arXiv subjects

András Vasy

Publications and source records attributed to András Vasy.

At least 19 recordsLinked to original sources

arXiv2026

Dual modes in Kerr spacetimes and the Whiting transform: Mode stability revisited

The purpose of the paper is to place Whiting's classical growing mode stability argument, extended to real frequencies by Shlapentokh-Rothman for the scalar wave equation and by Andersson, Ma, Paganini and Whiting in general, in the framework of classical PDE theory. The key steps are: a description of the dual or adjoint modes, a singular phase space pairing argument which is technically executed via the Fourier transform, followed by a standard unique continuation result. One part of our description of the dual modes connects them directly to the standard mode solutions which are smooth over the event horizon. In the zero spin case, there is a geometric interpretation of this: quasinormal modes (for non-zero real frequencies) which are smooth across the future event horizon can be extended as distributional QNM solutions by 0 through the past event horizon. We also describe the behavior of mode solutions at the bifurcate sphere.

math.AP↗

arXiv2026

An analogue of non-interacting quantum field theory in Riemannian signature

In this paper, we define a model of non-interacting quantum fields satisfying $(Δ_g-λ^2)ϕ=0$ on a Riemannian scattering space $(M,g)$ with two boundary components, i.e. a manifold with two asymptotically conic ends (meaning asymptotic to the "large end" of a cone). Our main result describes a canonical construction of two-point functions satisfying a version of the Hadamard condition.

hep-th↗

arXiv2025

The Feynman propagator for massive Klein-Gordon fields on radiative asymptotically flat spacetimes

On a large class of asymptotically flat spacetimes which includes radiative perturbations of Minkowski space, we define a distinguished global Feynman propagator for massive Klein-Gordon fields by means of the microlocal approach to non-elliptic Fredholm theory, working in the de,sc-pseudodifferential algebra due to Sussman. We extend the limiting absorption principle (the "$i\varepsilon$ prescription" for the Feynman propagator) to this setting. Motivated by the complicated Hamilton flow structure arising in this problem, we also prove a new localized radial point estimate in the spirit of Haber-Vasy which, under appropriate nondegeneracy assumptions, allows one to propagate microlocal regularity into a single radial point belonging to a larger radial set which can be a source, sink, or saddle for the Hamilton flow.

gr-qc↗

arXiv2025

On the relationship between the semiclassical and standard pseudodifferential algebras

In this short paper we discuss the precise relationship between the semiclassical and standard pseudodifferential algebras and explore implications such as for large spectral parameter elliptic estimates, even in the case of pseudodifferential spectral familes. We also explain the connection between second microlocalization and this relationship.

math.AP↗

arXiv2025

Conditional non-linear stability of Kerr-de Sitter spacetimes in the full subextremal range

We show the stability of Kerr-de Sitter black holes, in the full subextremal range, as solutions of the vacuum Einstein equation with a positive cosmological constant under the assumption that mode stability holds for these spacetimes. The method is similar to the (unconditional) proof in the slowly rotating case by Hintz and Vasy. The key novelties are the implementation of constraint damping in the full subextremal range as well as the verification of a subprincipal symbol condition at the trapped set.

gr-qc↗

arXiv2025

Linear stability of Kerr black holes in the full subextremal range

We prove, unconditionally, the linear stability of the Kerr family in the full subextremal range. On an analytic level, our proof is the same as that of our earlier paper in the slowly rotating case. The additional ingredients we use are, firstly, the mode stability result proved by Andersson, Whiting, and the first author and, secondly, computations related to the zero energy behavior of the linearized gauge-fixed Einstein equation in work by the second author.

gr-qc↗

arXiv2024

Dirac operators and local invariants on perturbations of Minkowski space

For small perturbations of Minkowski space, we show that the square of the Lorentzian Dirac operator $P= -D^2$ has real spectrum apart from possible poles in a horizontal strip. Furthermore, for $\varepsilon>0$ we relate the poles of the spectral zeta function density of $P-i\varepsilon$ to local invariants, in particular to the Lorentzian scalar curvature. The proof involves microlocal propagation and radial estimates in a resolved scattering calculus as well as high energy estimates in a further resolved classical-semiclassical calculus.

math.AP↗

arXiv2024

Stability of the expanding region of Kerr-de Sitter spacetimes and smoothness at the conformal boundary

We give a new proof of the recent result by Fournodavlos-Schlue on the nonlinear stability of the expanding region of Kerr-de Sitter spacetimes as solutions of the Einstein vacuum equations with positive cosmological constant. Our gauge is a modification of a generalized harmonic gauge introduced by Ringström in which the asymptotic analysis becomes particularly simple. Due to the hyperbolic character of our gauge, our stability result is local near points on the conformal boundary. We show furthermore that, in yet another gauge, the conformally rescaled metric is smooth down to the future conformal boundary, with the coefficients of its Fefferman-Graham type asymptotic expansion featuring a mild singularity at future timelike infinity of the black hole.

gr-qc↗

arXiv2024

Stationarity and Fredholm Theory in Subextremal Kerr-de Sitter Spacetimes

In a recent paper, we proved that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the results of Vasy (2013). One central ingredient in the argument was a new definition of quasinormal modes, where a non-standard choice of stationary Killing vector field had to be used in order for the Fredholm theory to be applicable. In this paper, we show that there is in fact a variety of allowed choices of stationary Killing vector fields. In particular, the horizon Killing vector fields work for the analysis, in which case one of the corresponding ergoregions is completely removed.

math.AP↗

arXiv2023

Analyticity of quasinormal modes in the Kerr and Kerr-de Sitter spacetimes

We prove that quasinormal modes (or resonant states) for linear wave equations in the subextremal Kerr and Kerr-de Sitter spacetimes are real analytic. The main novelty of this paper is the observation that the bicharacteristic flow associated to the linear wave equations for quasinormal modes with respect to a suitable Killing vector field has a stable radial point source/sink structure rather than merely a generalized normal source/sink structure. The analyticity then follows by a recent result in the microlocal analysis of radial points by Galkowski and Zworski. The results can then be recast with respect to the standard Killing vector field.

math.AP↗

arXiv2023

Wave equations in the Kerr-de Sitter spacetime: the full subextremal range

We prove that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the existing results. The main novelties are a different way of obtaining a Fredholm setup that defines the quasinormal modes and a new analysis of the trapping of lightlike geodesics in the Kerr-de Sitter spacetime, both of which apply in the full subextremal range. In particular, this reduces the question of decay for solutions to wave equations to the question of mode stability.

math.AP↗

arXiv2023

Microlocal analysis near null infinity in asymptotically flat spacetimes

We present a novel approach to the analysis of regularity and decay for solutions of wave equations in a neighborhood of null infinity in asymptotically flat spacetimes of any dimension. The classes of metrics and wave type operators we consider near null infinity include those arising in nonlinear stability problems for Einstein's field equations in $1+3$ dimensions. In a neighborhood of null infinity, in an appropriate compactification of the spacetime to a manifold with corners, the wave operators are of edge type at null infinity and totally characteristic at spacelike and future timelike infinity. On a corresponding scale of Sobolev spaces, we demonstrate how microlocal regularity propagates across or into null infinity via a sequence of radial sets. As an application, inspired by work of the second author with Baskin and Wunsch, we prove regularity and decay estimates for forward solutions of wave type equations on asymptotically flat spacetimes which are asymptotically homogeneous with respect to scaling in the forward timelike cone and have an appropriate structure at null infinity. These estimates are new even for the wave operator on Minkowski space. The results obtained here are also used as black boxes in a global theory of wave type equations on asymptotically flat and asymptotically stationary spacetimes developed by the first author.

math.DG↗

arXiv2022

The tensorial X-ray transform on asymptotically conic spaces

In this paper we show the invertibility of the geodesic X-ray transform on one forms and 2-tensors on asymptotically conic manifolds, up to the natural obstruction, allowing existence of certain kinds of conjugate points. We use the 1-cusp pseudodifferential operator algebra and its semiclassical foliation version introduced and used by Vasy and Zachos, who showed the same type invertibility on functions. The complication of the invertibility of the tensorial X-ray transform, compared with X-ray transform on functions, is caused by the natural kernel of the transform consisting of `potential tensors'. We overcome this by arranging a modified solenoidal gauge condition, under which we have the invertibility of the X-ray transform.

math.DG↗

arXiv2022

The X-ray transform on asymptotically conic spaces

In this paper, partly based on Zachos' PhD thesis, we show that the geodesic X-ray transform is stably invertible near infinity on a class of asymptotically conic manifolds which includes perturbations of Euclidean space. In particular certain kinds of conjugate points are allowed. Further, under a global convex foliation condition, the transform is globally invertible. The key analytic tool, beyond the approach introduced by Uhlmann and Vasy, is the introduction of a new pseudodifferential operator algebra, which we name the 1-cusp algebra, and its semiclassical version.

math.DG↗

arXiv2020

On the light ray transform of wave equation solutions

We study the light ray transform on Minkowski space-time and its small metric perturbations acting on scalar functions which are solutions to wave equations. We show that the light ray transform uniquely determines the function in a stable way. The problem is of particular interest because of its connection to inverse problems of the Sachs-Wolfe effect in cosmology.

math.AP↗

arXiv2020

A semiclassical approach to geometric X-ray transforms in the presence of convexity

In this short paper we introduce a variant of the approach to inverting the X-ray transform that originated in the author's work with Uhlmann. The new method is based on semiclassical analysis and eliminates the need for using sufficiently small domains and layer stripping for obtaining the injectivity and stability results, assuming natural geometric conditions are satisfied.

math.DG↗

arXiv2020

Stability of Minkowski space and polyhomogeneity of the metric

We study the nonlinear stability of the $(3+1)$-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linearized equation globally at each step; we use a generalized harmonic gauge and implement constraint damping to fix the geometry of null infinity. The linear analysis is largely based on energy and vector field methods originating in work by Klainerman. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Upon compactifying $\mathbb{R}^4$ to a manifold with corners, with boundary hypersurfaces corresponding to spacelike, null, and timelike infinity, we show, using the framework of Melrose's b-analysis, that polyhomogeneous initial data produce a polyhomogeneous spacetime metric. Finally, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.

math.AP↗

arXiv2020

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 Einstein's equations. In particular, the iteration scheme used to solve Einstein's 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.

math.DG↗