Schr\"{o}dinger operators on star graphs with singularly scaled potentials supported near the vertices

Stepan Man'ko

doi:10.1063/1.4769425

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arXiv · 1206.1263

Schr\"{o}dinger operators on star graphs with singularly scaled potentials supported near the vertices

Stepan Man'ko

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Abstract

We study Schr\"{o}dinger operators on star metric graphs with potentials of the form $\alpha\varepsilon^{-2}Q(\varepsilon^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions as $\varepsilon\to0$ the first derivative of the Dirac delta-function. We establish the convergence of the Schr\"{o}dinger operators in the uniform resolvent topology and show that the limit operator depends on $\alpha$ and $Q$ in a very nontrivial way.

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BibTeX RIS

Stepan Man'ko. 2012-06-06. Schr\"{o}dinger operators on star graphs with singularly scaled potentials supported near the vertices. https://doi.org/10.1063/1.4769425

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