Leaky quantum graphs: approximations by point interaction Hamiltonians

Abstract

We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma )$ in $L^2({\mathbb{R}}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an appropriate arrangement of the $\delta$ potentials. The result is illustrated on finding the spectral properties in cases when $\Gamma$ is a ring or a star. Furthermore, we use this method to indicate that scattering on an infinite curve $\Gamma$ which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunneling or repeated reflections.

LaTeX 2e, 31 pages with 18 postscript figures

References in corpus (2)

• Spectra of soft ring graphs
• Bound states in point-interaction star-graphs

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