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Asymptotic behavior of regularized scattering phases for long range perturbations

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Journées Équations aux dérivées partielles Forges-les-Eaux, 3–7 juin 2002 GDR 2434 (CNRS) Asymptotic behavior of regularized scattering phases for long range perturbations Jean-Marc Bouclet Abstract We define scattering phases for Schrödinger operators on Rd as limit of arguments of relative determinants. These phases can be defined for long range perturbations of the Laplacian and therefore they can replace the usual spectral shift function (SSF) of Birman-Krein’s theory, which can be defined for only special short range perturbations (relatively trace class perturba- tions). We prove the existence of asymptotic expansions for these phases, which generalize results on the SSF. 1. Introduction 1.1. Assumptions and statement of the problem In this article, we will mainly consider perturbations of the Laplacian operator H0 = − ∑d j=1 ∂ 2 j on L2(Rd), d ≥ 1. These perturbations are self-adjoint and defined by H1 = H0 + V = − ∑ 1≤j,k≤d gjk(x)∂j∂k + d∑ j=1 bj(x)∂j + c(x) which means that the perturbation V is a differential operator of order ≤ 2. We assume that H1 is uniformly elliptic, that is∑ 1≤j,k≤d gjk(x)ξjξk ≥ c|ξ|2, ∀ x, ξ ∈ Rd for some c > 0; its coefficients are smooth and satisfy for each multiindex α∑ j,k |∂α (gjk(x)− δjk) |+ ∑ j |∂αbj(x)|+ |∂αc(x)| ≤ Cα〈x〉−ρ−|α|, (1) for all x ∈ Rd; here δjk is the usual Kronecker’s symbol and 〈x〉 = (1 + |x|2)1/2. II–1 The number ρ > 0 is a real number, and the condition (1) is a long range condition. Our purpose is to define and study relative scattering determinants Dp(z) = Detp(1 + V (H0 − z)−1), z ∈ C \ R and the associated phases which are, in a sense, the limits of pi−1 argDp(z) when z approaches the real axis. The important point is that we will consider (1), with a general ρ > 0 in order to relax the usual condition ρ > d needed to use Birman- Krein’s theory. Our definition of Detp is an extension of the Fredholm determinants defined for compact perturbations of identity which are in Sp, th

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