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 in
itial data, we show a wave locally enjoys the decay rate $O(t^{-kappa-2+epsilon})$.
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.
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 show
s 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 $pgeq C_2 log k$, where $C_1$ is sufficiently small, $C_2$ is sufficiently large, and both are independent of $k,h,$ and $p$. 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.
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 Arnold, 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.
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 80s. 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 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 $pgeq 2$. 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 $|chi R(k) chi|_{L^2to L^2}leq C{k}^{-1}$ 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.
We consider the Schrodinger 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 singulari
ties 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, u_0 h log(1/h)]. ] The constant $ u_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.
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 anal
ytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2pi$. 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, $mathbb{R}mathbb{P}^2$.
We establish propagation of singularities for the semiclassical Schrodinger 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.
