Resonance Expansions Of Propagators In The Presence Of Potential Barriers

Introduction and statement of results The purpose of this note is to present an expansion of a semi-classical propagating state in terms of resonances. We consider energy levels at which there exists a potential barrier separating the interaction region from a non-trapping region near infinity. This assumption allows a significant strengthening of the results of Burq and the third author [3, Theorem 1]. It is motivated by the recent work of the first author on the spectral shift function in the presence of barriers [14]. Our results are applicable to the semi-classical Schrodinger equation for long range "black box" perturbations [18] -- see Sect.2 for a review of definitions, and for the barrier assumption in the general setting. We denote by Res(P (h)) the set of resonances of P (h), that is the set of poles of the meromorphic continuation of R(z, h) = (z P (h)) -1 from Im z > 0 to the lower half-plane. A typical operator to keep in mind is P (h) = #+V (x), where C|x| -#