Abstract
Abstract. Let j and Ej denote the eigenfunctions and eigenvalues of a S c hr odinger-type operator H~with discrete spectrum. Let (x ) bea coherent state centered at a point ( x ) belonging to an elliptic periodic orbit, of action S and Maslov index . We consider \weighted Weyl estimates" of the following form: we study the asymptotics, as~;! 0 along any sequence~= S 2 l; + l 2 N, 2 R xed, of X jE j ;Ej c~j
Abstract
We prove that the asymptotics depend strongly on -dependent arithmetical properties of c and on the angles of the Poincar e mapping of . 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 o f the limit as a function of c.

Weighted Weyl estimates near an elliptic trajectory Thierry Paul and Alejandro Uribe Abstract. Let j and Ej denote the eigenfunctions and eigenvalues of a S c hr odinger-type operator H~with discrete spectrum. Let (x ) bea coherent state centered at a point ( x ) belonging to an elliptic periodic orbit, of action S and Maslov index . We consider \weighted Weyl estimates" of the following form: we study the asymptotics, as~;! 0 along any sequence~= S 2 l; + l 2 N, 2 R xed, of X jE j ;Ej c~j ( (x ) h j ) j 2 
:
We prove that the asymptotics depend strongly on -dependent arithmetical properties of c and on the angles of the Poincar e mapping of . 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 o f the limit as a function of c.
Introduction and results.
Consider a Schr odinger operator H = ;~2 +V (x) with V smooth, either on M = R m (in which case we assume V tends to in nity a t i n nity and therefore H has discrete spectrum) or on a compact Riemannian manifold, M. In 7] we considered \trace formulae" associated to projectors on coherent states in the following sense. For (x ) 2 R 2m and a 2 S (R m ) de ne the coherent state a x as:
Introduction and results.
(1) a (x ) (y) = (y ; x) ( 2 ~) ;3m=4 2 ;m=4 e ;ix =2~ei y=~â y ; x p~ :
Introduction and results.
Here is a cut-o function near zero andâ is the Fourier transform of a, , the left-hand side tends to 0 rapidly in~.) Although the form of the asymptotic expansion does not depend on (x ), the coe cient c 0 (x ) is highly sensitive to the point (x ) being periodicor not with respect to the classical ow. In case (x ) is either not periodic or is on a hyperbolic trajectory, we proved in 7] (using a Tauberian theorem) that, for every c 2 R, remain bounded. The existence of these quasi-modes implies that part of the spectral density o f H concentrates near the quasi-energies de ned by (4) , but this doesn't say anything about E k l QM as j k j;! 1 and does not involve the rest of the spectrum. The results of this paper will indicate that the rescaled localized spectral density
Introduction and results.
(which is the rescaled spectral density microlocalized at the point in phase space (x )) has a certain semiclassical limit whose singularities are indeed precisely the quasi-energies (4), and this time with no restriction on j k j.
Introduction and results.
We will now state our results, valid for more general quantum Hamiltonians: Let H~= P L l=0~l P l (x D x ) where P l is a di erential operator of order l on R m (or M) of principal symbolP 0 l , sub-principal symbol P ;1 l (formally P l is regarded as acting on half-densities) and smooth coe cients. Let H(x ) = P L l=0 P 0 l (x ) and H sub (x ) = P L l=0 P ;1 l (x ) bethe principal and sub-principal symbolsof H~. We assume that P L is elliptic, H is positive, and in case M = R m , that H tends polynomially to in nity at in nity. We will also suppose for simplicity t h a t H sub (x ) = 0 .
Introduction and results.
Let Ej and j denote the eigenvalues and eigenvectors of H~. Weighted Weyl estimates near an elliptic trajectory 149
Introduction and results.
Remark. In the rational case the discontinuities of the function L are located exactly at the values of the E k l QM de ned before by (4) , for the values of~given by (7) . In the irrational case in order to prove that L (c) exists we need that c be at some distance from the quasi-energies E k l QM (unless the symbola of the quasi-mode is chosen very judiciously, in which case we can work with c in the complement of the set of all quasi-energies). In all cases this suggests that the weighted spectral measure, (5), in the semi-classical limit, is particularly singular exactly at the values of the E k l QM de ned before. We hope to provide a rigorous proof of a precise statement of this elsewhere.
Introduction and results.
The paper is organized as follows: In Section 3 we prove the existence of the functions L which are studied in Section 4. In Section 5
Introduction and results.
we nish the proof of the main Theorems, using a Tauberian argument that we recall in Section 2. Finally, i n the appendix we review and extend slightly a result on H older continuity of function such a s L using wavelets.
A Tauberian lemma.
In this section we re ne the Tauberian lemma of 2] and 7]. Consider an expression of the following form 
The existence of L (c).
In this section we prove the existence of the coe cients L (c) in the limits (8) and (10) we get (29) and the family fc k g is non-negative.
The existence of L (c).
It is also rapidly decreasing since the function b is Schwartz.
The existence of L (c).
Remark. For a given quantum Hamiltonian H, the coe cients fc k g depend only on the symbola of the coherent state. Observe that the proof shows that given any rapidly decreasing family fc k g one can nd an a giving rise to it.
The existence of L (c).
We next prove the existence of the limit (a n ) sin (n c T ) n T c k e in((k+1=2) + ) :
The existence of L (c).
We must then prove that the limit L (c) = l i m a!0 a (c) exists.
The existence of L (c).
To lighten up the notation a bit, let us de ne is absolutely convergent (for N su ciently large) and it dominates the absolute values of the terms of (41), we are done.
The existence of L (c).
We now turn to the irrational case. 
The existence of L (c).
The following result shows that such a decomposition is indeed valid. n T ( '(a n ) e incT ;'(;a n ) e ;incT )
The existence of L (c).
e ind kĝ (a n ) c k
The existence of L (c).
(a n ) g(a n ) s i n ( n c T ) e in ( 
End of proofs.
The convergence statements in boththeorems are immediate consequences of the Tauberian lemma of Section 2, applied to the following objects The fact that M is of full Lebesgue measure, is a classical result of Diophantine analysis (recall that the sequence fg k g in the remark 1, Section 3 is rapidly decreasing).

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