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

TL;DR

This paper investigates the behavior of Schr"{o}dinger operators on star graphs with highly scaled potentials near vertices, revealing complex dependence of the limit operator on parameters and potential shape.

Contribution

It establishes the convergence of these operators to a limit that depends intricately on the scaling parameter and potential profile, extending understanding of singularly scaled quantum graphs.

Findings

Operators converge in the uniform resolvent topology as \\varepsilon \\to 0

Limit operator's form depends nontrivially on \\alpha and Q

Potentials approximate derivatives of delta-functions in distributional sense

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.