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Andras Vasy

Publications and source records attributed to Andras Vasy.

At least 37 records · Page 2Linked to original sources
arXiv2013

Multi-scale discrete approximations of Fourier integral operators associated with canonical transformations and caustics

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation through the introduction of locally singularity-resolving diffeomorphisms, enabling the application of wave packet driven computation, and in the construction of the associated pseudo-differential joint-partition of unity on the canonical graphs. We apply the method to a parametrix of the wave equation in the vicinity of a cusp singularity.

math.NA↗
arXiv2012

Spectral theory for the Weil-Petersson Laplacian on the Riemann moduli space

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete. The second theorem is a Weyl asymptotic formula for the counting function for this spectrum.

math.DG↗
arXiv2011

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework is relatively simple given modern microlocal analysis, and only takes a bit over a dozen pages after the statement of notation. It resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis, including semiclassical analysis. The rest of the paper is devoted to applications. Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose's b-structures. These include asymptotically Minkowski metrics, asymptotically de Sitter-type metrics on a blow-up of the natural compactification and Kerr-de Sitter-type metrics. The simplest application, however, is to provide a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces). The results include, in particular, a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.

math.AP↗
arXiv2011

Microlocal analysis of asymptotically hyperbolic spaces and high energy resolvent estimates

In this paper we describe a new method for analyzing the Laplacian on asymptotically hyperbolic spaces, which was introduced recently by the author. This new method in particular constructs the analytic continuation of the resolvent for even metrics (in the sense of Guillarmou), and gives high energy estimates in strips. The key idea is an extension across the boundary for a problem obtained from the Laplacian shifted by the spectral parameter. The extended problem is non-elliptic -- indeed, on the other side it is related to the Klein-Gordon equation on an asymptotically de Sitter space -- but nonetheless it can be analyzed by methods of Fredholm theory. This method is a special case of a more general approach to the analysis of PDEs which includes, for instance, Kerr-de Sitter and Minkowski type spaces. The present paper is self-contained, and deals with asymptotically hyperbolic spaces without burdening the reader with material only needed for the analysis of the Lorentzian problems considered in the author's original work.

math.AP↗
arXiv2011

Diffraction of singularities for the wave equation on manifolds with corners

We consider the fundamental solution to the wave equation on a manifold with corners of arbitrary codimension. If the initial pole of the solution is appropriately situated, we show that the singularities which are diffracted by the corners (i.e., loosely speaking, are not propagated along limits of transversely reflected rays) are smoother than the main singularities of the solution. More generally, we show that subject to a hypothesis of nonfocusing, diffracted wavefronts of any solution to the wave equation are smoother than the incident singularities. These results extend our previous work on edge manifolds to a situation where the fibers of the boundary fibration, obtained here by blowup of the corner in question, are themselves manifolds with corners.

math.AP↗
arXiv2011

Semiclassical second microlocal propagation of regularity and integrable systems

We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold $X$ with respect to a Lagrangian submanifold of $T^*X.$ The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of $T^*\mathbb{R}^n,$ and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to Hörmander's theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g. eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.

math.AP↗
arXiv2011

The wave equation on asymptotically Anti-de Sitter spaces

In this paper we describe the behavior of solutions of the Klein-Gordon equation, (Box_g+lambda)u=f, on Lorentzian manifolds (X^o,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the sense that the metric is conformal to a smooth Lorentzian metric g^ on X, where C has a non-trivial boundary, in the sense that g=x^{-2}g^, with x a boundary defining function. The boundary is conformally time-like for these spaces, unlike asymptotically de Sitter spaces, which are similar but with the boundary being conformally space-like. Here we show local well-posedness for the Klein-Gordon equation, and also global well-posedness under global assumptions on the (null)bicharacteristic flow, for lambda below the Breitenlohner-Freedman bound, (n-1)^2/4. These have been known under additional assumptions, see the work of Breitenlohner-Freedman and Holzegel. Further, we describe the propagation of singularities of solutions and obtain the asymptotic behavior (at the boundary of X) of regular solutions. We also define the scattering operator, which in this case is an analogue of the hyperbolic Dirichlet-to-Neumann map. Thus, it is shown that below the Breitenlohner-Freedman bound, the Klein-Gordon equation behaves much like it would for the conformally related metric, g^, with Dirichlet boundary conditions, for which propagation of singularities was shown by Melrose, Sjöstrand and Taylor, though the precise form of the asymptotics is different.

math.AP↗
arXiv2009

Positive commutators at the bottom of the spectrum

Bony and Häfner have recently obtained positive commutator estimates on the Laplacian in the low-energy limit on asymptotically Euclidean spaces; these estimates can be used to prove local energy decay estimates if the metric is non-trapping. We simplify the proof of the estimates of Bony-Häfner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincaré inequality. Our main result is the positive commutator estimate $$ χ_I(H^2Δ_g)\frac{i}{2}[H^2Δ_g,A]χ_I(H^2Δ_g) \geq Cχ_I(H^2Δ_g)^2, $$ where $H\uparrow \infty$ is a \emph{large} parameter, $I$ is a compact interval in $(0,\infty),$ and $χ_I$ its indicator function, and where $A$ is a differential operator supported outside a compact set and equal to $(1/2)(r D_r +(r D_r)^*)$ near infinity. The Laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay--the same estimate then holds for the resulting Schrödinger operator.

math.AP↗
arXiv2009

Diffraction at corners for the wave equation on differential forms

In this paper we prove the propagation of singularities for the wave equation on differential forms with natural (i.e. relative or absolute) boundary conditions on Lorentzian manifolds with corners, which in particular includes a formulation of Maxwell's equations. These results are analogous to those obtained by the author for the scalar wave equation and for the wave equation on systems with Dirichlet or Neumann boundary conditions earlier. The main novelty is thus the presence of natural boundary conditions, which effectively make the problem non-scalar, even `to leading order', at corners of codimension at least 2.

math.AP↗
arXiv2008

Microlocal propagation near radial points and scattering for symbolic potentials of order zero

In this paper, the scattering and spectral theory of $H=Δ_g+V$ is developed, where $Δ_g$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V\in C^\infty(X)$ is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system; a general framework for microlocal analysis at these points forms the main part of the paper.

math.AP↗
arXiv2007

The wave equation on asymptotically de Sitter-like spaces

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the other manifold as the parameter goes to minus infinity, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y-.

math.AP↗
arXiv2004

Absence of super-exponentially decaying eigenfunctions on Riemannian manifolds with pinched negative curvature

Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the assumptions and conclusions localized near infinity.

math.AP↗
arXiv2004

Lipschitz domains, domains with corners and the Hodge Laplacian

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains and form domains. We obtain results valid for general Lipschitz domains, and stronger results for a special class of ``almost convex'' domains, which apply to domains with corners.

math.AP↗
arXiv2004

Propagation of singularities for the wave equation on manifolds with corners

In this paper we describe the propagation of smooth (C^\infty) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t^2-Δ_M, and u locally in H^1 solving Pu=0 with homogeneous Dirichlet or Neumann boundary conditions, we show that the wave front set of u is a union of maximally extended generalized broken bicharacteristics. This result is a smooth counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary. Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

math.AP↗
arXiv2003

Fixed energy inverse problem for exponentially decreasing potentials

In this paper we show that in two-body scattering the scattering matrix at a fixed energy determines real-valued exponentially decreasing potentials. This result has been proved by Novikov previously, see also the work of Novikov and Khenkin using a d-bar-equation. We present a different method, which combines a density argument and real analyticity in part of the complex momentum. The latter has been noted by Novikov and Khenkin; here we give a short proof using contour deformations. In the addendum to the paper we also supply a reference to the work of Eskin and Ralston that did not appear in the published paper since we were unaware of the relevant aspects of their work.

math.AP↗
arXiv2003

Inverse problems in N-body scattering

We survey finite energy inverse results in N-body scattering, and we also sketch the proof of the extension of our recent two-cluster to two-cluster three-body result to the many-body case: this requires only minor modifications. We also indicate how to extend a free-to-free inverse result in three-body scattering to the many-body case.

math.AP↗
arXiv2002

Scattering Theory on SL(3)/SO(3): Connections with Quantum 3-Body Scattering

In this paper we continue our program of extending the methods of geometric scattering theory to encompass the analysis of the Laplacian on symmetric spaces of rank greater than one and their geometric perturbations. Our goal here is to explain how analysis of the Laplacian on the globally symmetric space $\SL(3,\RR)/\SO(3,\RR)$ is very closely related to quantum three-body scattering. In particular, we adapt geometric constructions from recent advances in that field by one of us (A.V.), as well as from our previous work [Mazzeo-Vasy:Resolvents] concerning resolvents for product spaces, to give a precise description of the resolvent and the spherical functions on this space. Amongst the many technical advantages, these methods give results which are uniform up to the walls of the Weyl chambers.

math.AP↗