JournalsrmiVol. 14, No. 1pp. 145–165

Weighted Weyl estimates near an elliptic trajectory

  • Thierry Paul

    Ecole Polytechnique, Palaiseau, France
  • Alejandro Uribe

    University of Michigan, Ann Arbor, USA
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Abstract

Let ψj\psi^\hbar_j and EjE^\hbar_j denote the eigenfunctions and eigenvalues of a Schrödinger type operator HH_\hbar with discrete spectrum. Let ψ(x,ξ)\psi_{(x, \xi)} be a coherent state centered at a point (x,ξ)(x, \xi) belonging to an elliptic periodic orbit, γ\gamma of action SγS\gamma and Maslov index σγ\sigma_\gamma. We consider "weighted Weyl estimates" of the following form: we study the asymptotics, as 0\hbar \longrightarrow 0 along any sequence

=Sγ2πlα+σγ,\hbar = \frac {S_\gamma} {2\pi l – \alpha + \sigma_\gamma},

lN,αRl \in \mathbb N, \alpha \in \mathbb R fixed, of

EjEc(ψ(x,ξ),ψjh)2.\sum_{|Ej–E|≤c\hbar} | (\psi_{(x, \xi)}, \psi^h_j) |^2.

We prove that the asymptotics depend strongly on α\alpha-dependent arithmetical properties of cc and on the angles θ\theta of the Poincaré mapping of γ\gamma. In particular, under irrationality assumptions on the angles, the limit exists for a non-open set of full measure of cc's. We also study the regularity of the limit as a function of cc.

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