On ${\delta }^{\prime }$-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 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 ${\delta }^{\prime }$, i.e., the ${\delta }^{\prime }$ potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials $α\eps^{-2}Q(x/\eps)$ 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 ${\delta }^{\prime }$ 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.

16 pages, 2 figures

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Cited by in corpus (8)

• 1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials
• Single point potentials with total resonant tunneling
• Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices
• Quantum graphs: Coulomb-type potentials and exactly solvable models
• Some remarks on 1D Schrödinger operators with localized magnetic and electric potentials
• On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\prime }$-like potentials
• Hidden symmetries in non-self-adjoint graphs
• Conditions for realizing one-point interactions from a multi-layer structure model