On δ′-like potential scattering on star graphs

Abstract

We discuss the potential scattering on the noncompact star graph. The Schrödinger operator with the short-range potential localized in a neighborhood of the graph vertex is considered. We study the asymptotic behavior of 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 δ′, i.e. the δ′ potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials αε−2Q(x/ε) approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as ε → 0. We extend these results to star graphs with the point interaction, which is an analog of the δ′ 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.