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

TL;DR

This paper analyzes the asymptotic spectral distribution of the Dirichlet to Neumann operator on the two-sphere, revealing cluster structures and computing detailed eigenvalue asymptotics using Berezin symbols and Toeplitz operator calculus.

Contribution

It provides the first three terms of the eigenvalue asymptotics within spectral clusters for the Dirichlet to Neumann operator on the sphere, combining Berezin symbol analysis and Toeplitz calculus.

Findings

Eigenvalues form clusters of size O(1/k) around natural numbers

Computed the first three asymptotic terms of eigenvalue distribution

Established a symbolic calculus approach for the Berezin-Toeplitz operators

Abstract

We study the spectrum of the Dirichlet to Neumann operator of the two-sphere associated to a Schr\"odinger 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.