Smooth approximation of singular perturbations
One of the main advantages gained by the modern use of point interactions in mathematical physics is the exact solvability of the Schrödinger equation and, simultaneously, the possibility of understanding the underlying formulae simply in the language of the essentially self-adjoint extensions. Unfortunately, the empirically founded interpretation of real systems often prefers the smooth-function form of the interactions. The paper under review shows in which sense the two languages may be used as equivalent. More precisely, in one dimension and for both finite and infinite numbers \(N\) of ``points'', the authors show how the mutual equivalence can be understood in the strong resolvent sense, and also how one can keep the large-\(N\) convergence under strict control. Noticing, in such a context, that the recent development in the field moves towards a weakening of the Hermiticity to mere pseudo-Hermiticity [see, e.g., \textit{S. Albeverio, S. M. Fei} and \textit{P. Kurasov}, Lett. Math. Phys 59, 227-242 (2002; Zbl 1053.81026)], one can only hope that the rigorous studies of the presented type will be also extended in this direction in the nearest future.
scientific article; zbMATH DE number 1957231
scientific article