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Weighted Weyl estimates near an elliptic trajectory.

Thierry Paul, Alejandro Uribe (1998)

Revista Matemática Iberoamericana

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...

Weyl formula with optimal remainder estimate of some elastic networks and applications

Kaïs Ammari, Mouez Dimassi (2010)

Bulletin de la Société Mathématique de France

We consider a network of vibrating elastic strings and Euler-Bernoulli beams. Using a generalized Poisson formula and some Tauberian theorem, we give a Weyl formula with optimal remainder estimate. As a consequence we prove some observability and stabilization results.

Weyl type upper bounds on the number of resonances near the real axis for trapped systems

Plamen Stefanov (2001)

Journées équations aux dérivées partielles

We study semiclassical resonances in a box Ω ( h ) of height h N , N ≫ 1 . We show that the semiclassical wave front set of the resonant states (including the “generalized eigenfunctions”) is contained in the set 𝒯 of the trapped bicharacteristics. We also show that for a suitable self-adjoint reference operator P # ( h ) with discrete spectrum the number of resonances in Ω ( h ) is bounded by the number of eigenvalues of P # ( h ) in an interval a bit larger than the projection of Ω ( h ) on the real line. As an application, we prove a...

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