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

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 $O(1\mathrm{/}k)$ around the sequence of natural numbers $k=1,2,\dots$, and we compute the first three terms in the asymptotic distribution of the eigenvalues within the clusters, as $k\to \mathrm{\infty }$ (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.