Convergence of spectra of uniformly fattened open book structures

TL;DR

This paper proves that the spectrum of the Neumann Laplacian on a fattened open book structure converges to the spectrum of a differential operator on the original stratified surface as the fattening parameter approaches zero, extending previous results to singular structures.

Contribution

It extends spectral convergence results from fattened graphs and smooth manifolds to stratified 2D varieties with singularities, addressing the complexities introduced by lower-dimensional strata.

Findings

Spectral convergence of Neumann Laplacian on fattened structures

Extension of results to stratified 2D varieties with singularities

Application relevance to quantum graph models

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.