Eigenvalue falls in thin broken quantum strips

TL;DR

This paper investigates how the eigenvalues of the Dirichlet Laplacian in thin broken quantum strips depend on the angle, revealing that at certain angles eigenvalues rapidly decrease below a critical threshold, with detailed asymptotic analysis.

Contribution

It provides the first asymptotic expansion of eigenvalues in thin broken strips and characterizes the eigenvalue diving phenomenon at specific angles.

Findings

Eigenvalues dive below the threshold at particular angles.

Asymptotic expansion of eigenvalues as strip thickness tends to zero.

Eigenvalue behavior varies with the angle, milder at zero degrees.

Abstract

We are interesting in the spectrum of the Dirichlet Laplacian in thin broken strips with angle $\alpha$. Playing with symmetries, this leads us to investigate spectral problems for the Laplace operator with mixed boundary conditions in thin trapezoids characterized by a parameter $\varepsilon$ small. We give an asymptotic expansion of the first eigenvalues and corresponding eigenfunctions as $\varepsilon$ tends to zero. The new point in this work is to study the dependence with respect to $\alpha$. We show that for a small fixed $\varepsilon>0$, at certain particular angles $\alpha^\star_k$, $k=0,1,\dots$, that we characterize, an eigenvalue dives, i.e. moves down rapidly, below the normalized threshold $\pi^2/\varepsilon^2$ as $\alpha>0$ increases. We describe the way the eigenvalue dives below $\pi^2/\varepsilon^2$ and prove that the phenomenon is milder at $\alpha^\star_0=0$ than at $\alpha^\star_k$ for $k\ge1$.