strong limit w.r.t. β of the operators D_β⁺. For suitable sequence (β_n) we prove uniform convergence of D_βn⁻ towards D_∞ with optimal rate, namely β_n⁻¹. Convergence of the above mentioned operators against each other within Schatten–von Neumann ideals S_p, 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.