We prove a dispersive estimate for the time-independent Schrodinger operator H = -Δ+ V in three dimensions. The potential V(x) is assumed to lie in the intersection L^p(R^3) \cap L^q(R^3), p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay |V(x)| < C(1+|x|)^{-2-ε}, is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed.
We prove dispersive estimates for the linear Schrödinger evolution associated to an operator -Δ+ V, where the potential is a signed measure of fractal dimension at least 3/2.
We prove optimal (that is, without loss of derivatives) dispersive estimates for the Schrodinger group $e^{it(-Δ+V)}$ for a class of real-valued potentials $V\in C^k(R^n)$ with $k>(n-3)/2$, where $n=4,5$.
In this paper, we consider the Cauchy problem for the fractional Schrödinger equation $i D_t^αu + (-Δ)^{\fracβ{2}} u =0$ with $0<α<1$, $β>0$. We establish the dispersive estimates for the solutions. In particular, we prove that the decay rates are sharp.
In this paper we prove dispersive estimates for the system formed by two coupled discrete Schrödinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals.
In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow…
We study the dispersive behaviors of two-particles Schrödinger and wave equations in the Aharonov-Bohm field. In particular, we prove the Strichartz estimates for Schrödinger and wave equations in this setting. The key point is to construct spectral measure of Schrödinger operator with an Aharonov-Bohm type potential in $\R^4$. As applications, we finally prove a scattering theory for the nonlinear defocusing subcritical two-particles Schrödinger equation with Aharonov-Bohm potential.
Considered in this report is the one-dimensional fourth-order dispersive cubic nonlinear Schrödinger equation with mixed dispersion. Orbital stability, in the energy space, of a particular standing-wave solution is proved in the context of Hamiltonian systems. The main result is established by constructing a suitable Lyapunov function.
The initial value problem for the homogeneous Schrödinger equation is investigated for radially symmetric initial data with slow decay rates and not too wild oscillations. Our global wellposedness results apply to initial data for which Strichartz estimates fail.
We study resolvent estimates, spectral theory and long time dispersive properties of scalar and matrix Schrödinger-type operators on $\mathbb{H}^{n+1}$ for $n \geq 1$.
We prove L^1 --> L^\infty estimates for linear Schroedinger equations in dimensions one and three. The potentials are only required to satisfy some mild decay assumptions. No regularity on the potentials is assumed.
The present paper is dedicated to the proof of dispersive estimates on stratified Lie groups of step 2, for the linear Schrödinger equation involving a sublaplacian. It turns out that the propagator behaves like a wave operator on a space of the same dimension p as the center of the group, and like a Schrödinger operator on a space of the same dimension k as the radical of the canonical skew-symmetric form, which suggests a decay with exponant -(k+p-1)/2. In this article, we identify a property of the canonical skew-symmetric form under which we establish optimal dispersive estimates with this…
We consider the energy critical focusing NLS in R^3 and prove, for any $ν$ sufficiently small, the existence of radial finite energy solutions that as $t\to\infty$ behave as a sum of a dynamically rescaled ground state plus a radiation, the scaling law being of the form $t^ν$.
For nonlinear dispersive systems, the nonlinear Schrödinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally oscillating underlying carrier wave. Here, we justify the NLS approximation for a whole class of quasilinear dispersive systems, which also includes toy models for the waterwave problem. This is the first time that this is done for systems, where a quasilinear quadratic term is allowed to effectively lose more than one derivative. With effective loss we here mean the lo…
We prove a dispersive estimate for the evolution of Schroedinger operators H = -Δ+ V(x) in three dimensions. The potential should belong to the closure of bounded compactly-supported functions with respect to the golbal Kato norm. Some additional spectral conditions are imposed, namely that no resonances or eigenfunctions of H exist anywhere on the positive half-line. The proof is an application of a new version of Wiener's L^1 inversion theorem.
We study the $L^1-L^\infty$ dispersive estimate of the inhomogeneous fourth-order Schrödinger operator $H=Δ^{2}-Δ+V(x)$ with zero energy obstructions in $\mathbf{R}^{3}$. For the related propagator $e^{-itH}$, we prove that for $0<t\leq 1$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/4}$-estimate. For $t>1$, we prove that:\,\, 1) if zero is a regular point of $H$, then $e^{-itH}P_{ac}(H)$ satisfies the $|t|^{-3/2}$- dispersive estimate.\,\, 2) if zero is a resonance of $H$, there exists a time dependent operator $F_{t}$ such that $e^{-itH}P_{ac}(H)-F_{t}$ satisfies the $|t|^{-3/2}$- disper…
We study the fourth order Schrödinger operator $H=(-Δ)^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to L^\infty$ dispersive bounds. In all cases, we show that the natural $|t|^{-\frac34}$ decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schrödinger operator $-Δ+V$.
We prove smoothing estimates for Schrödinger equations $i\partial_t ϕ+\partial_x (a(x) \partial_x ϕ) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation.
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrödinger equation.
We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim 1/\log t$ for $t>2$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero energy. We …
In this paper we consider linear, time dependent Schrödinger equations of the form ${\rm i} \partial_t ψ= K_0 ψ+ V(t) ψ$, where $K_0$ is a strictly positive selfadjoint operator with discrete spectrum and constant spectral gaps, and $V(t)$ a time periodic potential. We give sufficient conditions on $V(t)$ ensuring that $K_0+V (t)$ generates unbounded orbits. The main condition is that the resonant average of $V(t)$, namely the average with respect to the flow of $K_0$, has a nonempty absolutely continuous spectrum and fulfills a Mourre estimate. These conditions are stable under perturbations.…
Let $H=-Δ+V$, where $V$ is a real valued potential on $\R^2$ satisfying $|V(x)|\les \la x\ra^{-3-}$. We prove that if zero is a regular point of the spectrum of $H=-Δ+V$, then $$ \|w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)}, |t| >2, $$ with $w(x)=\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights.
We study the fourth order Schrödinger operator $H=(-Δ)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different …
In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schrödinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schrödinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac3…
We consider the fourth order Schrödinger operator $H=Δ^2+V$ and show that if there are no eigenvalues or resonances in the absolutely continuous spectrum of $H$ that the solution operator $e^{-itH}$ satisfies a large time integrable $|t|^{-\frac54}$ decay rate between weighted spaces. This bound improves what is possible for the free case in two directions; both better time decay and smaller spatial weights. In the case of a mild resonance at zero energy, we derive the operator-valued expansion $e^{-itH}P_{ac}(H)=t^{-\frac34} A_0+t^{-\frac54}A_1$ where $A_0:L^1\to L^\infty$ is an operator of r…
We prove dispersive estimates for solutions to the Schrodinger equation with a real-valued potential $V\in L^\infty({\bf R}^n)$, $n\ge 4$, satisfying $V(x)=O(|x|^{-(n+2)/2-ε})$, $|x|>1$, $ε>0$.
In this work, we consider the long-time asymptotics for the Cauchy problem of a fourth-order dispersive nonlinear Schrödinger equation with nonzero boundary conditions at infinity. Firstly, in order to construct the basic Riemann-Hilbert problem associated with nonzero boundary conditions, we analysis direct scattering problem. Then we deform the corresponding matrix Riemann-Hilbert problem to explicitly solving models via using the nonlinear steepest descent method and employing the $g$-function mechanism to eliminate the exponential growths of the jump matrices. Finally, we obtain the asympt…
We consider the non-selfadjoint operator [\cH = [{array}{cc} -Δ+ μ-V_1 & -V_2 V_2 & Δ- μ+ V_1 {array}]] where $μ>0$ and $V_1,V_2$ are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain $L^1(\R^2)\times L^1(\R^2)\to L^\infty(\R^2)\times L^\infty(\R^2)$ dispersive decay estimates for the evolution $e^{it\cH}P_{ac}$. We also obtain the following weighted estimate $$ \|w^{-1} e^{it\cH}P_{ac}f\|_{L^\infty(\R^2)\times L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)\times L^1(\R…
We prove dispersive estimates at high frequency in dimension two for both the wave and the Schrodinger groupes for a very large class of real-valued potentials.
We prove dispersive estimates for Schroedinger operators in dimension three without any assumptions on zero energy. Ie, we allows resonances and/or eigenvalues at zero energy.
We prove dispersive estimates for the low frequency part of the Schrodinger group for a large class of potentials in dimensions greater or equal to four. As a consequence, we extend the result of Journe, Sofer and Sogge to a larger class of potentials. In this revised version a mistake in the proof of the estimate (B.4) is removed.
We consider a nonlinear dispersive equation with a quasilinear quadratic term. We establish two results. First, we show that solutions to this equation with initial data of order $\mathcal{O}(\varepsilon)$ in Sobolev norms exist for a time span of order $\mathcal{O}(\varepsilon^{-2})$ for sufficiently small $\varepsilon$. Secondly, we derive the Nonlinear Schrödinger (NLS) equation as a formal approximation equation describing slow spatial and temporal modulations of the envelope of an underlying carrier wave, and justify this approximation with the help of error estimates in Sobolev norms bet…
We prove a sharp dispersive estimate $$ |P_{ac}u(t,x)|\le C|t|^{-1/2}\|u(0)\|_{L^1(R)} $$ for the one dimensional Schrödinger equation $$ iu_{t}-u_{xx}+V(x)u+V_0 u=0, $$ where $(1+x^2)V\in L^1(R)$ and $V_0$ is a step function, real valued and consant on the positive and negative real axes.
In this paper, we prove the asymptotic stability of nonlinear Schrodiger equations on star graphs, which partially solves an open problem in D. Noja \cite{DN}. The essential ingredient of our proof is the dispersive estimate for the linearized operator around the soliton with Kirchhoff boundary condition. In order to obtain the dispersive estimates, we use the Born's series technique and scattering theory for the linearized operator.
We present a time-local existence theorem of the initial value problem for a third-order dispersive evolution equation for open curves on compact almost Hermitian manifolds arising in the geometric analysis of vortex filaments. This equation causes the so-called loss of one-derivative since the target manifold is not supposed to be a Kähler manifold. We overcome this difficulty by using a gauge transformation of a multiplier on the pull-back bundle to eliminate the bad first order terms essentially.
In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^α$, $α< \frac{n-3}2$, for which the solutions to the linear Schrödinger equation in $\R^n$, $$ -i\partial_t u = - Δu + Vu, \quad u(0)=f, $$ do not obey the usual $L^1\to L^{\infty}$ dispersive estimate. This contrasts with known results in dimensions $n \leq 3$, where a pointwise decay condition on $V$ is generally sufficient to imply dispersive bounds.
We use exponential sums to study the fractal dimension of the graphs of solutions to linear dispersive PDE. Our techniques apply to Schrödinger, Airy, Boussinesq, the fractional Schrödinger, and the gravity and gravity-capillary water wave equations. We also discuss applications to certain nonlinear dispersive equations. In particular, we obtain bounds for the dimension of the graph of the solution to cubic nonlinear Schrödinger and Korteweg-de Vries equations along oblique lines in space-time.
Using a Fourier spectral method, we provide a detailed numerically investigation of dispersive Schrödinger type equations involving a fractional Laplacian. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be computed in one spatial dimension, only. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states, and the long time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerical…
We show that for a one-dimensional Schrödinger operator with a potential whose (j+1)'th moment is integrable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case.
We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials $V(x)=O(|x|^{-δ})$, $δ>n-1$, and $V\in C^k({\bf R}^n$, $k>k_n$, where $n\ge 4$ and $(n-3)/2\le k_n<n/2$. We also give a sufficient condition in terms of $L^1\to L^\infty$ bounds for the formal iterations of Duhamel's formula, which might be satisfied for potentials of less regularity.
We prove local smoothing, local energy decay and weighted Strichartz inequalities for fractional Schrödinger equations with a Aharonov-Bohm magnetic field in 2D. Explicit representations of the flows in terms of spherical expansions of the Hamiltonians are involved in the study. An improvement of the free estimate is proved, when the total flux of the magnetic field through the unit sphere is not an integer.
