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.