On the Robin spectrum for the equilateral triangle

Zeév Rudnick*, Igor Wigman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The equilateral triangle is one of the few planar domains where the Dirichlet and Neumann eigenvalue problems were explicitly determined, by Lamé 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.

Original languageEnglish
Article number254004
JournalJournal of Physics A: Mathematical and Theoretical
Volume55
Issue number25
DOIs
StatePublished - 24 Jun 2022

Funding

FundersFunder number
Horizon 2020 Framework Programme786758
European Research Council
Israel Science Foundation1881/20

    Keywords

    • Laplacian
    • Robin boundary condition
    • Robin-Neumann gaps
    • equilateral triangle
    • level spacing distribution

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