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On the Robin spectrum for the hemisphere

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Zeev Rudnick, Igor Wigman

Original languageEnglish
JournalAnnales mathematiques du quebec
Published24 Dec 2020

King's Authors

Abstract

We study the spectrum of the Laplacian on the hemisphere with Robin boundary conditions. It is found that the eigenvalues fall into small clusters close to the Neumann spectrum, and satisfy a Szeg\H{o} type limit theorem. Sharp upper and lower bounds for the gaps between the Robin and Neumann eigenvalues are derived, showing in particular that these are unbounded. Further, it is shown that except for a systematic double multiplicity, there are no multiplicities in the spectrum as soon as the Robin parameter is positive, unlike the Neumann case which is highly degenerate. Finally, the limiting spacing distribution of the desymmetrized spectrum is proved to be the delta function at the origin.

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