Olaf Post

Convergence of spectra of graph-like thin manifolds

Abstract We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace–Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.

On the Spectra of Carbon Nano-Structures

An explicit derivation of dispersion relations and spectra for periodic Schrödinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided.

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

Periodic manifolds with spectral gaps

\Delta _{X_{\epsilon}}

EXISTENCE OF SPECTRAL GAPS, COVERING MANIFOLDS AND RESIDUALLY FINITE GROUPS

and the generalized Neumann (Kirchhoff) Laplacian

Anderson localization for radial tree-like random quantum graphs

\Delta _{X_0 }