Spectral Convergence of Quasi-One-Dimensional Spaces

Abstract

We consider a family of non-compact manifolds Xε (“graph-like manifolds”) approaching a metric graph X0 and establish convergence results of the related natural operators, namely the (Neumann) Laplacian $$\Delta _{X_{\epsilon}}$$ and the generalized Neumann (Kirchhoff) Laplacian $$\Delta _{X_0 } $$ 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.