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arXiv · 1007.0398

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

Stepan Man'ko
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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.

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Stepan Man'ko. 2010-07-02. On $\delta'$-like potential scattering on star graphs. https://doi.org/10.1088/1751-8113/43/44/445304

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