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

Abstract

The main goal of this paper is to compare two long-range perturbations of constant coefficient operators on R n such that their difference is short range. Typical examples are Schrödinger Hamiltonians: H j = − h 2 · Δ + V j with V j ( x) = O(| x| −δ), V 2( x) − V 1( 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 R n . For the scattering pair ( H 2, H 1) the average time-delay, τ D (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 τ D as λ ↗ + ∞ (high energy "régime") and as h ↘ 0 (semi-classical "régime").