arXiv subjects
Publications and source records attributed to James Ralston.
In this article we consider a Bloch electron in a crystal lattice subject to slowy varying external magnetic fields. We offer an explanation of de Haas-van Alfven oscillations in terms of energy levels of approximate eigenfunctions for the magnetic Schrodinger operator by using a gaussian beam construction for a small enough magnetic field.
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 to generate more general high frequency solutions to PDEs. We present a superposition of Gaussian beams over an arbitrary bounded set of dimension $m$ in phase space, and show that the tools recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp., 82: 919--952, 2013] can be applied to obtain the propagation error of order $k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is the spatial dimension. Moreover, we study the sharpness of this estimate in examples.
We consider Friedlander's wave equation in two space dimensions in the half-space x > 0 with the boundary condition u(x,y,t)=0 when x=0. For a Gaussian beam w(x,y,t;k) concentrated on a ray path that is tangent to x=0 at (x,y,t)=(0,0,0) we calculate the "reflected" wave z(x,y,t;k) in t > 0 such that w(x,y,t;k)+z(x,y,t;k) satisfies Friedlander's wave equation and vanishes on x=0. These computations are done to leading order in k on the ray path. The interaction of beams with boundaries has been studied for non-tangential beams and for beams gliding along the boundary. We find that the amplitude of the solution on the central ray for large k after leaving the boundary is very nearly one half of that of the incoming beam.
We show that in the absence of a magnetic field the spectrum of the magnetic Schrödinger operator in an annulus depends on the cosine of the flux associated with the magnetic potential. This result follows from an analysis of a singularity in the wave trace for this Schrödinger operator, and hence shows that even in the absence of a magnetic field the magnetic potential can change the asymptotics of the Schrödinger spectrum, i.e. the Aharonov-Bohm effect takes place. We also study the Aharonov-Bohm effect for the magnetic Schrödinger operator on a torus.
In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of non-trapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number $k$, both for single beams and superposition of beams. The main result is that the relative local $L^2$ error in the beam approximations decay as {$k^{-N/2}$ independent of dimension and presence of caustics, for $N$-th order beams.
We give a simple proof of Guillemin's theorem on the determination of the magnetic field on the torus by the spectrum of the corresponding Schrödinger operator.
We consider wave equations in domains with time-dependent boundaries (moving obstacles) contained in a fixed cylinder for all time. We give sufficient conditions for the determination of the moving boundary from the Cauchy data on part of the boundary of the cylinder. We also study the related problem of accessibility of the moving boundary by time-like curves from the boundary of the cylinder.
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 to generate 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. Following \cite{LR:2009} we present a systematic construction of asymptotic high frequency wave fields from computations in phase space for acoustic wave equations; the superposition of phase space based Gaussian beams over two moving domains is shown necessary. Moreover, we prove that the $k$-th order Gaussian beam superposition converges to the original wave field in the energy norm, at the rate of $\ep^{\frac{k}{2}+\frac{1-n}{4}}$ in dimension $n$.
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 to generate 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 $k^{th}$ order phase space based Gaussian beam superposition converges to the original wave field in $L^2$ at the rate of $\ep^{\frac{k}{2}-\frac{n}{4}}$ in dimension $n$.
We consider the problem of unique identification of dielectric coefficients for gratings and sound speeds for wave guides from scattering data. We prove that the "propagating modes" given for all frequencies uniquely determine these coefficients. The gratings may contain conductors as well as dielectrics and the boundaries of the conductors are also determined by the propagating modes.
We consider the Schrodinger operator in n-dimensional rectangular domains with either Dirichlet or Neumann boundary conditions on the faces and study the constraints on the potential imposed by fixing the spectrum of the operator.We study also the asymptotics of the heat kernel when t tends to 0.
We describe a simple way of constructing exponentially growing solutions of the second order systems with the Laplacian as the principal term.