# Spectra of "fattened" open book structures

## TL;DR

This paper proves that the spectra of the Neumann Laplacian in thin neighborhoods of 3D branching structures converge to a 2D operator spectrum, extending quantum graph concepts to physical applications.

## Contribution

It introduces a new spectral convergence result for Neumann Laplacians in 3D structures, generalizing quantum graph models to more complex geometries.

## Key findings

- Spectral convergence of Neumann Laplacian established
- Extension of quantum graph analogy to 3D structures
- Applicable to physics and engineering problems

## Abstract

We establish convergence of spectra of Neumann Laplacian in a thin neighborhood of a branching 2D structure in 3D to the spectrum of an appropriately defined operator on the structure itself. This operator is a 2D analog of the well known by now quantum graphs. As in the latter case, such considerations are triggered by various physics and engineering applications.