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We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of $z$, the norm of the solution operator is bounded by that function. This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows superalgebraically, with these frequencies often called $\textit{quasi-resonances}$. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations.

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated., Comment: v2: 31 pages, added a Morse Index Theorem for composition of paths

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.

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., Comment: 58 pgs; Lemma 9.3 has been updated

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 consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML width and the scaling angle, and the approximation was recently proved to be exponentially accurate in the wavenumber $k$ in [Galkowski, Lafontaine, Spence, 2021]. We show that the $hp$-FEM applied to this problem does not suffer from the pollution effect, in that there exist $C_1,C_2>0$ such that if $hk/p\leq C_1$ and $p \geq C_2 \log k$ then the Galerkin solutions are quasioptimal (with constant independent of $k$), under the following two conditions (i) the solution operator of the original Helmholtz problem is polynomially bounded in $k$ (which occurs for "most" $k$ by [Lafontaine, Spence, Wunsch, 2021]), and (ii) either there is no obstacle and the coefficients are smooth or the obstacle is analytic and the coefficients are analytic in a neighbourhood of the obstacle and smooth elsewhere. This $hp$-FEM result is obtained via a decomposition of the PML solution into "high-" and "low-frequency" components, analogous to the decomposition for the original Helmholtz solution recently proved in [Galkowski, Lafontaine, Spence, Wunsch, 2022]. The decomposition is obtained using tools from semiclassical analysis (i.e., the PDE techniques specifically designed for studying Helmholtz problems with large $k$).

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).

A static rotating cosmic string metric is singular along a timelike line and fails to be globally hyperbolic; these features make it difficult to solve the wave equation by conventional energy methods. Working on a single angular mode at a time, we use microlocal methods to construct forward parametrices for wave and Klein--Gordon equations on such backgrounds., Comment: 15 pages; improvements to exposition in response to referee comments

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

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.

The Dirac equation in $\mathbb{R}^{1,3}$ 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 1-0 derivatives smoother than the main singularities and is a conormal singularity., Comment: 51 pages, 1 figure; version 2: minor changes to introduction

A convergence theory for the $hp$-FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [Melenk-Sauter, 2010], [Melenk-Sauter, 2011], [Esterhazy-Melenk, 2012], [Melenk-Parsania-Sauter, 2013]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber $k$, then the Galerkin method is quasioptimal provided that $hk/p \leq C_1$ and $p\geq C_2 \log k$, where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both are independent of $k,h,$ and $p$. The significance of this result is that if $hk/p= C_1$ and $p=C_2\log k$, then quasioptimality is achieved with the total number of degrees of freedom proportional to $k^d$; i.e., the $hp$-FEM does not suffer from the pollution effect. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable-coefficient) Helmholtz equation, posed in $\mathbb{R}^d$, $d=2,3$, with the Sommerfeld radiation condition at infinity, and $C^\infty$ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem. These are the first ever results on the wavenumber-explicit convergence of the $hp$-FEM for the Helmholtz equation with variable coefficients.

We study semiclassical sequences of distributions $u_h$ associated to a Lagrangian submanifold of phase space $\lag \subset T^*X$. If $u_h$ is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on $\lag,$ then the asymptotics of $u_h$ are well-understood by work of Arnol'd, provided $\lag$ projects to $X$ with a stable Lagrangian singularity. We establish sup-norm estimates on $u_h$ under much more general hypotheses on the rate at which it is concentrating on $\lag$ (again assuming a stable projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians., Comment: 33 pages, 3 tables, 1 figure

For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth $h$ must decrease with the frequency $k$ to maintain accuracy as $k$ increases has been studied since the mid 80's. Nevertheless, there still do not exist in the literature any $k$-explicit bounds on the relative error of the FEM solution (the measure of the FEM error most often used in practical applications), apart from in one dimension. The main result of this paper is the sharp result that, for the lowest fixed-order conforming FEM (with polynomial degree, $p$, equal to one), the condition "$h^2 k^3$ sufficiently small" is sufficient for the relative error of the FEM solution in 2 or 3 dimensions to be controllably small (independent of $k$) for scattering of a plane wave by a nontrapping obstacle and/or a nontrapping inhomogeneous medium. We also prove relative-error bounds on the FEM solution for arbitrary fixed-order methods applied to scattering by a nontrapping obstacle, but these bounds are not sharp for $p\geq 2$. A key ingredient in our proofs is a result describing the oscillatory behaviour of the solution of the plane-wave scattering problem, which we prove using semiclassical defect measures.

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.

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/\pi$ times the length of the longest generalized bicharacteristic of $|\xi|_g^2-1$ remaining in the support of $\chi.$ We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation., Comment: 40 pages

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

A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We generalize this question somewhat and study cones $[0,\infty)\times Y$ which do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2\pi$. Moreover, we show that if $\dim Y=2$, then $Y$ is isometric to either the sphere of radius 1 or its $\mathbb{Z}^2$ quotient, $\mathbb{R}\mathbb{P}^2$., Comment: Updated in response to referee comments. 13 pages

We establish propagation of singularities for the semiclassical Schr\"odinger equation, where the potential is conormal to a hypersurface. We show that semiclassical wavefront set propagates along generalized broken bicharacteristics, hence reflection of singularities may occur along trajectories reaching the hypersurface transversely. The reflected wavefront set is weaker, however, by a power of $h$ that depends on the regularity of the potential. We also show that for sufficiently regular potentials, wavefront set may not stick to the hypersurface, but rather detaches from it at points of tangency to travel along ordinary bicharacteristics., Comment: 99 pages; exposition revised in response to referee comments

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

In this survey, we review some applications and extensions of the author's results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet)., Comment: 15 pages; contribution to Seminaire Laurent Schwartz proceedings

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 $u$ solve the damped Klein--Gordon equation $$ \big( \partial_t^2-\sum \partial_{x_j}^2 +m \text{Id} +\gamma(x) \partial_t \big) u=0 $$ on $\mathbb{R}^n$ with $m>0$ and $\gamma\geq 0$ bounded below on a $2 \pi \mathbb{Z}^n$-invariant open set by a positive constant. We show that the energy of the solution $u$ decays at a polynomial rate. This is proved via a periodic observability estimate on $\mathbb{R}^n.$, Comment: 7 pages; final version takes into account referee comments

We consider three problems for the Helmholtz equation in interior and exterior domains in R^d (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., Comment: Version 3: 42 pages; improved exposition in response to referee comments and added several references

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