Convergence of resonances on thin branched quantum wave guides

TL;DR

This paper establishes a general criterion for resolvent convergence of operators in different Hilbert spaces and applies it to show that resonances in thin branched quantum waveguides approximate those of the limiting skeleton graph as the waveguides become thinner.

Contribution

It introduces an abstract resolvent convergence criterion and demonstrates its application to quantum waveguides, linking their resonances to those of the associated skeleton graph.

Findings

Resonances in thin quantum waveguides approximate those of the skeleton graph.

The abstract criterion facilitates analysis of operator convergence in different Hilbert spaces.

Application of exterior complex scaling aids in resonance approximation.

Abstract

We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on $X_\eps$ approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of $X_\eps$.