Convergence of spectra of uniformly fattened open book structures

Abstract

We consider a compact $C^{\infty }$ stratified 2D variety $M$ in ${\mathbb{R}}^3$ and its $\epsilon$ neighborhood $M_{\epsilon }$, which we call a "fattened open book structure". Assuming absence of zero-dimensional strata, i.e. "corners", we show that the (discrete) spectrum of the Neumann Laplacian in $M_{\epsilon }$ converges when $\epsilon$ tends to $0$ to the spectrum of a differential operator on $M$. Similar results have been obtained before for the case of fattened graphs, i.e. $M$ being one dimensional. In the case of a 2D smooth submanifold $M$, the problem has been studied well. However, having singularities along strata of lower dimensions significantly complicates considerations. As in the quantum graph case, such considerations are triggered by various applications.

46 pages, 9 figures