L. I. Ignat, D. Stan
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.
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Dispersive decay for the mass-critical nonlinear Schrödinger equation
该研究证明了在空间维度1、2、3中，质量临界非线性薛定谔方程的解具有逐点时间上的色散衰减性质。
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 $n=4,5$.
Dispersive estimates of fourth order Schrödinger operators with scaling-critical magnetic potentials in dimension two
本文通过构造预解核并运用稳相方法，获得了二维中具有尺度临界磁势的四阶Schrödinger算子的色散估计。
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.
Dispersive decay for the energy-critical nonlinear Schrödinger equation
本文针对空间维数 d=3,4 的能量临界非线性薛定谔方程，证明了其解在初值问题和终态问题中均具有逐时间点的色散衰减。
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 the dispersive blow-up phenomena for the Schrödinger-Korteweg-de Vries (S-KdV) system. Roughly, dispersive blow-up has being called to the development of point singularities due to the focussing of short or long waves. In mathematical terms, we show that the existence of this kind of singularities is provided by the linear dispersive solution by proving that the Duhamel term is smoother. It seems that this result is the first regarding systems of nonlinear dispersive equations. To obtain our results we use, in addition to smoothing properties, persistence properties for solutions of t…
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 $t^ν$.
In this expository note, we prove some extensions and refinements of classical Kato type estimates with elementary techniques.
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.
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.
Dispersive estimates for 1D matrix Schrödinger operators with threshold resonance
本文研究含阈值共振的一维非自伴矩阵 Schrödinger 算子，通过减去有限秩共振项证明加权色散与局部衰减率恢复至正则情形，并发现聚焦三次 NLS 扰动方程二次非线性的有利结构。
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.
Dispersive decay for the Inter-critical nonlinear Schrödinger equation in $\mathbb{R}^3$
本文研究了三维空间中非临界非线性薛定谔方程的色散衰减问题，得出了初值在临界Sobolev空间中解的长期衰减估计。
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 …
Dispersive estimates and optimality for Schrödinger equations on product cones
本文研究乘积锥上薛定谔传播子的时间衰减估计，证明当半径$\rho\geq1$时通常色散估计成立，否则不成立，方法基于调和分析，补充并优化了Jia-Zhang的微局部分析结果。
Dispersive estimates for wave and Schrödinger equations with a potential in non-trapping exterior domains
本文研究非捕获外部区域中带短程位势的波动方程与薛定谔方程，通过证明预解估计，得到局部能量可积性、光滑效应，并将自由方程的局部 Strichartz 估计推广为全局估计。
Dispersive effects for the Schrödinger equation on finite metric graphs with infinite ends
该研究针对带无穷端点的有限度量图上的自由薛定谔方程，给出了$L^1$到$L^\ty$时间衰减率至少为$t^{-1/2}$的充分条件，并探究了有界子图与无界端点间的概率流动态。
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.…
Dispersive estimates for fractional order Schrödinger operators
研究分数阶薛定谔算子\(H = (-\Delta)^{\alpha}+V\)的色散界，通过推导预解算子逐点界、定量极限吸收原理等，在\(n\geq 2\)且\(\frac{n + 1}{4}\leq \alpha <\frac{n}{2}\)范围建立全局色散估计。
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 …
On the fractional abstract Schrodinger-type evolution equations on the Hilbert space and its applications to the fractional dispersive equations
本文基于自伴算子谱定理和 Mittag-Leffler 函数渐近行为，证明了 Hilbert 空间上时间分数阶抽象 Schrödinger 型发展方程及分数阶色散方程的局部和全局适定性，并给出了线性方程温和解的 Hölder 正则性与渐近行为。
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…
Dispersive and Strichartz estimates for 3D wave equation with a class of many-electric potentials
本文针对三维空间中具有一类多电势的波动方程，基于Schrödinger算子谱性质证明其色散估计，并结合Keel-Tao标准论证得出Strichartz估计。
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.
Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential
本文证明带反平方势的三维能量临界非线性薛定谔方程的色散估计，涵盖非负势端点情形和负势强衰减及Lorentz估计，限制尖锐，方法结合自举、实插值和端点Sobolev-Lorentz估计。
Global solutions for 1D cubic defocusing dispersive equations, Part IV: general dispersion relations
本文证明了一类具有一般色散关系（非 Schrödinger 型）的一维三次散焦半线性色散方程在临界正则性小初值条件下具有全局解，并得到全局 Strichartz 估计和双线性界，首次将相关猜想推广到非 Schrödinger 型问题。
Dispersive blow-up for a coupled Schrödinger-fifth order KdV system
本文针对耦合薛定谔-五阶KdV系统的初值问题，通过证明Bourgain空间中的局部适定性及解非线性部分的正则性，构造出可引发色散爆破现象的初值，建立了该系统的色散爆破结果。
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…
Dispersive decay estimates for the magnetic Schrödinger equations
本文通过引入带磁势的分数阶畸变傅里叶变换和分数阶微分算子，结合 Strichartz 估计，证明了线性和非线性磁薛定谔方程的色散衰减界，衰减率为 t^{-n/2}，从而揭示了解的扩散性质与长期动力学。
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, $$ where $(1+x^2)V\in L^1(R)$ and $V_0$ is a step function, real valued and consant on the positive and negative real axes.
In this paper, we prove the asymptotic stability of nonlinear Schrodiger equations on star graphs, which partially solves an open problem in D. Noja \cite{DN}. The essential ingredient of our proof is the dispersive estimate for the linearized operator around the soliton with Kirchhoff boundary condition. In order to obtain the dispersive estimates, we use the Born's series technique and scattering theory for the linearized operator.
In dimension $n>3$ we show the existence of a compactly supported potential in the differentiability class $C^α$, $α< \frac{n-3}2$, for which the solutions to the linear Schrödinger equation in $\R^n$, $$ -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.
Global solutions for 1D cubic dispersive equations, Part III: the quasilinear Schrödinger flow
本文研究一维三次拟线性薛定谔方程，证明其尖锐局部适定性，并在相位旋转对称、保守且散焦条件下建立小 Sobolev 数据的全局散射解，同时证明无散焦条件时 ε 大小数据在 ε^{-8} 时间尺度上的长时间解猜想。
Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials
本文针对具有尺度临界位势的高阶薛定谔方程，利用广义 Kato 范数和 Wiener 反演定理建立了 $L^1$ 到 $L^\infty$ 的色散估计，并证明了无正共振及更一般算子族的色散估计。
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.
Dispersive estimates for Schrödinger's and wave equations on Riemannian manifolds
本文证明了三维 Riemannian 流形上带标量势的 Schrödinger 和波动方程的 $L^p$ 衰减估计，主要针对常负截面曲率度规的小扰动，并基于波动传播子的直接估计在 $\mathbb S^3$ 和 $\mathbb H^3$ 上得出结论。
Dispersive equations with invariant measures
本文以带 Ornstein-Uhlenbeck 算子的 Schrödinger 方程 Cauchy 问题为模型，倡导研究具有不变测度、现有理论无法处理的新型可能退化色散方程。
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