On norm resolvent convergence of Schr\"odinger operators with $\delta'$-like potentials

TL;DR

This paper establishes the uniform resolvent convergence of Schrödinger operators with regularized potentials approaching a $ abla$-like distribution, revealing non-trivial limit operators with interface conditions in resonant cases.

Contribution

It proves the resolvent convergence of Schrödinger operators with short-range potentials to a limit operator with interface conditions, clarifying the case of resonant potentials.

Findings

Resonant potentials lead to non-trivial limit operators with interface conditions.

In non-resonant cases, the limit operator is the free Schrödinger operator with Dirichlet boundary.

Partial wave transmission occurs in the resonant case, indicating non-trivial scattering behavior.

Abstract

We address the problem on the right definition of the Schroedinger operator with potential $\delta'$, where $\delta$ is the Dirac delta-function. Namely, we prove the uniform resolvent convergence of a family of Schroedinger operators with regularized short-range potentials $\epsilon^{-2}V(x/\epsilon)$ tending to $\delta'$ in the distributional sense as $\epsilon\to 0$. In 1986, P. Seba claimed that the limit coincides with the direct sum of free Schroedinger operators on the semi-axes with the Dirichlet boundary condition at the origin, which implies that in dimension one there is no non-trivial Hamiltonians with potential $\delta'$. Our results demonstrate that, although the above statement is true for many V, for the so-called resonant V the limit operator is defined by the non-trivial interface condition at the origin determined by some spectral characteristics of V. In this resonant case, we show that there is a partial transmission of the wave package for the limiting Hamiltonian.