Spectral convergence of quasi-one-dimensional spaces. (English) Zbl 1187.81124
Summary: We consider a family of non-compact manifolds \(X_\varepsilon\) (“graph-like manifolds”) approaching a metric graph \(X_{0}\) and establish convergence results of the related natural operators, namely the (Neumann) Laplacian \(\Delta_{X_\varepsilon}\) 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.
MSC:
81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
47A10 | Spectrum, resolvent |
47A55 | Perturbation theory of linear operators |
47N50 | Applications of operator theory in the physical sciences |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |