Spectral distributions for long range perturbations

Abstract

We study distributions which generalize the concept of spectral shift function, for pseudodifferential operators on R d : We call such distributions spectral distributions. Relations between relative scattering determinants and spectral distributions are established; they lead to the definition of regularized scattering phase. These relations are analogous to the usual one for the standard spectral shift function. We give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered. In particular, we prove Breit-Wigner formulae for the regularized scattering phases, for semiclassical Schro¨dinger operators with long-range potentials. r

On global Strichartz estimates for non-trapping metrics

“…The proof of Proposition 4.2 follows from the considerations in [1,2,25]. By Proposition 4.2 and the Duhamel formula…”

“…Note also that this parametrix has already been used globally in time, i.e. for t ∈ [0, ±∞), for L 2 problems [2,12,16,24,25]. Here we want to prove L 1 → L ∞ estimates and control them globally in time.…”

Resolvent Estimates for the Laplacian on Asymptotically Hyperbolic Manifolds

“…In particular, they are known to be useful to obtain Weyl formulas for scattering phases in Euclidean scattering [20,21,2,3] and the present paper was motivated by similar considerations in the hyperbolic context [4,5]. Actually, high energy estimates are important tools to get semiclassical approximations of the Schrödinger group by the techniques of Isozaki-Kitada [13,14].…”

“…We first note that the right-hand side of (3.5) is nothing but 2D r a k D r + 2a k μ k e −2r − a (3) k /2. Since a k (r) ≥ χ R (r)ξ S (r − log ν k ) and a k (r) ≥ Sχ R (r)ξ S (r − log ν k ), we get…”

Resolvent expansions and trace regularizations for Schrödinger operators

“…In the first part of this section we shall briefly discuss relations between the trace regularization of Section 2 and a different approach to regularized traces for long range Schrödinger operators, developed in [4], [5], following [10]. When V ∈ S −ε (R n ), for some ε > 0, it is proved in [4] that the operator…”