Spectral and scattering theory for symbolic potentials of order zero

The spectral and scattering theory is investigated for a generalization, to scattering metrics on two-dimensional compact manifolds with boundary, of the class of smooth potentials on which are homogeneous of degree zero near infinity. The most complete results require the additional assumption that the restriction of the potential to the circle(s) at infinity be Morse. Generalized eigenfunctions associated to the essential spectrum at non-critical energies are shown to originate both at minima and maxima, although the latter are not germane to the L² spectral theory. Asymptotic completeness is shown, both in the traditional L² sense and in the sense of tempered distributions. This leads to a definition of the scattering matrix, the structure of which will be described in a future publication.     

Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rⁿ, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rⁿ) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N(r)?A_nrⁿ+o(rⁿ) with an explicit A_n. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.     

Scattering theory for the Schrödinger equation in some external time-dependent magnetic fields

We study the theory of scattering for a Schrödinger equation in an external time-dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schrödinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least asymptotic results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of space derivatives.     

A precise pseudodifferential Foldy-Wouthuysen transform for the Dirac equation

We will discuss existence of a unitary pseudodifferential operator U in our algebra of strictly classical pseudodifferential operators on such that U precisely decouples the electronic and positronic part of the Dirac equation, for rather general potentials, and without supersymmetry. Interestingly, an obstruction appears: On may have to remove a finite dimensional space of electronic states, and declare them as positronic, or, vice versa, depending on a certain deficiency index. Possibly, this index is nonzero if electronic bound states penetrate into the positronic continuous spectrum, or vice versa.     

Weighted energy decay for 3D Klein-Gordon equation

We obtain a dispersive long-time decay in weighted energy norms for solutions of the 3D Klein-Gordon equation with generic potential. The decay extends the results obtained by Jensen and Kato for the 3D Schrödinger equation. For the proof we modify the spectral approach of Jensen and Kato to make it applicable to relativistic equations.     

Energy decay for Timoshenko systems of memory type

Linear systems of Timoshenko type equations for beams including a memory term are studied. The exponential decay is proved for exponential kernels, while polynomial kernels are shown to lead to a polynomial decay. The optimality of the results is also investigated.     

An approximation problem in inverse scattering theory

In [3] a new method was introduced for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. This method is based on the solution of a new class of boundary value problems for the reduced wave equation called interior transmission problems. In this paper it is shown that if there is absorption there exists at most one solution to the interior transmission problem and an approximate solution can be found such that the metaharmonic part is a Herglotz wave function. These results provide the necessary theoretical basis for the inverse scattering method introduced in [3]     

Singularity of eigenfunctions at the junction of shrinking tubes,Part II

In continuation with [17], we investigate the asymptotic behavior of weighted eigenfunctions in two half-spaces connected by a thin tube. We provide several improvements about some convergences stated in [17]; most of all, we provide the exact asymptotic behavior of the implicit normalization for solutions given in [17] and thus describe the (N−1)$(N-1)$-order singularity developed at a junction of the tube (where N is the space dimension).     

Preparing materials with a desired refraction coefficient

A recipe is given for creating material with a desired refraction coefficient by embedding many small particles in a given material. To implement this recipe practically, some technological problems have to be solved. These problems are formulated.     

A probabilistic framework for the continuous observation of a quantum process

This paper treats quantum measurement within von Neumann's abstract framework. Specifically, observation is defined as a fixed self-adjoint operator with countable spectrum and nondegenerate eigenstates. Suppose scenarios for the observation of a quantum process over time are expanded by adding extra observations at time points interspersed among those of a previous scenario. If each observation leads to a mixture of eigenstates rather than a pure state, then the naturally defined joint probability measures on observed results are not consistent as scenarios vary. Nevertheless, we characterize the limiting subprobability measure when the times of observation become infinitely dense in any finite interval. This limiting measure corresponds to a continuous-time sub-stochastic process which decays with exponential rate out of any initial state and never reappears in any other state. Thus the process loses probability exponentially over time, and this loss occurs equally fast in the case of nonselective observation as for selective observation.Previous treatments of this problem have concentrated on the special case when Zeno's Paradox is in force, i.e. the rate of decay out of any state is zero and the process is immobilized by continuous observation. This situation exists, for instance, when the initial state is in the domain of the generator for the unitary group underlying the quantum process.     

Some remarks on Sobolev type inequalities

We extend two Sobolev type inequalities for balls to arbitrary smooth bounded domains. In the case of balls, one inequality is due to Brezis and Lieb and another is due to Escobar. The extension has been achieved by analyzing the asymptotic behaviour of solutions of certain semilinear Neumann problems.     

The inverse scattering problem for Schrödinger and Klein–Gordon equations with a nonlocal nonlinearity

We study the inverse scattering problem for the nonlinear Schrödinger equation and for the nonlinear Klein–Gordon equation with the generalized Hartree type nonlinearity. We reconstruct the nonlinearity from knowledge of the scattering operator, which improves the known results.     

A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

A New Look at Mourre’s Commutator Theory

Mourre’s commutator theory is a powerful tool to study the continuous spectrum of self-adjoint operators and to develop scattering theory. We propose a new approach of its main result, namely the derivation of the limiting absorption principle (LAP) from a so called Mourre estimate. We provide a new interpretation of this result. Received: September 20, 2006. Revised: February 1, 2007.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in where the linear term is given by Schr?dinger operators H = − Δ + V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|². Dedicated to Professor Jun Uchiyama on the occasion of his sixtieth birthday Received: May 4, 2004     

Exponential decay for semilinear distributed systems with damping

The stability of second order abstract distributed systems with damping and nonlinear perturbations is considered. Sufficient conditions, including unique continuation property assumptions, are formulated to obtain (local, non-uniform and uniform) exponential stability. Applications to the wave and Euler-Bernoulli equations are given.     

Uniqueness in inverse obstacle scattering with conductive boundary conditions

We consider the uniqueness of the inverse obstacle scattering with conductive boundary conditions. This work is based on the original idea of Isakov for transmission boundary conditions, which utilize the solvability of the direct problem, orthogonality relations, approximations to solution of the direct problem and singular solutions. The methodology used is constructive and allows an extension to more general conditions and numerical methods.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.