The Robin problem on rectangles

TL;DR

This paper investigates the Robin spectral properties of rectangles, revealing how the spectrum's multiplicities, gaps, and pair correlations depend on the rectangle's aspect ratio and Robin parameter, with notable differences from Neumann spectra.

Contribution

It provides new insights into the Robin spectrum of rectangles, including multiplicity behavior, gap bounds, and Poissonian pair correlations, especially highlighting differences from Neumann spectra.

Findings

No multiplicities for Robin spectrum of the square with small Robin parameter

Established bounds for Robin-Neumann spectral gaps

Proved Poissonian pair correlation for Robin spectrum on Diophantine rectangles

Abstract

We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum, depending on the Diophantine nature of the aspect ratio. In particular, it is shown that for the square, unlike the case of Neumann eigenvalues where there are unbounded multiplicities of arithmetic origin, there are no multiplicities in the Robin spectrum for sufficiently small (but nonzero) Robin parameter except a systematic symmetry. In addition, uniform lower and upper bounds are established for the Robin-Neumann gaps in terms of their limiting mean spacing. Finally, that the pair correlation function of the Robin spectrum on a Diophantine rectangle is shown to be Poissonian.