WEIGHTED TRACE FORMULA NEAR A HYPERBOLIC TRAJECTORY AND COMPLEX ORBITS

In this pager we consider a weighted trace formula for Schrodinger ope rators. Mon precisely, let psi(j)((h) over bar) and E-j((h) over bar) denote the eigenfunctions and eigenvalues of a Schrodinger-type operat or H-(h over bar) with a discrete spectrum. Let psi((x, xi)) be a cohe rent state centered at a point (x,xi) of a hyperbolic closed orbit gam ma. We show that, as (h) over bar-->0 the leading term of Sigma(j)phi[ {E-j((h) over bar)- E]/(h) over bar}/(psi((x,xi)), psi(j)(($) over bar ))/(2) can be expressed in terms of the analytic continuation on the u pper and lower half-planes of the positive and negative frequencies pa rt of phi. The result is also related to complex trajectories surround ing gamma. (C) 1998 American Institute of Physics. [S0022-2488(98)0190 8-2].