Band asymptotics with non-smooth potentials

Abstract

Let s” denote the n-dimensional sphere with its canonical Riemannian metric, and A its Laplace-Beltrami operator. If q is a real-valued function on S”, we can consider the self-adjoint operator S = A + q. For the case when q is smooth, Weinstein proved in [W] the following very beautiful theorem on the asymptotic behavior of the spectrum of S. Let dk = k(k + n 1) be the kth eigenvalue of A, and let d, be its multiplicity. By the minimax principle, the eigenvalues of S can be written in the form & = Ak + P~,~, i= l,..., dk, where IpcLk,il is bounded by a constant independent of k and i. Let 0 denote the space of oriented, closed geodesics on S”. and let dp be the unique probability measure on 0 which is rotationally invariant. By integrating q along closed geodesics we obtain a smooth function, S, on 0, which we call the x-ray transform of q. Then Weinstein’s theorem is