Scattering theory for the Laplacian on symmetric spaces of noncompact type and its application to a conjecture of Strichartz

In this paper we develop the scattering theory for the Laplacian on symmetric spaces of noncompact type. We study the asymptotic properties of the resolvent in the framework of the Agmon–Hörmander space. Our approach is based on a detailed analysis of the Helgason Fourier transform and generalized spherical functions on symmetric spaces of noncompact type. As an application of our scattering theory, we prove a conjecture by Strichartz concerning a characterization of a family of generalized eigenfunctions of the Laplacian.     

Semiclassical approximation of the Dirac equation in a central field

We develop a semiclassical approximation of the Dirac equation in a central field with a Coulomb asymptotic behavior. We obtain relativistic semiclassical scattering phases, energy levels of hydrogen-like ions, and a semiclassical expression for the multiplicative constant in the asymptotic expansion of the wave function of the valence electron in a relativistic multiply charged ion, which plays an important role in quantum defect theory.     

N-body quantum scattering theory in two Hilbert spaces. III. Theory of approximations

A rigorous mathematical theory of approximations is developed for abstract nonrelativistic quantum scattering systems within the two-Hilbert-space framework. An approximate space of asymptotic states and an approximate asymptotic Hamiltonian must be specified initially. An approximate N-particle Hamiltonian is then constructed and proved to be self-adjoint. Approximate wave operators are shown to exist and, in certain interesting cases, to be asymptotically complete. Certain sequences of the approximate wave operators are proved to converge to the exact wave operators in an appropriate limit. Thus approximate scattering operators are unitary and converge to the exact scattering operator.     

Scattering theory for dissipative hyperbolic systems

An abstract theory of scattering is developed for dissipative hyperbolic systems, a typical example is the wave equation: u_tt = Δu in an exterior domain with lossy boundary conditions: u_n + αu_t = 0, α ? 0. In this theory, as in an earlier theory developed by the authors for conservative systems, a central role is played by two distinguished subspaces of data common to both the perturbed and unperturbed problems. Associated with each subspace is a translation representation of the unperturbed system. When these representations coincide they provide a convenient tool for extending the data so as to include a large class of generalized eigenfunctions for both the perturbed and unperturbed generators. The scattering matrix is characterized in terms of these generalized eigenfunctions; it is shown to be meromorphic in the whole complex plane and holomorphic in the lower half plane. The zeroes and poles of the scattering matrix correspond, respectively, to incoming and outgoing generalized eigenfunctions of the perturbed generator. In the second part of the paper the assumptions introduced in the abstract theory are verified for the wave equation problem cited above.     

A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order “rogue wave” solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.     

Time-dependent potential scattering in chiral media

Electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium are considered. We define a chiral operator and study potential scattering relative to this operator. A spectral analysis of associated operators is obtained, based on the Plancherel theory of the Fourier transform. Using the generalised eigenfunction expansion theory, we give an integral representation of the solution. A discussion of asymptotic equality of solutions is provided and the associated wave operator introduced.     

Transmission Eigenvalues and the Riemann Zeta Function in Scattering Theory for Automorphic Forms on Fuchsian Groups of Type I

We introduce the concept of transmission eigenvalues in scattering theory for automorphic forms on fundamental domains generated by discrete groups acting on the hyperbolic upper half complex plane. In particular, we consider Fuchsian groups of type Ⅰ. Transmission eigenvalues are related to those eigen-parameters for which one can send an incident wave that produces no scattering. The notion of transmission eigenvalues, or non-scattering energies, is well studied in the Euclidean geometry, where in some cases these eigenvalues appear as zeros of the scattering matrix. As opposed to scattering poles, in hyperbolic geometry such a connection between zeros of the scattering matrix and non-scattering energies is not studied, and the goal of this paper is to do just this for particular arithmetic groups. For such groups, using existing deep results from analytic number theory, we reveal that the zeros of the scattering matrix, consequently non-scattering energies, are directly expressed in terms of the zeros of the Riemann zeta function. Weyl's asymptotic laws are provided for the eigenvalues in those cases along with estimates on their location in the complex plane.     

Vector potential theory on nonsmooth domains in R<Superscript>3</Superscript> and applications to electromagnetic scattering

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for L^p-integrable bounday data for optimal values of p's. We also discuss a number of relevant applications in electromagnetic scattering.     

Vector Potential Theory on Nonsmooth Domains in R3 and Applications to Electromagnetic Scattering

We study boundary value problems for the time-harmonic form of the Maxwell equations, as well as for other related systems of equations, on arbitrary Lipschitz domains in the three-dimensional Euclidean space. The main goal is to develop the corresponding theory for L^p-integrable bounday data for optimal values of p’s. We also discuss a number of relevant applications in electromagnetic scattering.     

Long range scattering for the Wave–Schrödinger system revisited

We reconsider the theory of scattering for the Wave–Schrödinger system and more precisely the local Cauchy problem with infinite initial time, which is the main step in the construction of the wave operators. Using a method due to Nakanishi, we eliminate a loss of regularity between the Schrödinger asymptotic data and the Schrödinger solution in the treatment of that problem, in the special case of vanishing asymptotic data for the wave field.     

Two-dimensional Coulomb scattering of a quantum particle: Low-energy asymptotic behavior

We consider a charged quantum particle moving in a two-dimensional plane in the three-dimensional coordinate space and scattering on an immovable Coulomb center in the same plane. We derive and investigate expansions of the wave function and of all radial wave functions of the particle in integer powers of the wave number and in Bessel functions of a real order. We prove that finite sums of such expansions are asymptotic in the low-energy limit.     

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.     

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H _m+H_om+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H _m, H _om) is established and asymptotic expansions of the scattering matrix are derived as the energy parameter tends to the lowest Landau threshold.     

Nonlinear Cusped Caustics for Dispersive Waves

A multiple-scale perturbation analysis for slowly varying weakly nonlinear dispersive waves predicts that the wave number breaks or folds and becomes triple-valued. This theory has some difficulties, since the wave amplitude becomes infinite. Energy first focuses along a cusped caustic (an envelope of the rays or characteristics). The method of matched asymptotic expansions shows that a thin focusing region with relatively large wave amplitudes, valid near the cusped caustic, is described by the nonlinear Schrödinger equation (NSE). Solutions of the NSE are obtained from an asymptotic expansion of an equivalent linear singular integral equation related to a Riemann-Hilbert problem. In this way connection formulas before and after focusing are derived. We show that a slowly varying nearly monochromatic wave train evolves into a triple-phased slowly varying similarity solution of the NSE. Three weakly nonlinear waves are simultaneously superimposed after focusing, giving meaning to a triple-valued wave number. Nonlinear phase shifts are obtained which reduce to the linear phase shifts previously described by the asymptotic expansion of a Pearcey integral.     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Given a normally hyperbolic invariant manifold Λ for a map f, whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future.We show that when f and Λ are symplectic (respectively exact symplectic) then, the scattering map is symplectic (respectively exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions.We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometrically natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type.We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [Amadeu Delshams, Rafael de la Llave, Tere M. Seara, Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows, Adv. Math. 202 (1) (2006) 64-188] using methods of fast-slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.     

Long Range Scattering and Modified Wave Operators for the Wave-Schrödinger System II

We continue the study of scattering theory for the system consisting of a Schrödinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper, we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the different propagation properties of the wave and Schrödinger equations. Here we eliminate that condition by using an improved asymptotic form for the solutions. Communicated by Bernard Helffer submitted 20/02/03, accepted: 24/06/03     

From the AKNS system to the matrix Schrödinger equation with vanishing potentials: Direct and inverse problems

We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schrödinger equation. The corresponding Miura transformation, which allows this connection, converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.     

微分代数系统的渐近性

从动力系统的角度研究微分代数系统,利用单调流理论中的结果和方法讨论微分代数系统渐近性态.首先,我们把所考察的系统嵌入到一族相关的系统,引进使得系统族中的每个系统生成单调流的相应偏序和条件.然后给出了若干关于解收敛于平衡点的一般性结果,并对一类微分代数系统的渐近性作了较为精细的讨论.我们的结果是Hirch等关于常微分方程的相关结果的推广和改进.