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arXiv · 2412.16652

Spectral cluster asymptotics of the Dirichlet to Neumann operator on the two-sphere

S. Pérez-Esteva·A. Uribe·C. Villegas-Blas
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Abstract

We study the spectrum of the Dirichlet to Neumann operator of the two-sphere associated to a Schrödinger operator in the unit ball. The spectrum forms clusters of size $O(1/k)$ around the sequence of natural numbers $k=1,2,\ldots$, and we compute the first three terms in the asymptotic distribution of the eigenvalues within the clusters, as $k\to\infty$ (band invariants). There are two independent aspects of the proof. The first is a study of the Berezin symbol of the Dirichlet to Neumann operator, which arises after one applies the averaging method. The second is the use of a symbolic calculus of Berezin-Toeplitz operators on the manifold of closed geodesics of the sphere.

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S. Pérez-Esteva, A. Uribe, C. Villegas-Blas. 2024-12-21. Spectral cluster asymptotics of the Dirichlet to Neumann operator on the two-sphere. https://arxiv.org/abs/2412.16652

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