We develop the wave packet decomposition to study the Schrodinger evolution with rough potential. As an application, we obtain the improved bound on the wave propagation for the generic value of a parameter.
The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimesional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. And, for large time, a new period for the dynamics appears : the wave packets revivals.
We consider semiclassically scaled Schrodinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow obtained by Peierl's substitution, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrodinger equation with time-dependent effective mass. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid u…
In this paper, we give a characterization of the ranges of the wave operators for Schrodinger equations with time-dependent short-range potentials by using wave packet transform. We also give an alternative proof of the existence of the wave operators for time-dependent potentials and the asymptotic completeness for timeindependent potentials.
Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent papers, including pseudodifferential, Fourier integral and metaplectic operators, with applications to time-frequency analysis of Schrödinger-type evolution equations. It has been proved that such operators are approximately diagonalized by Gabor wave packets. While the latter are expected to undergo some spreading phenomenon, there is no record of this issue in the aforementioned results. In this paper we prove refined estimates for the Gabor matrix of metaplectic operator…
We consider a class of linear Schroedinger equations in R^d, with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic…
In this note, we consider wave packet parametrices for Schrodinger-like evolution equations. Under an integrability condition along the flow, we prove that the flow is then globally well-defined and bilipschitz. Under an additional smallness assumption, we prove that the kernel of the phase space operator decays rapidly away from the graph of the Hamilton flow. This can be used to prove that smoothness is preserved for finite times by the flow of the equation.
In this article, we study wave dynamics in the fractional nonlinear Schrödinger equation with a modulated honeycomb potential. This problem arises from recent research interests in the interplay between topological materials and nonlocal governing equations. Both are current focuses in scientific research fields. We first develop the Floquet-Bloch spectral theory of the linear fractional Schrödinger operator with a honeycomb potential. Especially, we prove the existence of conical degenerate points, i.e., Dirac points, at which two dispersion band functions intersect. We then investigate the d…
We establish dispersive and Strichartz estimates for solutions to the linear time-dependent Schrödinger equations with potential in three dimensions. Our main focus is on the small rough time-dependent potentials. Examples of such potentials are of the form $V(t,x)=T(t) V_0(x)$, where $T$ is quasiperiodic in time and $V_0$ is essentially an $L^{3/2}$ function of the spatial variables. We also prove the dispersive estimates for small time-independent potentials which belong to the interestion of the Rollnik and global Kato classes. Finally, we settle the question posed by Journe, Soffer, Sogge …
This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable coefficient versions of fractional Schrödinger equations.
We consider the propagation of wave packets for the nonlinear Schrodinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave…
We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.
We study the propagation of wave packets for nonlinear nonlocal Schrodinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.
In this paper, we consider the existence and the asymptotic completeness of the wave operators for Schrodinger equations with time-dependent potentials which are short-range in space.
We prove a global well-posedness result for defocusing nonlinear Schrodinger equations with time dependent potential. We then focus on time dependent harmonic potentials. This aspect is motivated by Physics (Bose--Einstein condensation), and appears also as a preparation for the analysis of the propagation of wave packets in a nonlinear context. The main aspect considered here is the growth of high Sobolev norms of the solution.
For the Schrödinger equation, $ (i \partial_t + Δ) u = 0 $ on a torus, an arbitrary non-empty open set $ Ω$ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T] \times Ω)} $. We show that the same result remains true for $ (i \partial_t + Δ- V) u = 0 $ where $ V \in L^2 (\T^2) $, and $ \T^2 $ is a (rational or irrational) torus. That extends the results of \cite{AM}, and \cite{BZ4} where the observability was proved for $ V \in C (\T^2) $ and conjectured for $ V \in L^\infty (\T^2) $. The higher dimensional generalization remains ope…
We study the scattering theory for the Schrödinger and wave equations with rough potentials in a scale of homogeneous Sobolev spaces. The first half of the paper concerns with an inverse-square potential in both of subcritical and critical constant cases, which is a particular model of scaling-critical singular perturbations. In the subcritical case, the existence of the wave and inverse wave operators defined on a range of homogeneous Sobolev spaces is obtained. In particular, we have the scattering to a free solution in the homogeneous energy space for both of the Schrödinger and wave equati…
In a recent article [10], the authors proved that the non-relativistic Schrödinger operator with a generic honeycomb lattice potential has conical (Dirac) points in its dispersion surfaces. These conical points occur for quasi-momenta, which are located at the vertices of the Brillouin zone, a regular hexagon. In this paper, we study the time-evolution of wave-packets, which are spectrally concentrated near such conical points. We prove that the large, but finite, time dynamics is governed by the two-dimensional Dirac equations.
We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter $\a>0$. The high-frequency (or: semi-classical) parameter is $\eps>0$. We let $\eps$ and $\a$ go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution $u^\eps$ radiates in the outgoing direction, {\bf uniformly} in $\eps$. In particular, the function $u^\ep…
We study an inverse scattering problem associated with a Schrödinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single re…
In this paper, we consider the Schrödinger equation in one space-dimension with potential and we aim at exhibiting dynamic interaction phenomena produced by the potential. To this end, we focus our attention on the time-asymptotic behaviour of the two first terms of the Dyson-Phillips series, which gives a representation of the solution of the equation according to semigroup theory. The first term is actually the free wave packet while the second term corresponds to the wave packet resulting from a first interaction between the free solution and the potential. In order to follow a method devel…
We study, for times of order 1/h, solutions of Maxwell's equations in an O(h^2) modulation of an h-periodic medium. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order h. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation. A weak ray average hypothesis guarantees stability. Compared to earlier work on scalar wave equations, the generator is no longer ellipt…
