On the Robin spectrum for the equilateral triangle

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