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

Abstract

We consider a metric graph consisting of two edges, one of which has length $\epsilon$ 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 $\epsilon$. In contrast, the negative eigenvalues escape to minus infinity at rates that could be fractional, namely, ${\epsilon }^0$, ${\epsilon }^{-2/3}$ or ${\epsilon }^{-1}$. These rates take place when the corresponding eigenfunction localizes, respectively, only on the long edge, on both edges, or only on the short edge.

17 pages, 1 figure

References in corpus (1)

• A Quantum Graph Approach to Metamaterial Design