Exotic eigenvalues and analytic resolvent for a graph with a shrinking edge

TL;DR

This paper analyzes how the spectrum and resolvent of a Laplacian on a two-edge graph behave as one edge shrinks to zero, revealing analytic dependence of the resolvent and fractional escape rates of eigenvalues.

Contribution

It demonstrates the analytic dependence of the resolvent on the shrinking parameter and characterizes the fractional rates at which eigenvalues diverge.

Findings

Resolvent depends analytically on the shrinking parameter.

Negative eigenvalues escape to minus infinity at fractional rates.

Eigenfunction localization determines the escape rate.

Abstract

We consider a metric graph consisting of two edges, one of which has length $\varepsilon$ which we send to zero. On this graph we study the resolvent and spectrum of the Laplacian subject to a general vertex condition at the connecting vertex. Despite the singular nature of the perturbation (by a short edge), we find that the resolvent depends analytically on the parameter $\varepsilon$. In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, $\varepsilon^0$, $\varepsilon^{-2/3}$ or $\varepsilon^{-1}$. These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge.