Relative Time-Delay for Perturbations of Elliptic Operators and Semiclassical Asymptotics

The main goal of this paper is to compare two long-range perturbations of constant coefficient operators on Rn such that their difference is short range. Typical examples are Schrödinger Hamiltonians: Hj = −h2 · Δ + Vj with Vj(x) = O(|x|−δ), V2(x) − V1(x) = O(|x|−ρ) with δ > 0, ρ > n. We can also consider perturbations by magnetic fields and cases where Δ is the Laplace-Beltrami operator for asymptotically flat metrics on Rn. For the scattering pair (H2, H1) the average time-delay, τD(λ, h), depending on the energy λ and the parameter h, is well defined. It is related to the spectral shift function and also to the scattering phase. Under suitable assumptions we prove asymptotic results on τD as λ ↗ + ∞ (high energy "régime") and as h ↘ 0 (semi-classical "régime").