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Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing high-frequency Helmholtz solutions into "low"- and "high"-frequency components have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficient Helmholtz equation in either the exterior of a Dirichlet obstacle or an interior domain with an impedance boundary condition. Using the Helffer-Sj\"ostrand functional calculus, this paper proves analogous decompositions for scattering problems fitting into the black-box scattering framework of Sj\"ostrand-Zworski, thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allow us to prove new frequency-explicit convergence results for (i) the $hp$-finite-element method applied to the variable coefficient Helmholtz equation in the exterior of a Dirichlet obstacle, when the obstacle and coefficients are analytic, and (ii) the $h$-finite-element method applied to the Helmholtz penetrable-obstacle transmission problem. In particular, the result in (i) shows that the $hp$-FEM applied to this problem does not suffer from the pollution effect.

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: 16 pages, 3 figures, comments are welcome!

A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber $k$? The recent paper [Ganesh, Kuo, Sloan 2021] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine dependence of the coefficients on the stochastic parameters that the solution operator can be analytically continued a distance $\sim k^{-1}$ into the complex plane. In this paper, we generalise the result in [Ganesh, Kuo, Sloan 2021] about $k$-explicit parametric holomorphy to a much wider class of Helmholtz problems with arbitrary (holomorphic) dependence on the stochastic parameters; we show that in all cases the region of parametric holomorphy decreases with $k$, and show how the rate of decrease with $k$ is dictated by whether the unperturbed Helmholtz problem is trapping or nontrapping. We then give examples of both trapping and nontrapping problems where these bounds on the rate of decrease with $k$ of the region of parametric holomorphy are sharp, with the trapping examples coming from the recent results of [Galkowski, Marchand, Spence 2021]. An immediate implication of these results is that the $k$-dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo (QMC) methods in [Ganesh, Kuo, Sloan 2021] arise from a genuine feature of the Helmholtz equation with $k$ large (and not, for example, a suboptimal bound).

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

It is well known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.

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 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 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 Schr��dinger equation $iu_t + {1/2} ��_M u = 0$, 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 Schr��dinger 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., 50 pages, 2 figures

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 Schr\"odinger equation $iu_t + {1/2} \Delta_M u = 0$. 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 Schr\"odinger solutions including a new one proved here with weight $r^{-1}$ at infinity and with the gradient term involving only one angular derivative., Comment: 43 pages, 1 figure; minor corrections to earlier version

We discuss the form of the propagator $U(t)$ for the time-dependent Schr\"odinger equation on an asyptotically Euclidean, or, more generally, asymptotically conic, manifold with no trapped geodesics. In the asymptotically Euclidean case, if $\chi \in \mathcal{C}_0^\infty$, and with $\mathcal{F}$ denoting Fourier transform, $\mathcal{F}\circ e^{-ir^2/2t} U(t) \chi$ 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 Schr\"odinger 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.