1,720,974 research outputs found
Structure Formulas for Wave Operators
Get PDFWe establish quantitative estimates on the structure function arising in the representation of the intertwining wave operators of a Schroedinger operator in three dimensions. Regularity of zero energy is assumed throughout. This paper is related to, and corrects some inaccuracies in, the first author\u27s work this https UR
A Centre-Stable Manifold for the Focussing Cubic NLS in R 1+3
Get PDFConsider the focussing cubic nonlinear Schr\ odinger equation in R 3 : iψ t +Δψ=−|ψ| 2 ψ.
It admits special solutions of the form e itα ϕ , whereϕ is a Schwartz function and a positive (ϕ\u3e0 ) solution of −Δϕ+αϕ=ϕ 3 .
The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form e i(v⋅+Γ) ϕ(⋅−y,α) . We prove that any solution starting sufficiently close to a standing wave in the Σ=W 1,2 (R 3 )∩|x| −1 L 2 (R 3 ) norm and situated on a certain codimension-one local Lipschitz manifold exists globally in time and converges to a point on the manifold of standing waves. Furthermore, we show that \mc N is invariant under the Hamiltonian flow, locally in time, and is a centre-stable manifold in the sense of Bates, Jones. The proof is based on the modulation method introduced by Soffer and Weinstein for the L 2 -subcritical case and adapted by Schlag to the L 2 -supercritical case. An important part of the proof is the Keel-Tao endpoint Strichartz estimate in R 3 for the nonselfadjoint Schr\ odinger operator obtained by linearizing around a standing wave solution
A Critical Centre-Stable Manifold for Schroedinger\u27s Equation in R^3
Get PDFConsider the focusing cubic semilinear Schroedinger equation in R^3 i \partial_t \psi + \Delta \psi + | \psi |^2 \psi = 0. It admits an eight-dimensional manifold of special solutions called ground state solitons. We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant. Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a soliton and radiation. Conversely, in a general setting, any solution that stays close to the soliton manifold for all time is in N. The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time-dependent linearized equation. The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here --- of the focusing cubic NLS in R^3 --- by the work of Marzuola-Simpson and Costin-Huang-Schlag
New Estimates for a Time-Dependent Schroedinger Equation
Get PDFThis paper establishes new estimates for linear Schroedinger equations in R^3 with time-dependent potentials. Some of the results are new even in the time-independent case and all are shown to hold for potentials in scaling-critical, translation-invariant spaces. The proof of the time-independent results uses a novel method based on an abstract version of Wiener\u27s Theorem
A Critical Centre-Stable Manifold for the Cubic Focusing Schroedinger Equation in Three Dimensions
Get PDFConsider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t \psi + \Delta \psi + |\psi|^2 \psi = 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(\Gamma + vx - t|v|^2 + \alpha^2 t)} \phi(x-2tv-D, \alpha), where \phi(x, \alpha) is a positive ground state solution of the semilinear elliptic equation -\Delta \phi + \alpha^2\phi = \phi^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a solution starting on N remains on N for all positive time and for some finite negative time and N can be identified as the centre-stable manifold for this equation. The proof is based on the method of modulation, introduced by Soffer and Weinstein and adapted by Schlag to the L^2-supercritical case. Novel elements include a different linearization and new Strichartz-type estimates for the linear Schroedinger equation. The main result depends on a spectral assumption concerning the absence of embedded eigenvalues. We also establish several new estimates for solutions of the time-dependent and time-independent linear Schroedinger equation, which hold under sharper or more general conditions than previously known. Several of these estimates are based on a new approach that makes use of Wiener\u27s Theorem in the context of function spaces
Decay Estimates for the Wave Equation in Two Dimensions
Get PDFWe establish Strichartz estimates (both reversed and some direct ones), pointwise decay estimates, and weighted decay estimates for the linear wave equation in dimension two with an almost scaling-critical potential, in the case when there is no resonance or eigenvalue at the edge of the spectrum. We also prove some simple nonlinear applications
A Centre-Stable Manifold for the Energy-Critical Wave Equation in R^3 in the Symmetric Setting
Get PDFConsider the focusing semilinear wave equation in R^3 with energy-critical nonlinearity \partial_t^2 \psi - \Delta \psi - \psi^5 = 0, \psi(0) = \psi_0, \partial_t \psi(0) = \psi_1. This equation admits stationary solutions of the form \phi(x, a) := (3a)^{1/4} (1+a|x|^2)^{-1/2}, called solitons, which solve the elliptic equation -\Delta \phi - \phi^5 = 0. Restricting ourselves to the space of symmetric solutions \psi for which \psi(x) = \psi(-x), we find a local centre-stable manifold, in a neighborhood of \phi(x, 1), for this wave equation in the weighted Sobolev space ^{-1} \dot H^1 \times\u3c x\u3e^{-1} L^2. Solutions with initial data on the manifold exist globally in time for t \geq 0, depend continuously on initial data, preserve energy, and can be written as the sum of a rescaled soliton and a dispersive radiation term. The proof is based on a new class of reverse Strichartz estimates, introduced in Beceanu-Goldberg and adapted here to the case of Hamiltonians with a resonance
Dispersive Estimates in R^3 with Threshold Resonances
Get PDFWe prove dispersive estimates in R^3 for the Schroedinger evolution generated by the Hamiltonian H = -\Delta+V, under optimal decay conditions on V, in the presence of zero energy eigenfunctions and resonances
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