# Weighted Weyl estimates near an elliptic trajectory.

Revista Matemática Iberoamericana (1998)

- Volume: 14, Issue: 1, page 145-165
- ISSN: 0213-2230

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topPaul, 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 -

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