Weighted Weyl estimates near an elliptic trajectory

  • Thierry Paul

    Ecole Polytechnique, Palaiseau, France
  • Alejandro Uribe

    University of Michigan, Ann Arbor, USA


Let and denote the eigenfunctions and eigenvalues of a Schrödinger type operator with discrete spectrum. Let be a coherent state centered at a point  belonging to an elliptic periodic orbit, of action  and Maslov index . We consider "weighted Weyl estimates" of the following form: we study the asymptotics, as along any sequence

fixed, of

We prove that the asymptotics depend strongly on -dependent arithmetical properties of 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 's. We also study the regularity of the limit as a function of .

Cite this article

Thierry Paul, Alejandro Uribe, Weighted Weyl estimates near an elliptic trajectory. Rev. Mat. Iberoam. 14 (1998), no. 1, pp. 145–165

DOI 10.4171/RMI/238