Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems.
Weighted Weyl estimates near an elliptic trajectory.
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...