arXiv subjects

Alden Waters

Publications and source records attributed to Alden Waters.

At least 19 recordsLinked to original sources

arXiv2026

The scattering matrix for the p-form Laplacian on asymptotically conic manifolds

We give an explicit description of the scattering matrix for the Hodge-Laplacian on co-closed p-forms on asymptotically conic manifolds of dimension $n\geq 3$. We develop generalized eigenfunctions and a functional calculus to describe the result, a departure from the Fourier Integral Operators used in the scalar case. Starting from the Hodge decomposition at infinity, we construct generalized eigenforms which are co-closed and establish a spectral representation for the p-form Hodge Laplacian. The special case of dimension 3 for $p=1$ characterises the electric field for Maxwell's equations.

math.AP↗

arXiv2025

[Dispersion for the wave equation in the exterior of the torus]{Dispersion for the wave equation in the exterior of the torus in three dimensions}

We prove dispersive estimates for the wave equation in the exterior of a torus. Because no separation of variables into a basis of eigenfunctions and eigenvalues exists for the time harmonic problem, we introduce a related approximate operator for the Dirichlet Laplacian in the exterior of a torus. The approximate operator coincides with the Schr\"odinger operator with a P\"oschl-Teller potential and agrees with the Dirichlet Laplacian to leading order. The operator here which we develop is related to the so-called Mehler-Fock kernel. Using the known solution to the eigenvector and eigenvalue problem of P\"oschl-Teller, a high-frequency analysis of the approximate operator for the wave equation can be made accurately. The operator for this problem gives a close approximation to the $L^1\rightarrow L^{\infty}$ dispersive estimate at a suitable small distance from the torus for the corresponding exterior wave operator with Dirichlet Laplacian.

math.AP↗

arXiv2023

Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere

The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in $L^1-L^{\infty}$ operator norm in terms of time $t$ and powers of $h$. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator $L^1-L^{\infty}$ norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem.

math.AP↗

arXiv2023

Approximation of the non-linear water hammer problem by a Lax-Wendroff finite difference scheme

We study the water hammer problem in the case of a sudden closing of a valve upstream, and we consider a Lax-Wendroff finite difference scheme in order to obtain a numerical solution of this problem. In order to establish the approximation of this scheme to the original case, we rigorously show some properties such as consistency, stability and weak convergence of the scheme under reasonable conditions. In addition, we present some numerical simulations in order to show some features of the numerical method.

math.NA↗

arXiv2022

A relative trace formula for obstacle scattering

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. In this setting the absolutely continuous part of the Laplace operator $Δ$ with Dirichlet boundary conditions and the free Laplace operator $Δ_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(Δ)-g(Δ_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this paper we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles we consider the Laplace operators $Δ_1$ and $Δ_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(Δ) - g(Δ_1) - g(Δ_2) + g(Δ_0)$ is a trace class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman-Krein formula. In case $g(x)=x^\frac{1}{2}$ the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using non-rigorous path integral derivations and our formula provides both a rigorous justification as well as a generalisation.

math.SP↗

arXiv2022

Observability for Schrödinger equations with quadratic Hamiltonians

We consider time dependent harmonic oscillators and construct a parametrix to the corresponding Schrödinger equation using Gaussian wavepackets. This parametrix of Gaussian wavepackets is precise and tractable. Using this parametrix we prove $L^2$ and $L^2-L^{\infty}$ observability estimates on unbounded domains $ω$ for a restricted class of initial data. This data includes a class of compactly supported piecewise $C^1$ functions which have been extended from characteristic functions. Initial data of this form which has the bulk of its mass away from $ω^c=Ω$, a connected bounded domain, is observable, but data centered over $Ω$ must be very nearly a single Gaussian to be observable. We also give counterexamples to established principles for the simple harmonic oscillator in the case of certain time dependent harmonic oscillators.

math.AP↗

arXiv2022

The Birman-Krein formula for differential forms and electromagnetic scattering

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean near infinity. Allowing for compact boundaries of low regularity we prove a Birman-Krein formula on the space of co-closed differential forms. In the case of dimension three this reduces to a Birman-Krein formula in Maxwell scattering.

math.SP↗

arXiv2022

Analytic properties of heat equation solutions and reachable sets

There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result holds in any dimension. For the heat equation on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ at positive time all solutions are analytically extendable to a geometrically determined subdomain $\mathcal{E}(Ω)$ of $\mathbb{C}^d$ containing $Ω$. This domain is sharp in the sense that there is no larger domain for which this is true. If $Ω$ is a ball we prove an almost converse of this theorem. Any function that is analytic in an open neighborhood of $\mathcal{E}(Ω)$ is reachable in the sense that it can be obtained from a solution of the heat equation at positive time. This is based on an analysis of the convergence of heat equation solutions in the complex domain using the boundary layer potential method for the heat equation. The converse theorem is obtained using a Wick rotation into the complex domain that is justified by our results. This gives a simple explanation for the shapes appearing in the one-dimensional analysis of the problem in the literature. It also provides a new short and conceptual proof in that case.

math.AP↗

arXiv2021

The relative trace formula in electromagnetic scattering and boundary layer operators

This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in $\mathbb{R}^3$. Spectral and scattering theory of the absolute and relative Laplacian is equivalent to the spectral analysis and scattering theory for Maxwell equations. The trace-formulae allow for unbounded functions in the functional calculus that are not admissible in the Birman-Krein formula. In special cases the trace-formula reduces to a determinant formula for the Casimir energy that is being used in the physics literature for the computation of the Casimir energy for objects with metallic boundary conditions. Our theorems justify these formulae in the case of electromagnetic scattering on Lipschitz domains, give a rigorous meaning to them as the trace of certain trace-class operators, and clarifies the function spaces on which the determinants need to be taken.

math.AP↗

arXiv2020

Geometric and Obstacle Scattering at Low Energy

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case of functions eigenvalues generally exist at the bottom of the continuous spectrum and the corresponding eigenforms represent cohomology classes. We show that these eigenforms appear in the expansion of the resolvent, the scattering matrix, and the spectral measure in terms of the spectral parameter $λ$ near zero, and we determine the first terms in this expansion explicitly. In dimension two an additional cohomology class appears as a resonant state in the presence of an obstacle. In even dimensions the expansion is in terms of $λ$ and $\log λ$. The theory of Hahn holomorphic functions is used to describe these expansions effectively. We also give a Birman-Krein formula in this context. The case of one forms with relative boundary conditions has direct applications in physics as it describes the scattering of electromagnetic waves.

math.AP↗

arXiv2019

Rank optimality for the Burer-Monteiro factorization

When solving large scale semidefinite programs that admit a low-rank solution, an efficient heuristic is the Burer-Monteiro factorization: instead of optimizing over the full matrix, one optimizes over its low-rank factors. This reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points. The article [Boumal, Voroninski, and Bandeira, 2018] shows that when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: for almost any cost matrix, second-order critical points are global solutions. In this article, we show that this result is essentially tight: for smaller values of the size, second-order critical points are not generically optimal, even when the global solution is rank 1.

math.OC↗

arXiv2018

Unique Determination of Sound Speeds for Coupled Systems of Semi-linear Wave Equations

We consider coupled systems of semi-linear wave equations with different sound speeds on a finite time interval $[0,T]$ and a bounded Lipschitz domain $Ω$ in $\mathbb{R}^3$, with boundary $\partialΩ$. We show the coupled systems are well posed for variable coefficient sounds speeds and short times. Under the assumption of small initial data, we prove the source to solutions map on $[0,T]\times\partialΩ$ associated with the nonlinear problem is sufficient to determine the source-to-solution map for the linear problem. We can then reconstruct the sound speeds in $Ω$ for the coupled nonlinear wave equations under certain geometric assumptions. In the case of the full source to solution map in $Ω\times[0,T]$ this reconstruction could also be accomplished under fewer geometric assumptions.

math.AP↗

arXiv2018

Asymptotics for optimal design problems for the Schrödinger equation with a potential

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential in $L^{\infty}(Ω)$, with $Ω\subset \mathbb{R}^d$, using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.

math.AP↗

arXiv2018

Low Regularity Ray Tracing for Wave Equations with Gaussian beams

We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and observability regions are in $\mathbb{R}^n$, and the GCC applies. This work generalizes previous observability estimates to higher dimensions and time dependent coefficients. The construction for the Gaussian beamlets solving $C^{1,1}$ wave equations represents an improvement and simplification over Waters (2011).

math.AP↗

arXiv2016

Recovery of the sound speed for the Acoustic wave equation from phaseless measurements

We recover the higher order terms for the acoustic wave equation from measurements of the modulus of the solution. The recovery of these coefficients is reduced to a question of stability for inverting a Hamiltonian flow transform, not the geodesic X-ray transform encountered in other inverse boundary problems like the determination of conformal factors. We obtain new stability results for the Hamiltonian flow transform, which allow to recover the higher order terms.

math.AP↗

arXiv2016

A deterministic optimal design problem for the heat equation

For the heat equation on a bounded subdomain $Ω$ of $\mathbb{R}^d$, we investigate the optimal shape and location of the observation domain in observability inequalites. A new decomposition of $L^2(\mathbb{R}^d)$ into heat packets allows us to remove the randomisation procedure and assumptions on the geometry of $Ω$ in previous works. The explicit nature of the heat packets gives new information about the observability constant in the inverse problem.

math.AP↗

arXiv2015

Lower resolvent bounds and Lyapunov exponents

We prove a new polynomial lower bound on the scattering resolvent. For that, we construct a quasimode localized on a trajectory $γ$ which is trapped in the past, but not in the future. The power in the bound is expressed in terms of the maximal Lyapunov exponent on $γ$, and gives the minimal number of derivatives lost in exponential decay of solutions to the wave equation.

math.AP↗

arXiv2015

Stability for Time Dependent X-ray Transforms and Applications

We prove a logarithmic stability estimate for the time dependent X-ray transform on $\mathbb{R}_t^+\times\mathbb{R}^n$. To do so, we extend a known result by Begmatov for the stability of the time dependent X-ray transform in $\mathbb{R}^+_t\times\mathbb{R}^2$. We give some examples of stability and injectivity results in relationship to the Dirichlet-to-Neumann problem. In particular, under the Geometric Control Condtion, we derive inverse logarithmic stability estimates for time dependent conformal factors.

math.AP↗