Weighted trace formula near a hyperbolic trajectory and complex orbits

Abstract

In this paper we consider a weighted trace formula for Schrödinger operators. More precisely, let ψjℏψjℏ and EjℏEjℏ denote the eigenfunctions and eigenvalues of a Schrödinger-type operator HℏHℏ with a discrete spectrum. Let ψ(x,ξ)ψ(x,ξ) be a coherent state centered at a point (x,ξ)(x,ξ) of a hyperbolic closed orbit γ. We show that, as ℏ→0, the leading term of ∑jφ{[Ej(ℏ)−E]/ℏ}∣(ψ(x,ξ),ψjℏ)∣2∑jφ{[Ej(ℏ)−E]/ℏ}∣(ψ(x,ξ),ψjℏ)∣2 can be expressed in terms of the analytic continuation on the upper and lower half-planes of the positive and negative frequencies part of φ. The result is also related to complex trajectories surrounding γ.© 1998 American Institute of Physics.