András Vasy

SEMICLASSICAL ESTIMATES IN ASYMPTOTICALLY EUCLIDEAN SCATTERING

Abstract: We consider long range semiclassical perturbations of the Laplacian on asymptotically Euclidean manifolds. We obtain precise resolvent estimates under non-trapping assumptions. The novelty lies in a systematic use of geometric microlocal methods.

Asymptotics of Solutions of the Wave Equation on de Sitter-Schwarzschild Space

Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.

The spectral projections and the resolvent for scattering metrics

In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x−4 dx2+x−2 h’ near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x2C∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX associated to an intersecting pair with conic points. In this paper, we describe the kernel of the spectral projections and the resolvent,R(σ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched productXb2 (the blowup ofX2 about the corner, (ϖX)2). The structure of the resolvent is only slightly more complicated.As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.

Spectral and scattering theory for symbolic potentials of order zero

The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on R2 which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at non-critical energies are shown to originate both at minima and maxima, although the latter are not germane to the L2 spectral theory. Asymptotic completeness is shown, both in the traditional L2 sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.

Propagation of singularities in many-body scattering

Abstract In this paper we describe the propagation of singularities of tempered distributional solutions u∈ S ′ of (H−λ)u=0, λ>0, where H is a many-body Hamiltonian H=Δ+V, Δ⩾0, V=∑aVa, under the assumption that no subsystem has a bound state and that the two-body interactions Va are real-valued polyhomogeneous symbols of order −1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term ‘singularity’ provides a microlocal description of the lack of decay at infinity. We use this result to prove that the wave front relation of the free-to-free S-matrix (which, under our assumptions, is all of the S-matrix) is given by the broken geodesic flow, broken at the ‘singular directions’, on S n−1 at time π. We also present a natural geometric generalization to asymptotically Euclidean spaces.

COMPLEX POWERS AND NON-COMPACT MANIFOLDS

Abstract We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called “Guillemin algebras, ” whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyls formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131–160]. A Guillemin algebra can be thought of as an algebra of “abstract pseudodifferential operators.” Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,…) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for A z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melroses work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

The global non-linear stability of the Kerr-de Sitter family of black holes

We establish the full global non-linear stability of the Kerr-de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: We develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einsteins equations. In particular, the iteration scheme used to solve Einsteins equations automatically finds the parameters of the Kerr-de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.

Propagation Through Trapped Sets and Semiclassical Resolvent Estimates

Let \(P = -{h}^{2}\Delta + V (x)\), \(V \in {C}_{0}^{\infty }({\mathbb{R}}^{n})\).We are interested in semiclassical resolvent estimates of the form

Resolvents and Martin boundaries of product spaces

\|\chi {(P - E - i0)}^{-1}{\chi \|}_{{L}^{2}({\mathbb{R}}^{n})\rightarrow {L}^{2}({\mathbb{R}}^{n})} \leq \frac{a(h)}{h} ,\qquad h \in (0,{h}_{0}],