Searcharxiv⌕ Search

arXiv subjects

Jared Wunsch

Publications and source records attributed to Jared Wunsch.

At least 55 records · Page 3Linked to original sources
arXiv2012

Resolvent estimates and local decay of waves on conic manifolds

We consider manifolds with conic singularites that are isometric to $\mathbb{R}^{n}$ outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via a doubling process. The proof of the resolvent estimate relies on the propagation of singularities theorems of Melrose and the second author to establish a "very weak" Huygens' principle, which may be of independent interest. As applications of the estimate, we obtain a exponential local energy decay and a resonance wave expansion in odd dimensions, as well as a lossless local smoothing estimate for the Schr{ö}dinger equation.

math.AP↗
arXiv2012

Resolvent estimates with mild trapping

We discuss recent progress in understanding the effects of certain trapping geometries on cut-off resolvent estimates, and thus on the qualititative behavior of linear evolution equations. We focus on trapping that is unstable, so that strong resolvent estimates hold on the real axis, and large resonance-free regions can be shown to exist beyond it.

math.AP↗
arXiv2012

From resolvent estimates to damped waves

In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order \emph{complex absorbing potential} gives an operator enjoying polynomial resolvent bounds on the real axis, then the "resolvent" associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.

math.AP↗
arXiv2012

Local smoothing for the Schrödinger equation with a prescribed loss

We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schrödinger equation with a loss that depends on the degeneracy, and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale. As a byproduct of our proof, we obtain a cutoff resolvent estimate with a sharp polynomial loss.

math.AP↗
arXiv2011

Morawetz estimates for the wave equation at low frequency

We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact part of the manifold. We further prove a new type of Morawetz estimate in this context, with both hypotheses and conclusion localized inside the forward light cone. This result allows us to gain a 1/2 power of $t$ decay relative to what would be dictated by energy estimates, in a small part of spacetime.

math.AP↗
arXiv2011

Non-concentration of quasimodes for integrable systems

We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set $WF_h$ of such a sequence is invariant under the Hamilton flow. In principle this may allow concentration of $WF_h$ along positive codimension sub-tori of a Liouville torus $\mathcal{L}$ if there exist rational relations among the frequencies of the flow on $\mathcal{L}.$ We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. The main tools are the spreading of Lagrangian regularity on $\mathcal{L}$ previously shown by Vasy and the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode.

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↗
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↗
arXiv2007

Spreading of Lagrangian regularity on rational invariant tori

Let $P_h$ be a self-adjoint semiclassical pseudodifferential operator on a manifold $M$ such that the bicharacteristic flow of the principal symbol on $T^*M$ is completely integrable and the subprincipal symbol of $P_h$ vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes $u_h$ of $P_h.$ We show that on a nondegenerate rational invariant torus, Lagrangian regularity of $u_h$ (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional ``diffractive'' manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics.

math.AP↗
arXiv2006

The semiclassical resolvent and the propagator for nontrapping scattering metrics

Consider a compact manifold with boundary $M$ with a scattering metric $g$ or, equivalently, an asymptotically conic manifold $(M^\circ, g)$. (Euclidean $\mathbb{R}^n$, with a compactly supported metric perturbation, is an example of such a space.) Let $Δ$ be the positive Laplacian on $(M,g)$, and $V$ a smooth potential on $M$ which decays to second order at infinity. In this paper we construct the kernel of the operator $(h^2 Δ+ V - (λ_0 \pm i0)^2)^{-1}$, at a nontrapping energy $λ_0 > 0$, uniformly for $h \in (0, h_0)$, $h_0 > 0$ small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, $e^{-it(Δ/2 + V)}$, $t \in (0, t_0)$ as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

math.AP↗
arXiv2005

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions $u_λ$ of the Bunimovich stadium $S$, satisfying $(Δ- λ^2) u_λ= 0$. Write $S = R \cup W$ where $R$ is the central rectangle and $W$ denotes the ``wings,'' i.e. the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in $R$ as $λ\to \infty$. We obtain a lower bound $C λ^{-2}$ on the $L^2$ mass of $u_λ$ in $W$, assuming that $u_λ$ itself is $L^2$-normalized; in other words, the $L^2$ norm of $u_λ$ is controlled by $λ^2$ times the $L^2$ norm in $W$. Moreover, if $u_λ$ is a $o(λ^{-2})$ quasimode, the same result holds, while for a $o(1)$ quasimode we prove that $L^2$ norm of $u_λ$ is controlled by $λ^4$ times the $L^2$ norm in $W$. We also show that the $L^2$ norm of $u_λ$ may be controlled by the integral of $w \abs{\partial_N u}^2$ along $\partial S \cap W$, where $w$ is a smooth factor on $W$ vanishing at $R \cap W$. These results complement recent work of Burq-Zworski which shows that the $L^2$ norm of $u_λ$ is controlled by the $L^2$ norm in any pair of strips contained in $R$, but adjacent to $W$.

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

Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$, and $2 < q, r \leq \infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > 2$). These estimates imply existence theorems for semi-linear Schrödinger equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}. This result improves on our previous result in \cite{HTW}, which was an $L^4_{t,x}$ Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.

math.AP↗
arXiv2004

A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds

We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$ of the solution with itself. We also use smoothing estimates for Schrödinger solutions including a new one proved here with weight $r^{-1}$ at infinity and with the gradient term involving only one angular derivative.

math.AP↗
arXiv2003

The radiation field is a Fourier integral operator

We exhibit the form of the ``radiation field,'' describing the large-scale, long-time behavior of solutions to the wave equation on a manifold with no trapped rays, as a Fourier integral operator. We work in two different geometric settings: scattering manifolds (a class which includes asymptotically Euclidean spaces) and asymptotically hyperbolic manifolds. The canonical relation of the radiation field operator is a map from the cotangent bundle of the manifold to a cotangent bundle over the boundary at infinity; it is associated to a sojourn time, or Busemann function, for geodesic rays. In non-degenerate cases, the symbol of the operator can be described explicitly in terms of the geometry of long-time geodesic flow. As a consequence of the above result, we obtain a description of the (distributional) high-frequency asymptotics of the scattering-theoretic Poisson operator, better known as the Eisenstein function in the asymptotically hyperbolic case.

math.AP↗
arXiv2003

On the structure of the Schrödinger propagator

We discuss the form of the propagator $U(t)$ for the time-dependent Schrödinger equation on an asyptotically Euclidean, or, more generally, asymptotically conic, manifold with no trapped geodesics. In the asymptotically Euclidean case, if $χ\in \mathcal{C}_0^\infty$, and with $\mathcal{F}$ denoting Fourier transform, $\mathcal{F}\circ e^{-ir^2/2t} U(t) χ$ is a Fourier integral operator for $t\neq 0.$ The canonical relation of this operator is a ``sojourn relation'' associated to the long-time geodesic flow. This description of the propagator follows from its more precise characterization as a ``scattering fibered Legendrian,'' given by the authors in a previous paper and sketched here. A corollary is a propagation of singularities theorem that permits a complete description of the wavefront set of a solution to the Schrödinger equation, restricted to any fixed nonzero time, in terms of the oscillatory behavior of its initial data. We discuss two examples which illustrate some extremes of this propagation behavior.

math.AP↗