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Leaky quantum graphs: approximations by point interaction Hamiltonians

P. Exner·K. Nemcova
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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.

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P. Exner, K. Nemcova. 2003-06-12. Leaky quantum graphs: approximations by point interaction Hamiltonians. https://doi.org/10.1088/0305-4470/36/40/004

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