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In this article, we study the propagation of defect measures for Schrödinger operators - h 2 Δ 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 tangential 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 tangential to Y at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold. [ABSTRACT FROM AUTHOR]

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 \operatorname {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. [ABSTRACT FROM AUTHOR]

A rotating cosmic string spacetime has a singularity along a timelike curve corresponding to a one-dimensional source of angular momentum. Such spacetimes are not globally hyperbolic: they admit closed timelike curves near the string. This presents challenges to studying the existence of solutions to the wave equation via conventional energy methods. In this work, we show that semi-global forward solutions to the wave equation do nonetheless exist, but only in a microlocal sense. The main ingredient in this existence theorem is a propagation of singularities theorem that relates energy entering the string to energy leaving the string. The propagation theorem is localized in the fibers of a certain fibration of the blown-up string, but global in time, which means that energy entering the string at one time may emerge previously. [ABSTRACT FROM AUTHOR]

The Dirac equation in R 1 , 3 with potential Z / r is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is 1 - ϵ derivatives smoother than the main singularities, for all ϵ > 0 , and is a conormal singularity. [ABSTRACT FROM AUTHOR]

Maas (J. Fluid Mech., vol. 684, 2011, pp. 5–24) showed that, for an oscillating two-dimensional barotropic tide flowing over sub-critical topography of compact support, some topographic forms existed that produced non-radiating baroclinic disturbances. The problem is related to 'stealth' and 'cloaking' problems. Here Maas's result is derived using a simpler approach, not involving complicated mappings, but formally restricted to perturbation topography. Wider results come from the discussion of nearly compact support topographic disturbances provided by Schwartz functions with weak high-wavenumber radiation and by exploiting both a known functional equation formulation and Fourier methods. The problem is extended to disturbances on uniform slopes. A variety of non-radiating topographies can be found, although they are mathematically delicate and unlikely to be found in nature. Topography with weak radiation at high wavenumber is a much wider class of structures. Application of these solutions would lie with the ability to estimate dissipation over and near the topography from motions observed at a distance. [ABSTRACT FROM AUTHOR]

We study semiclassical sequences of distributions u h associated with a Lagrangian submanifold of phase space L ⊂ T ∗ X . If u h is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on L , then the asymptotics of u h are well understood by work of Arnol'd, provided L projects to X with a stable simple Lagrangian singularity. We establish sup-norm estimates on u h under much more general hypotheses on the rate at which it is concentrating on L (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator P = h²Δg + V on ℝn 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 - α along a compact hypersurface Y. For α > 2 (or even α > 1 if the classical flow is unique), we show that if E0 is a non-trapping energy for the classical flow, then the operator P has no resonances in a region [E0 - δ, E0 +δ] - i[0, ν0h log(1/h)]. The constant ν0 is explicit in terms of α 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. [ABSTRACT FROM AUTHOR]

Let H denote the harmonic oscillator Hamiltonian on Rd, perturbed by an isotropic pseudodifferential operator of order 1. We consider the Schrödinger propagator U(t)=e-itH, and find that while sing-suppTrU(t)⊂2πZ as in the unperturbed case, there exists a large class of perturbations in dimensions d≥2 for which the singularities of TrU(t) at nonzero multiples of 2π are weaker than the singularity at t = 0. The remainder term in the Weyl law is of order o(λd-1), improving in these cases the o(λd-1) remainder previously established by Helffer-Robert. [ABSTRACT FROM AUTHOR]

We consider three problems for the Helmholtz equation in interior and exterior domains in Rd (d = 2, 3): the exterior Dirichlet-to-Neumann and Neumann-to-Dirichlet problems for outgoing solutions, and the interior impedance problem. We derive sharp estimates for solutions to these problems that, in combination, give bounds on the inverses of the combined-field boundary integral operators for exterior Helmholtz problems. [ABSTRACT FROM AUTHOR]

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, our asymptotic expansions yield a rate of decay that improves on the Klainerman-Sobolev estimates. [ABSTRACT FROM AUTHOR]

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 undamped set. We show that if replacing the damping term with a higher-order 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 a single closed orbit if all frequencies of the flow are rationally related. We show that, subject to non-degeneracy hypotheses, this concentration may not in fact occur. Indeed, in the two-dimensional case, we show that WF h must fill out an entire Lagrangian torus. The main tools are the spreading of Lagrangian regularity previously shown by the author, and an analysis of higher order transport equations satisfied by the principal symbol of a Lagrangian quasimode. These yield a unique continuation theorem for the principal symbol of Lagrangian quasimode, which is the principal new result of the paper. [ABSTRACT FROM PUBLISHER]

We give pole free strips and estimates for resolvents of semiclassical operators which, on the level of the classical flow, have normally hyperbolic smooth trapped sets of codimension two in phase space. Such trapped sets are structurally stable and our motivation comes partly from considering the wave equation for Kerr black holes and their perturbations, whose trapped sets have precisely this structure. We give applications including local smoothing effects with epsilon derivative loss for the Schrödinger propagator as well as local energy decay results for the wave equation. [ABSTRACT FROM AUTHOR]

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 *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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We obtain the Strichartz inequalities ∥u∥LtqLxr([0,1]×M) ≤ C∥u(0)∥L²(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 nontrapping, where u is a solution to the Schrödinger equation iut + ½ΔMu = 0, and 2 < q, r ≤ ∞ are admissible Strichartz exponents (2⁄2 + n⁄r = n⁄2). This corresponds with the estimates available for Euclidean space (except for the endpoint (q, r) = (2, (Multiple line equation(s) cannot be represented in ASCII text)) when n > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on M. by adapting arguments from Cazenave and Weissler and Kato. This result improves on our previous result, which was an Lt,x4 Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically fiat manifolds. [ABSTRACT FROM AUTHOR]

We obtain the Strichartz inequality for any smooth three-dimensional Riemannian manifold ( M , g ) which is asymptotically conic at infinity and nontrapping, where u is a solution to the Schrödinger equation iu t + (1/2)? M u = 0. The exponent H 1/4 ( M ) is sharp, by scaling considerations. In particular our result covers asymptotically flat nontrapping 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 one (proved here) with weight r -1 at infinity and with the gradient term involving only one angular derivative. [ABSTRACT FROM AUTHOR]

For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under appropriate regularity assumptions the diffracted, non-direct, wave produced by the boundary is shown to have Sobolev regularity greater than the incoming wave. [ABSTRACT FROM AUTHOR]