Spectral convergence of non-compact quasi-one-dimensional spaces

TL;DR

This paper proves that the spectral properties of non-compact, graph-like manifolds converge to those of a metric graph, including eigenvalues and spectral gaps, under natural uniformity conditions.

Contribution

It establishes spectral convergence of Laplacians on non-compact, graph-like manifolds to those on metric graphs, including resolvents, eigenfunctions, and spectral projections, in a general abstract setting.

Findings

Spectral gaps can occur in the essential spectrum of the manifolds.

Discrete eigenvalues can appear within spectral gaps.

Manifolds can approach fractal spectra under the studied limits.

Abstract

We consider a family of non-compact manifolds $X_\eps$ (``graph-like manifolds'') approaching a metric graph $X_0$ and establish convergence results of the related natural operators, namely the (Neumann) Laplacian $\laplacian {X_\eps}$ and the generalised Neumann (Kirchhoff) Laplacian $\laplacian {X_0}$ on the metric graph. In particular, we show the norm convergence of the resolvents, spectral projections and eigenfunctions. As a consequence, the essential and the discrete spectrum converge as well. Neither the manifolds nor the metric graph need to be compact, we only need some natural uniformity assumptions. We provide examples of manifolds having spectral gaps in the essential spectrum, discrete eigenvalues in the gaps or even manifolds approaching a fractal spectrum. The convergence results will be given in a completely abstract setting dealing with operators acting in different spaces, applicable also in other geometric situations.