Large coupling limit for delta sphere interactions

Article type: Research Article

Authors: BelHadjAli, Hichem^; | BenAmor, Ali

Affiliations: Institut Préparatoire aux Etudes d'Ingénieurs de Nabeul, Nabeul, Tunisia. E-mail: [email protected] | Faculty of Sciences of Gafsa, Gafsa, Tunisia. E-mail: [email protected]

Note: [] Corresponding author: Hichem BelHadjAli, Institut Préparatoire aux Etudes d'Ingénieurs de Nabeul, Merazka, 8000 Nabeul, Tunisia. E-mail: [email protected].

Abstract: Let Hβ±:=−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 Dβ±:=(−Δ+1)−1−(Hβ±+1)−1 and D∞ be the strong limit w.r.t. β of the operators Dβ+. For suitable sequence (βn) we prove uniform convergence of Dβn− towards D∞ with optimal rate, namely βn−1. 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.

Keywords: rate of convergence, Schatten–von Neumann convergence, uniform convergence

DOI: 10.3233/ASY-2011-1085

Journal: Asymptotic Analysis, vol. 79, no. 1-2, pp. 1-15, 2012