Let X be a compact d-dimensional manifold and 
a first order positive
self-adjoint elliptic pseudodifferential operator. Let
,
be the eigenvalues of Q with corresponding eigenfunctions
, and let 
be the orthogonal projection of 
onto the space spanned
by 
.
Let B be a zero order pseudodifferential operator on X.
We are interested in asymptotic trace formulas as 
:
for 
and
In this talk, I will describe the following general results, which I obtained recently in joint work with Victor Guillemin.
Case 1. When the set of closed bicharacteristics under
the Hamiltonian flow generated by 
has measure zero,
we obtain the first 2 terms in (*).
Case 2. (Zoll Case)
When all bicharacteristics of the Hamiltonian flow generated by
are closed of length 
,
the eigenvalues of Q clump together in thin bands
which, after a translation, can be centered at the points
, 
.
We obtain the complete asymptotics (*), as through
the integer values.