The Robin problem on rectangles
- 1 November 2021
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 62 (11), 113503
- https://doi.org/10.1063/5.0061763
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 a 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, the pair correlation function of the Robin spectrum on a Diophantine rectangle is shown to be Poissonian.Funding Information
- H2020 European Research Council (786758)
- Israel Science Foundation (1881/20)
This publication has 5 references indexed in Scilit:
- The Robin Laplacian—Spectral conjectures, rectangular theoremsJournal of Mathematical Physics, 2019
- On the eigenvalues of a Robin problem with a large parameterMathematica Bohemica, 2014
- Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-toriAnnals of Mathematics, 2005
- Level clustering in the regular spectrumProceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences, 1977
- Some problems of diophantine approximation: Part I. The fractional part of nkθActa Mathematica, 1914