On \(\delta'\)-like potential scattering on star graphs. (English) Zbl 1202.81201

Summary: 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 \(\delta'\), 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 \(\alpha \varepsilon^{-2}Q(x/\varepsilon)\) approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as \(\varepsilon\to 0\). We extend these results to star graphs with the point interaction, which is an analog of the \(\delta'\) 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.


81U05 \(2\)-body potential quantum scattering theory
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34B24 Sturm-Liouville theory
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