On the Robin spectrum for the equilateral triangle

TL;DR

This paper investigates the Robin eigenvalue spectrum of the equilateral triangle, providing bounds on spectral gaps, analyzing multiplicities, and exploring the distribution of eigenvalue spacings, revealing complex spectral properties.

Contribution

It offers new bounds on Robin-Neumann gaps, studies spectral multiplicities, and characterizes the eigenvalue spacing distribution for the equilateral triangle's Robin spectrum.

Findings

Robin-Neumann gaps are uniformly bounded by their mean value.

The spectrum exhibits arbitrarily large and small gaps.

The eigenvalue spacing distribution is a delta function at zero.

Abstract

The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lam\'e in 1833, despite not admitting separation of variables. In this paper, we study the Robin spectrum of the equilateral triangle, which was determined by McCartin in 2004 in terms of a system of transcendental coupled secular equations. We give uniform upper bounds for the Robin-Neumann gaps, showing that they are bounded by their limiting mean value, which is hence an almost sure bound. The spectrum admits a systematic double multiplicity, and after removing it we study the gaps in the resulting desymmetrized spectrum. We show a spectral gap property, that there are arbitrarily large gaps, and also arbitrarily small ones, moreover that the nearest neighbour spacing distribution of the desymmetrized spectrum is a delta function at the origin. We show that for sufficiently small Robin parameter, the desymmetrized spectrum is simple.