Weighted Weyl estimates near an elliptic trajectory.

Paul, Thierry, and Uribe, Alejandro. "Weighted Weyl estimates near an elliptic trajectory.." Revista Matemática Iberoamericana 14.1 (1998): 145-165. <http://eudml.org/doc/39545>.

@article{Paul1998,
abstract = {Let Ψjh and Ejh denote the eigenfunctions and eigenvalues of a Schrödinger-type operator Hh with discrete spectrum. Let Ψ(x,ξ) be a coherent state centered at a point (x,ξ) belonging to an elliptic periodic orbit, γ of action Sγ and Maslov index σγ. We consider weighted Weyl estimates of the following form: we study the asymptotics, as h → 0 along any sequenceh = Sγ / (2πl - α + σγ), l ∈ N, α ∈ R fixed, ofΣ|Ej - E| ≤ ch |(Ψ(x,ξ), Ψjh)|2.We prove that the asymptotics depend strongly on α-dependent arithmetical properties of c and on the angles θ of the Poincaré mapping of γ. 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.},
author = {Paul, Thierry, Uribe, Alejandro},
journal = {Revista Matemática Iberoamericana},
keywords = {Ecuación de Schrödinger; Autofunciones; Autovalores; Sistema hamiltoniano; Trayectoria elíptica; elliptic trajectory; Weyl estimates; Schrödinger operator; discrete spectrum; Poincaré mapping; Fourier transform},
language = {eng},
number = {1},
pages = {145-165},
title = {Weighted Weyl estimates near an elliptic trajectory.},
url = {http://eudml.org/doc/39545},
volume = {14},
year = {1998},
}

TY - JOUR
AU - Paul, Thierry
AU - Uribe, Alejandro
TI - Weighted Weyl estimates near an elliptic trajectory.
JO - Revista Matemática Iberoamericana
PY - 1998
VL - 14
IS - 1
SP - 145
EP - 165
AB - Let Ψjh and Ejh denote the eigenfunctions and eigenvalues of a Schrödinger-type operator Hh with discrete spectrum. Let Ψ(x,ξ) be a coherent state centered at a point (x,ξ) belonging to an elliptic periodic orbit, γ of action Sγ and Maslov index σγ. We consider weighted Weyl estimates of the following form: we study the asymptotics, as h → 0 along any sequenceh = Sγ / (2πl - α + σγ), l ∈ N, α ∈ R fixed, ofΣ|Ej - E| ≤ ch |(Ψ(x,ξ), Ψjh)|2.We prove that the asymptotics depend strongly on α-dependent arithmetical properties of c and on the angles θ of the Poincaré mapping of γ. 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.
LA - eng
KW - Ecuación de Schrödinger; Autofunciones; Autovalores; Sistema hamiltoniano; Trayectoria elíptica; elliptic trajectory; Weyl estimates; Schrödinger operator; discrete spectrum; Poincaré mapping; Fourier transform
UR - http://eudml.org/doc/39545
ER -