Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices

Abstract

We study Schrödinger operators on star metric graphs with potentials of the form $\alpha {\epsilon }^{-2}Q({\epsilon }^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions as $\epsilon \to 0$ the first derivative of the Dirac delta-function. We establish the convergence of the Schrödinger operators in the uniform resolvent topology and show that the limit operator depends on $\alpha$ and $Q$ in a very nontrivial way.

References in corpus (2)

• Quantum wires with magnetic fluxes
• Coupling in the singular limit of thin quantum waveguides

Cited by in corpus (6)

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• Single point potentials with total resonant tunneling
• Approximations of quantum-graph vertex couplings by singularly scaled potentials
• Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions
• Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core
• Conditions for realizing one-point interactions from a multi-layer structure model