Weighted Weyl estimates near an elliptic trajectory

Abstract

Let ?jh and Ejh denote the eigenfunctions and eigenvalues of a Schrödinger-type operator Hh with discrete spectrum. Let ?(x,?) be a coherent state centered at a point (x,?) belonging to an elliptic periodic orbit, ? of action S? and Maslov index s?. We consider weighted Weyl estimates of the following form: we study the asymptotics, as h ? 0 along any sequence h = S? / (2pl - a + s?), l Î N, a Î R fixed, of S|Ej - E| = ch |(?(x,?), ?jh)|2. We prove that the asymptotics depend strongly on a-dependent arithmetical properties of c and on the angles ? of the Poincaré mapping of ?. In particular, under irrationality assumptions on the angles, the limit exists for a non-open set of full measure of c's. We also study the regularity of the limit as a function of c