On the Robin spectrum for the hemisphere

TL;DR

This paper investigates the spectral properties of the Laplacian with Robin boundary conditions on the hemisphere, revealing clustering behavior, bounds on eigenvalue gaps, and the distribution of eigenvalue spacings.

Contribution

It provides new bounds on eigenvalue gaps, characterizes multiplicities, and establishes the limiting spacing distribution for Robin eigenvalues on the hemisphere.

Findings

Eigenvalues cluster around the Neumann spectrum.

Eigenvalue gaps are unbounded with positive Robin parameter.

The spacing distribution converges to a delta function at zero.

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 around 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.