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In this article, we study propagation of defect measures for Schr\"odinger operators, $-h^2\Delta_g+V$, on a Riemannian manifold $(M,g)$ of dimension $n$ with $V$ having conormal singularities along a hypersurface $Y$ in the sense that derivatives along vector fields tangent to $Y$ preserve the regularity of $V$. We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface $Y$ whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangent to $Y$ at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold.

We prove explicit asymptotics for the location of semiclassical scattering resonances in the setting of $h$-dependent delta-function potentials on $\mathbb{R}$. In the cases of two or three delta poles, we are able to show that resonances occur along specific lines of the form $\Im z \sim -\gamma h \log(1/h).$ More generally, we use the method of Newton polygons to show that resonances near the real axis may only occur along a finite collection of such lines, and we bound the possible number of values of the parameter $\gamma.$ We present numerical evidence of the existence of more and more possible values of $\gamma$ for larger numbers of delta poles., Comment: 17 pages. New version incorporating suggestions from referee and miscellaneous minor corrections. To appear in Trans. AMS

We obtain estimates on the rate of decay of a solution to the wave equation on a stationary spacetime that tends to Minkowski space at a rate $O(\lvert x \rvert^{-\kappa}),$ $\kappa \in (1,\infty) \backslash \mathbb{N}.$ Given suitably smooth and decaying initial data, we show a wave locally enjoys the decay rate $O(t^{-\kappa-2+\epsilon})$., Comment: 36 pages; updated with clarifying exposition and fixed typos

We consider the Schr\"odinger operator \[ P=h^2 \Delta_g + V \] on $\mathbb{R}^n$ equipped with a metric $g$ that is Euclidean outside a compact set. The real-valued potential $V$ is assumed to be compactly supported and smooth except at conormal singularities of order $-1-\alpha$ along a compact hypersurface $Y.$ For $\alpha>2$ (or even $\alpha>1$ if the classical flow is unique), we show that if $E_0$ is a non-trapping energy for the classical flow, then the operator $P$ has no resonances in a region \[ [E_0 - \delta, E_0 + \delta] - i[0,\nu_0 h \log(1/h)]. \] The constant $\nu_0$ is explicit in terms of $\alpha$ and dynamical quantities. We also show that the size of this resonance-free region is optimal for the class of piecewise-smooth potentials on the line., Comment: 20 pages; added Section 2.4 on applications to quantum evolution

Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schr\"odinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp} \operatorname{Tr} U(t) \subset 2 \pi \mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d \geq 2$ for which the singularities of $\operatorname{Tr} U(t)$ at nonzero multiples of $2 \pi$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(\lambda^{d-1})$, improving in these cases the $O(\lambda^{d-1})$ remainder previously established by Helffer--Robert., Comment: 28 pages; new section added on propagation of singularities

We describe the resonances closest to the real axis generated by diffraction of waves among cone points on a manifold with Euclidean ends. These resonances lie asymptotically evenly spaced along a curve of the form $$\frac{\Im \lambda}{\log \left |\Re \lambda\right |}= -\nu;$$ here $\nu=(n-1)/2 L_0$ where $n$ is the dimension and $L_0$ is the length of the longest geodesic connecting two cone points. Moreover there are asymptotically no resonances below this curve and above the curve $$ \frac{\Im \lambda}{\log \left |\Re \lambda\right |}= -\Lambda $$ for a fixed $\Lambda>\nu.$, Comment: Slight correction to main theorem: finitely many different values of the constants $C_\Re$ and $C_\Im$ may be possible if there is more than one maximal diffracted closed orbit. Final version to appear in Ann. Inst. Fourier

We show the existence of the full compound asymptotics of solutions to the scalar wave equation on long-range non-trapping Lorentzian manifolds modeled on the radial compactification of Minkowski space. In particular, we show that there is a joint asymptotic expansion at null and timelike infinity for forward solutions of the inhomogeneous equation. In two appendices we show how these results apply to certain spacetimes whose null infinity is modeled on that of the Kerr family. In these cases the leading order logarithmic term in our asymptotic expansions at null infinity is shown to be nonzero., Comment: Version 2: 52 pages, 1 figure. Added a remark about Price's law to Section 9.2 and made minor expository changes requested by referee

Let $(X,g)$ be a compact manifold with conic singularities. Taking $\Delta_g$ to be the Friedrichs extension of the Laplace-Beltrami operator, we examine the singularities of the trace of the half-wave group $e^{- i t \sqrt{ \smash[b]{\Delta_g}}}$ arising from strictly diffractive closed geodesics. Under a generic nonconjugacy assumption, we compute the principal amplitude of these singularities in terms of invariants associated to the geodesic and data from the cone point. This generalizes the classical theorem of Duistermaat-Guillemin on smooth manifolds and a theorem of Hillairet on flat surfaces with cone points., Comment: 46 pages; 3 figures. New version reflects correction to trace formula (factor of primitive length rather than length), as well as some corrections to discussions of Jacobi fields in section 1.3

We consider a non-trapping $n$-dimensional Lorentzian manifold endowed with an end structure modeled on the radial compactification of Minkowski space. We find a full asymptotic expansion for tempered forward solutions of the wave equation in all asymptotic regimes. The rates of decay seen in the asymptotic expansion are related to the resonances of a natural asymptotically hyperbolic problem on the "northern cap" of the compactification. For small perturbations of Minkowski space that fit into our framework, we show a rate of decay that improves on the Klainerman--Sobolev estimates., Comment: 67 pages, 2 figures; version 2 was a substantial revision with the main result (Theorem 1.1) improved; version 3 fixes a minor error in the iterative argument establishing the asymptotic expansion; version 4 includes a few cosmetic changes and adds a few references

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., Comment: 16 pages, 1 figure

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 Schr\"odinger 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., Comment: 26 pages, incorporated referee's comments

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., Comment: One addition to acknowledgments; to appear in Comm. PDE

Bony and H\"afner 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-H\"afner and generalize them to the setting of scattering manifolds (i.e. manifolds with large conic ends), by applying a sharp Poincar\'e inequality. Our main result is the positive commutator estimate $$ \chi_I(H^2\Delta_g)\frac{i}{2}[H^2\Delta_g,A]\chi_I(H^2\Delta_g) \geq C\chi_I(H^2\Delta_g)^2, $$ where $H\uparrow \infty$ is a \emph{large} parameter, $I$ is a compact interval in $(0,\infty),$ and $\chi_I$ 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 Schr\"odinger operator., Comment: One new reference added

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 H\"ormander'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., Comment: Updated with an erratum, detailing an error in the proof of Corollary 6.2 and substituting a weaker result. The erratum now precedes the bulk of the manuscript in the pdf file

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 $\Delta$ 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 $(h^2 \Delta + V - (\lambda_0 \pm i0)^2)^{-1}$, at a nontrapping energy $\lambda_0 > 0$, 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, $e^{-it(\Delta/2 + V)}$, $t \in (0, t_0)$ as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.

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., Comment: Revised version: proof of Theorem A pruned, some examples added, hypotheses clarified

We consider Dirichlet eigenfunctions $u_\lambda$ of the Bunimovich stadium $S$, satisfying $(\Delta - \lambda^2) u_\lambda = 0$. 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 $\lambda \to \infty$. We obtain a lower bound $C \lambda^{-2}$ on the $L^2$ mass of $u_\lambda$ in $W$, assuming that $u_\lambda$ itself is $L^2$-normalized; in other words, the $L^2$ norm of $u_\lambda$ is controlled by $\lambda^2$ times the $L^2$ norm in $W$. Moreover, if $u_\lambda$ is a $o(\lambda^{-2})$ quasimode, the same result holds, while for a $o(1)$ quasimode we prove that $L^2$ norm of $u_\lambda$ is controlled by $\lambda^4$ times the $L^2$ norm in $W$. We also show that the $L^2$ norm of $u_\lambda$ 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 $u_\lambda$ is controlled by the $L^2$ norm in any pair of strips contained in $R$, but adjacent to $W$.

Let $X$ be a compact Riemannian manifold with conic singularities, i.e. a Riemannian manifold whose metric has a conic degeneracy at the boundary. Let $\Delta$ be the Friedrichs extension of the Laplace-Beltrami operator on $X.$ There are two natural ways to define geodesics passing through the boundary: as ``diffractive'' geodesics which may emanate from $\partial X$ in any direction, or as ``geometric'' geodesics which must enter and leave $\partial X$ at points which are connected by a geodesic of length $\pi$ in $\partial X.$ Let $\DIFF=\{0\} \cup \{\pm lengths of closed diffractive geodesics\}$ and $\GEOM=\{0\} \cup \{\pm lengths of closed geometric geodesics\}.$ We show that $$ \Tr \cos t \sqrt\Delta \in C^{-n-0}(\RR) \cap C^{-1-0}(\RR\backslash \GEOM) \cap C^\infty(\RR\backslash \DIFF).$$ This generalizes a classical result of Chazarain and Duistermaat-Guillemin on boundaryless manifolds, which in turn follows from Poisson summation in the case $X=S^1.$, Comment: Exposition substantially improved. 1 figure added. Title changed