Distorted Plane Waves Research Articles

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

We consider the semilinear wave equation on a Lorentzian manifold (M, g) with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric g and the coefficient a up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.

An Inverse Boundary Value Problem for a Semilinear Wave Equation on Lorentzian Manifolds

Abstract We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness.

Semiclassical Limits of Distorted Plane Waves in Chaotic Scattering Without a Pressure Condition

Abstract In this paper, we study the semiclassical behavior of distorted plane waves, on manifolds that are Euclidean near infinity or hyperbolic near infinity, and of non-positive curvature. Assuming that there is a strip without resonances below the real axis, we show that distorted plane waves are bounded in $L^2_{loc}$ independently of $h$ and that they admit a unique semiclassical measure and we prove bounds on their $L^p_{loc}$ norms.

Lower bounds for the number of nodal domains for sums of two distorted plane waves in non-positive curvature

In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees an optimal lower bound, of the same order as Courant's upper bound. As an application, we will obtain optimal lower bounds for the number of nodal domains of distorted plane waves on some families of surfaces of non-positive curvature.

Nonlinear interaction of waves in elastodynamics and an inverse problem

We consider nonlinear elastic wave equations generalizing Gol'dberg's five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map.

Nucleon Associations in Direct Nuclear Reactions

Based on the model of nucleon associations, the two-nucleon transfer reaction (p, t) is considered in intermediate nuclei. The interaction between the nucleons is local and short-range, and the bineutron association already exists in the nucleon. The internal states of the incident proton, departing triton, and any intermediate association states are assumed to be completely symmetric S-states. Investigations are performed with distorted plane waves of the proton and triton, and the assumption is made that there is no exchange by nucleons among nucleon fragments. The differential cross sections calculated on the basis of this approach are presented, and these results demonstrate that the association model allows one to explain qualitatively the experimental data on particle scattering. Some possible directions of further study of direct nuclear reactions on the basis of the model are discussed.

Distorted plane waves in chaotic scattering

Distorted plane waves, sometimes called Eisenstein functions, are a family of eigenfunctions of a Schr\"odinger operator that are not square integrable. More precisely, they can be written as the sum of a plane wave and an outgoing wave. We shall study distorted plane waves in the semiclassical limit, in a general setting which includes manifolds that are Euclidean near infinity, under the hypothesis that the classical dynamics is hyperbolic close to the trapped set, and that some topological pressure is negative.

Distorted Plane Waves on Manifolds of Nonpositive Curvature

We will consider the high frequency behaviour of distorted plane waves on manifolds of nonpositive curvature which are Euclidean or hyperbolic near infinity, under the assumption that the curvature is negative close to the trapped set of the geodesic flow and that the topological pressure associated to half the unstable Jacobian is negative. We obtain a precise expression for distorted plane waves in the high frequency limit, similar to the one in Guillarmou and Naud (Am J Math 136:445–479, 2014) in the case of convex co-compact manifolds. In particular, we will show \({L_{loc}^\infty}\) bounds on distorted plane waves that are uniform with frequency. We will also show a small-scale equidistribution result for the real part of distorted plane waves, which implies sharp bounds for the volume of their nodal sets.

Localized input fields in rigorous coupled-wave analysis.

We introduce a new treatment for localized input fields in coupled-wave theory. It consists of a combination of angular spectrum decomposition and a cyclic formulation of rigorous coupled-wave analysis (RCWA), which does not require the computation of several standard RCWA calculations. The cyclic approach especially concerns the definition of the convolution matrix of the permittivity and enables the processing of multiple input plane waves in a single RCWA calculation. At the same time it ensures physically meaningful simulation results for finite truncation limits and thus inherently conserves energy. By adapting the three-dimensional (3D)-RCWA algorithm to efficiently simulate the effects of localized and arbitrarily polarized input fields to 3D volume gratings, various advanced diffraction problems such as fiber coupling, the influence of distorted plane waves, or the effects of focused beams on diffraction gratings can be investigated. Therefore, the impact of this work should concern many fields of application.

Scattering, Homogenization, and Interface Effects for Oscillatory Potentials with Strong Singularities

We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schr\"odinger equation \[ H_\epsilon \psi := (-\partial_x^2 + V_0(x) + q(x,x/\epsilon)) \psi = k^2 \psi, \] for $k\in\RR$ and $\epsilon << 1$. Here, $q(.,y+1)=q(.,y)$, has mean zero and $|V_0(x)+q(x,.)|$ goes to zero as $|x|$ goes to infinity. The distorted plane waves of $H_\epsilon$ are solutions of the form: $e^{\pm ikx}+u^s_\pm(x;k)$, $u^s_\pm$ outgoing as $|x|$ goes to infinity. We derive their $\epsilon$ small asymptotic behavior, from which the asymptotic behavior of scattering quantities such as the transmission coefficient, $t^\epsilon (k)$, follow. Let $t_0^{hom}(k)$ denote the homogenized transmission coefficient associated with the average potential $V_0$. If the potential is smooth, then classical homogenization theory gives asymptotic expansions of, for example, distorted plane waves, and transmission and reflection coefficients. Singularities of $V_0$ or discontinuities of $q_\epsilon $, that our theory admits, are "interfaces" across which a solution must satisfy interface conditions (continuity or jump conditions). To satisfy these conditions it is necessary to introduce interface correctors, which are highly oscillatory in $\epsilon$. A consequence of our main results is that $t^\epsilon (k)-t_0^{hom}(k)$, the error in the homogenized transmission coefficient is (i) $O(\epsilon ^2)$ if $q_\epsilon $ is continuous and (ii) $O(\epsilon)$ if $q_\epsilon $ has discontinuities. Moreover, in the discontinuous case the correctors are highly oscillatory in $\epsilon$, so that a first order corrector is not well-defined. The analysis is based on a (pre-conditioned) Lippman-Schwinger equation, introduced in [SIAM J. Mult. Mod. Sim. (3), 3 (2005), pp. 477--521].

Inverse scattering for the Laplace–Beltrami operator with complex electromagnetic potentials and embedded obstacles

We consider the direct and inverse obstacle scattering problem for the Laplace–Beltrami operator with complex-valued electromagnetic potentials where outside of some large ball the operator is simply the negative Laplacian. It is assumed there are an arbitrary number of impenetrable obstacles with various boundary conditions (including mixed boundary conditions, i.e., partially coated obstacles). We prove that given the scattering amplitude a(θ, ω, k) for all θ, ω ∊ Sn−1 and any fixed k > 0, the location of the obstacles and their respective boundary conditions are uniquely determined. In addition, we give a new proof of the asymptotic expansion of the Green's function in terms of distorted plane waves and, as an immediate corollary, prove the determination of the Dirichlet-to-Neumann operator from the scattering data.

Distorted plane waves for the 3 body Schrödinger operator

Distorted plane waves, incoming in the free region, are constructed for generalized 3 body Schrodinger operators on \( {\Bbb R}^N \) under the assumption that the 'two-body' potential functions are rapidly decreasing. The construction yields the form of the complete asymptotic expansion of the plane waves at infinity away from the collision planes, and gives an explicit construction and geometric characterization of the singularities of the free channel scattering matrix.

The Coulomb interaction betweenpion wavepackets: the puzzle

The time-dependent Schrödinger equation for pairs, which are emitted from the interaction zone in relativistic nuclear collisions, is solved using wavepacket states. It is shown that the Coulomb enhancement in the momentum correlation function of such pairs is smaller than obtained in earlier calculations based on Coulomb distorted plane waves. These results suggest that the experimentally observed positive correlation signal cannot be caused by the Coulomb interaction between pions emitted from the interaction zone. But other processes which involve long-living resonances and pions emitted with a large coherence length could provide a possible explanation for the observed signal.

Reciprocity relations and completeness of far‐field pattern vectors for obstacle scattering of acoustic wave in a stratified medium

AbstractUnder a generalized Sommerfeld radiation condition, we proved the uniqueness and existence of the direct obstacle scattering problem of time‐harmonic acoustic waves in a stratified medium [8]. In this paper, we study the asymptotic behaviour of the scattered waves and prove three reciprocity relations among the free‐wave far‐field patterns and the guided‐wave far‐field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far‐field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse obstacle scattering problems.

Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity

In this paper, a generalized Sommerfeld radiation condition is presented for the scattering waves in a stratified medium with a nonstratified inhomogeneity. Using integral equation methods, the uniqueness and existence of the direct scattering problem are proved. Relations between the scattered acoustic waves in the far field and the sound profile of the inhomogeneity are obtained. Using these relations, the author proves three reciprocity relations between the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then conditions under which a set of far-field patterns is complete in a Hilbert space are determined using the reciprocity relation. These properties are important in investigating inverse scattering problems

Regularized inner products of distorted plane waves and near‐forward inverse scattering

AbstractRegularized inner products, r(s, e1, e2,) of the distorted plane waves fór the plasma wave equation uu − Δu + qu = 0 are introduced for potentials with compact support. The regularized inner product may be represented in terms of the far‐field pattern. The use of the WKB approximation shows that the behaviour of r as e2 → − e1 is closely related to the Radon transform of q. This observation indicates the possibility of finding the Radon transform of q in terms of r(s, e1, e2) and then of recovering q by means of the inverse Radon transform.

Perturbation theory for Coulomb scattering

Perturbation series for the ’’renormalized’’ complex-energy distorted plane waves (RCEW’s) are defined and shown to converge to the series expansions for the physical distorted plane waves (PW’s) for two-particle scattering via Coulomb-like potentials V (x) =e1e2‖x‖−1+Vs(x), where Vs ∈ ℒ2 has compact support. A perturbation series for the ’’renormalized’’ half-off-shell T matrix is defined and shown to converge to the series expansion for the pure Coulomb physical T matrix.

Renormalized off-energy-shell Coulomb scattering

The complex-energy distorted plane waves for scattering via a general class of two-body Coulomb-like potentials are shwn to satisfy off-energy-shell Lippmann-Schwinger equaions in the case of two particle scattering and appropriately iterated Weinberg-Van Winter and Faddeev equations in the case of three particle scattering. A renormalized off-energy-shell formalism is defined for scattering involving more than one charged fragment in either the incoming or outgoing channel. The existence of limit to real energies of the normalized off-energy-shell formalism is verified.

Eigenfunction expansions for stationary scattering theory in spaces with negative norm

The space with “negative” norm L - with which the Hilbert space L of wave-pockets is equipped is used as a natural carrier of non-square integrable functions which appear in scattering theory, such as the plane waves and distorted plane waves in potential scattering. The concept of a complete set of observables is formulated within this framework. The S-matrix formulae leading to the expression for the scattering cross-section are derived on a general level, and then are applied to the case of potential scattering.

Electromagnetic Production of Pion Pairs

The Pauli-Weisskopf theory of pion pair production by photons was modified to include the strong pion-nucleus interaction in the form of a complex optical potential, and the matrix element was obtained by using "exact" distorted plane waves which were expanded into angular momentum eigenstates. Numerical calculations were performed for lead to examine the effect of a nuclear charge distribution and nuclear optical potential on the cross section in the energy region just above threshold. The calculations show that when the charge distribution and nuclear potential are included, the cross section for lead which leaves the nucleus in an unexcited state increases slowly just above threshold until approximately 295 Mev, where it starts to increase almost linearly, attaining a value of 1.07\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}32}$ ${\mathrm{cm}}^{2}$ at 310 Mev.