On $\delta'$-like potential scattering on star graphs

TL;DR

This paper investigates the scattering behavior of $ abla'$-like potentials on star graphs, revealing that such potentials are generally opaque with specific resonant cases allowing partial transmission, extending known results from line to graph structures.

Contribution

It extends the analysis of $ abla'$-like potentials from line to star graphs, demonstrating generic opacity and identifying resonant intensities where partial transmission occurs.

Findings

Most $ abla'$-like potentials on star graphs are opaque.

A countable set of resonant intensities allows partial transmission.

Resonances are characterized by the spectrum of an auxiliary Sturm-Liouville problem.

Abstract

We discuss the potential scattering on the noncompact star graph. The Schr\"{o}dinger operator with the short-range potential localizing in a neighborhood of the graph vertex is considered. We study the asymptotic behavior the corresponding scattering matrix in the zero-range limit. It has been known for a long time that in dimension 1 there is no non-trivial Hamiltonian with the distributional potential $\delta'$, i.e., the $\delta'$ potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials $\alpha\eps^{-2}Q(x/\eps)$ approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as $\eps\to0$. We extend these results to star graphs with the point interaction, which is an analogue of $\delta'$ potential on the line. We prove that generically such a potential on the graph is opaque. We also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs. This set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary Sturm-Liouville problem associated with $Q$ on the graph.