James Ralston

INVERSE SCATTERING PROBLEM FOR THE SCHRODINGER EQUATION WITH MAGNETIC POTENTIAL AT A FIXED ENERGY

In this article we consider the Schrödinger operator inRn,n≧3, with electric and magnetic potentials which decay exponentially as |x|→∞. We show that the scattering amplitude at fixed positive energy determines the electric potential and the magnetic field.

A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition

Abstract:The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator as Plancks constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of . Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof.

Mountain Waves and Gaussian Beams

Gaussian beams are approximate solutions to hyperbolic partial differential equations that are concentrated on a curve in space-time. In this paper, we present a method for computing the stationary in time wave field that results from steady air flow over topography as a superposition of Gaussian beams. We derive the system of equations that governs these mountain waves as a linearization of the basic equations of fluid dynamics and show that this system is well-posed. Furthermore, we show that the approximate Gaussian beam stationary solution is close to a true time-dependent solution of the linearized system.

The inverse backscattering problem in three dimensions

This article is a study of the mapping from a potentialq(x) onR3 to the backscattering amplitude associated with the Hamiltonian −Δ+q(x). The backscattering amplitude is the restriction of the scattering amplitudea(θ, ω, k), (θ, ω, k)εS2×S2×ℝ+, toa(θ,−θ, k). We show that in suitable (complex) Banach spaces the map fromq(x) toa(x/|x|, −x/|x|, |x|) is usually a local diffeomorphism. Hence in contrast to the overdetermined problem of recoveringq from the full scattering amplitude the inverse backscattering problem is well posed.

A Lyapunov functional for the evolution of solutions to the porous medium equation to self‐similarity. II

This article continues the analysis of the preceding article, showing that the Lyapunov functional introduced by Newman can be used to prove the stability of the Barenblatt–Pattle solutions of the porous medium equation.

On the inverse boundary value problem for linear isotropic elasticity

We derive three results on the inverse problem of determining the Lame parameters λ(x) and μ(x) for an isotropic elastic body from its Dirichlet-to-Neumann map.

Semi-classical asymptotics in solid state physics

This article studies the Schrödinger equation for an electron in a lattice of ions with an external magnetic field. In a suitable physical scaling the ionic potential becomes rapidly oscillating, and one can build asymptotic solutions for the limit of zero magnetic field by multiple scale methods from “homogenization.” For the time-dependent Schrödinger equation this construction yields wave packets which follow the trajectories of the “semiclassical model.” For the time-independent equation one gets asymptotic eigenfunctions (or “quasimodes”) for the energy levels predicted by Onsagers relation.

Isospectral sets for boundary value problems on the unit interval

We analyse isospectral sets of potentials associated to a given ‘generalized periodic’ boundary condition in SL(2, R ) for the Sturm-Liouville equation on the unit interval. This is done by first studying the larger manifold M of all pairs of boundary conditions and potentials with a given spectrum and characterizing the critical points of the map from M to the trace a + d Isospectral sets appear as slices of M whose geometry is determined by the critical point structure of the trace function. This paper completes the classification of isospectral sets for all real self-adjoint boundary conditions.

Recovery of High Frequency Wave Fields from Phase Space–Based Measurements

Computation of high frequency solutions to wave equations is important in many applications, and notoriously difficult in resolving wave oscillations. Gaussian beams are asymptotically valid high frequency solutions concentrated on a single curve through the physical domain, and superposition of Gaussian beams provides a powerful tool for generating more general high frequency solutions to PDEs. An alternative way to compute Gaussian beam components such as phase, amplitude, and Hessian of the phase is to capture them in phase space by solving Liouville-type equations on uniform grids. In this work we review and extend recent constructions of asymptotic high frequency wave fields from computations in phase space. We give a new level set method of computing the Hessian and higher derivatives of the phase. Moreover, we prove that the kth order phase space–based Gaussian beam superposition converges to the original wave field in

RECOVERY OF HIGH FREQUENCY WAVE FIELDS FOR THE ACOUSTIC WAVE EQUATION

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