Schroedinger operators with (\alpha\delta'+\beta \delta)-like potentials: norm resolvent convergence and solvable models

TL;DR

This paper studies the convergence of a family of one-dimensional Schrödinger operators with scaled potentials to limit operators, revealing shape-dependent limits and the impossibility of a single self-adjoint operator for certain pseudo-Hamiltonians.

Contribution

It proves norm resolvent convergence of scaled Schrödinger operators with elta' and elta-like potentials and shows the shape-dependent nature of the limit operators.

Findings

Limit operators depend on the shape of the potentials.

It is impossible to assign a unique self-adjoint operator to the pseudo-Hamiltonian -D^2 + elta' + elta.

The convergence results do not rely on distributional convergence of potentials.

Abstract

For real functions \Phi and \Psi that are integrable and compactly supported, we prove the norm resolvent convergence, as \epsilon\ goes to 0, of a family S(\epsilon) of one-dimensional Schroedinger operators on the line of the form S(\epsilon)= -D^2 + \alpha \epsilon^{-2} \Phi(x/\epsilon) + \beta \epsilon^{-1} \Psi(x/\epsilon). The limit results are shape-dependent: without reference to the convergence of potentials in the sense of distributions the limit operator S(0) exists and strongly depends on the pair (\Phi,\Psi). We show that it is impossible to assign just one self-adjoint operator to the pseudo-Hamiltonian -D^2 + \alpha \delta'(x) + \beta \delta(x), which is a symbolic notation only for a wide variety of quantum systems with quite different properties.