Dispersive estimates of solutions to the Schrodinger equation in dimensions $n\ge 4$
Georgi Vodev
证明了具有实值位势$V\in L^\infty({\bf R}^n)$，$n\ge 4$的薛定谔方程解的色散估计，满足$V(x)=O(|x|^{-(n+2)/2-ε})$，$|x|>1$，$ε>0$ .
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 $V(x)=O(|x|^{-(n+2)/2-ε})$, $|x|>1$, $ε>0$.
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Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I
我们证明了色散估计的薛定谔算子在三维零能量没有任何假设。也就是说，我们允许共振和/或本征值在零能量。
Low frequency dispersive estimates for the Schrodinger group in higher dimensions
我们证明色散估计的低频部分的薛定谔组的一大类潜在的尺寸大于或等于四。因此，我们将Cornene，Sofer和Sogge的结果推广到更大的一类势。在本修订版中，删除了估算证明（B.4）中的一个错误。
Dispersive estimate for the 1D Schrödinger equation with a steplike potential
本文证明了一维薛定谔方程$$ iu_{t}-u_{xx}+V(x)u+V_0 u=0, $$的一个锐色散估计$$ |P_{ac}u(t,x)|\le C|t|^{-1/2}\|u(0)\|_{L^1(R)} $$，其中$(1+x^2)V\in L^1(R)$和$V_0$是一个实值的阶梯函数，在正负实轴上一致.
A counterexample to dispersive estimates for Schrödinger operators in higher dimensions
在$n>3$维中，我们证明了在可微类$C^α$，$α< \frac{n-3}2$中存在一个紧支势，对于这个紧支势，线性薛定谔方程在$\R^n$，$$ -i\partial_t u = - Δu + Vu, \quad u(0)=f, $$中的解不服从通常的$L^1\to L^{\infty}$色散估计.这与已知的结果在尺寸$n \leq 3$，其中逐点衰减条件$V$一般足以暗示色散边界。
High frequency dispersive estimates for the Schrodinger equation in high dimensions
我们证明了最佳色散估计在高频的薛定谔群与实值电位$V(x)=O(|x|^{-δ})$，$δ>n-1$，和$V\in C^k({\bf R}^n$，$k>k_n$，其中$n\ge 4$和$(n-3)/2\le k_n<n/2$。我们还给出了Duhamel公式形式迭代的$L^1\to L^\infty$界的一个充分条件，该条件对于正则性较差的势函数可能满足。
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.
We consider the defocusing nonlinear Schr{ö}dinger equation with a gauge invariant power-like nonlinearity. We prove global dispersive estimates in a semi-classical scaling, after rescaling the solution thanks to a suitable distorsion of the natural dispersion rate. We show a uniform bound on the modulus of the solution in some space providing compactness properties. We discuss the consequences of these estimates in the light of both scattering theory and semi-classical analysis.
Dispersive Estimates for Nonlinear Schrödinger Equations with External Potentials
该研究针对带外势的三维及以上维数小初值Hartree型非线性薛定谔方程，通过控制解的高阶Sobolev范数估计Duhamel公式项，证明了与自由解衰减相当的最优色散衰减估计。
Zero Energy Scattering for One-Dimensional Schrödinger Operators and Applications to Dispersive Estimates
我们证明了对于一个具有（j+1）阶矩可积的势的一维薛定谔算子，散射矩阵的j阶导数在具有可积傅里叶变换的函数的Wiener代数中.我们利用这个结果改进了共振情形下一维薛定谔方程已知的具有可积时间衰减的色散估计。
Weak Dispersive estimates for Schrödinger equations with long range potentials
我们证明了薛定谔初值问题的局部光滑估计的数据在$L^2(\mathbb{R}^d)$，$d \geq 2$和一般类的潜力。在排斥的情况下，我们必须假设，对于某些$γ>0$，只有一个类似于衰减$(1+|x|)^{-γ}$的幂。也被认为是有吸引力的扰动。的估计保持所有的时间，因此得到的解决方案的弱色散。证明是基于相应的固定Helmholtz方程和加藤H-光滑理论的类似估计。
Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential
本文证明了带吸引库仑势的薛定谔方程在三维空间中的尖锐色散估计，揭示了长时间衰减率由阈值决定，且与势强度$Z$成正比。
Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy
我们调查$L^1(\R^2)\to L^\infty(\R^2)$色散估计薛定谔运营商$H=-Δ+V$时，有障碍物，共振或本征值，在零能量。特别是，我们表明，在零能量的s波共振的存在下，不破坏的$t^{-1}$衰变率。我们还证明了，如果存在p波共振或零能量本征值，则存在一个不定常算子$F_t$满足$\|F_t\|_{L^1\to L^\infty} \lesssim 1$，使得$$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{-1}, \text{for} |t|>1.$$。我们还建立了一个加权色散估计，当存在零能量本征值但没有共振时，其衰减率为$t^{-1}$。
Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension
本文研究了一维Schrödinger算子带有负Coulomb型衰减势的色散估计，推导了标准和正交归一Strichartz估计。通过推导WKB表达式和退化稳态相位公式，克服了快速衰减势的扰动方法不可用的困难。
Asymptotic Stability and Completeness in the Energy Space for Nonlinear Schrödinger Equations with Small Solitary Waves
本文研究了一类非线性薛定谔方程的小孤波解族。我们考虑在能量空间$H^1$中小的解，并将其分解为孤立波和色散波分量。目的是建立孤立波的渐近稳定性和色散波的渐近完备性。也就是说，我们表明，作为$t \to \infty$，孤立波分量收敛到一个固定的孤立波，色散分量收敛到一个自由的薛定谔方程的解决方案。
Weighted dispersive estimates for two-dimensional Schrödinger operators with Aharonov-Bohm magnetic field
考虑具有Aharonov-Bohm磁场和附加电势的二维薛定谔算子$H$.我们得到了酉群$e^{-i t H}$在加权$L^2$空间中对$t\to\infty$的渐近展开式的一个显式首项.特别是，我们表明，磁场改善了$e^{-i t H}$的衰减相对于由非磁性薛定谔算子产生的酉群，并在时间上的衰减率是由磁通量。
We investigate the dispersive properties of solutions to the Schrödinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schrödinger flow on each eigenspace of the link manifold satisfies a weighted $L^1\to L^\infty$ dispersive estimate. In odd dimensions, the decay rate we compute is consistent with that of the Schrödinger equation in a Euclidean space of the same dimension, but the spatial weights reflect the more complicated regularity issues in frequency that we face in the form of the spectral m…
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。
Dispersive Estimates for Schrödinger Operators with Measure-Valued Potentials in R^3
我们证明了与算子-Δ+ V相关的线性薛定谔发展的色散估计，其中势是分维至少为3/2的带号测度。
Dispersive estimates for the Schrodinger equation in dimensions four and five
我们证明了薛定谔群$e^{it(-Δ+V)}$的最佳（即没有损失的衍生物）色散估计为一类实值的潜力$V\in C^k(R^n)$与$k>(n-3)/2$，其中$n=4,5$。
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。
Dispersive Properties for Discrete Schrodinger Equations
本文证明了由两个耦合离散薛定谔方程组成的系统的色散估计。我们得到了离散算子的预解式的估计，并证明了它满足极限吸收原理。利用振荡积分的经典结果和一些新的结果证明了解的衰减性。
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.
The Fourth-order dispersive nonlinear Schrödinger equation: orbital stability of a standing wave
本文研究一维四阶色散立方非线性薛定谔方程的混合色散问题。在哈密顿系统的背景下，证明了一个特定的驻波解在能量空间中的轨道稳定性。通过构造适当的李雅普诺夫函数，建立了主要结果.
Optimal dispersion for discrete periodic Schrödinger operators
该研究证明了一维周期离散薛定谔算子的最优衰减速率色散估计，并借助标准方法推导了周期哈密顿量下小初值、合适非线性项的离散非线性薛定谔方程的色散估计。
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.
Dispersive estimates for the Landau Hamiltonian on the hyperbolic plane
本文针对双曲平面上均匀磁场下的薛定谔方程，利用传播子核的显式公式获得色散估计，进而证明所有允许对的 Strichartz 估计。
Dispersive Estimates for Scalar and Matrix Schrödinger operators on $\mathbb{H}^{n+1}$
研究了$n \geq 1$的$\mathbb{H}^{n+1}$上的标量和矩阵Schrödinger型算子的预解估计、谱理论和长时间色散性质.
Semi-classical dispersive estimates
我们证明了色散估计的波和薛定谔集团相关的二阶椭圆自伴算子依赖于一个半经典参数。应用非捕获度规扰动和扰动的磁位。
Dispersive estimates for Schrodinger operators in dimensions one and three
证明了一维和三维线性Schroedinger方程的L^1 --> L^ \infty估计.势只需要满足一些温和的衰变假设。没有规则性的潜在的假设。
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…
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…
Non-dispersive vanishing and blow up at infinity for the energy critical nonlinear Schrödinger equation in R^3
我们考虑R^3中的能量临界聚焦NLS，并证明了对于任何足够小的$ν$，径向有限能量解的存在性，当$t\to\infty$表现为一个动态重新标度的基态加上一个辐射的和时，标度律的形式为$t^ν$。
Elementary Proofs for Kato Smoothing Estimates of Schrödinger-like Dispersive Equations
在本说明性注释中，我们用初等技巧证明了经典Kato型估计的一些扩展和改进。
Dispersive Bounds for the three-dimensional Schrodinger equation with almost critical potentials
证明了三维空间中与时间无关的Schrodinger算子H = -Δ+ V的色散估计.假设势V（x）位于交点L^p（R^3）\cap L^q（R^3），p < 3/2 < q，并且还满足一般的零能谱条件。这个类，包括具有逐点衰减的势|V（x）|< C（1+ |x|）^{-2-ε}对于拉普拉斯算子的自然缩放几乎是关键的。没有额外的规律性，衰减，或积极的V被假定。
In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-Δ+\sum_{j=1}^N\langle\cdot\,, φ_j\rangle φ_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $φ_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, φ_1,\ldots, φ_N)>0$ such that $$ \|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0. $$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ estimates for finite rank perturbations of the Laplacian in…
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…
Schrödinger dispersive estimates for a scaling-critical class of potentials
本文证明了三维Schroedinger算子H = -Δ+ V（x）演化的色散估计.势函数应属于有界紧支函数关于golbal Kato范数的闭包。一些额外的谱条件，即没有共振或H的本征函数存在于任何地方的正半线。这个证明是维纳L^1反演定理的一个新版本的应用。
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 $-Δ+V$.
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。
Time decay for solutions of Schrödinger equations with rough and time dependent potentials
我们建立了三维空间中具有位势的线性含时薛定谔方程解的色散估计和Schehartz估计。我们的主要重点是小粗糙的时间依赖的潜力。这种势的例子是$V(t,x)=T(t) V_0(x)$形式，其中$T$在时间上是准周期的，而$V_0$基本上是空间变量的$L^{3/2}$函数。我们还证明了属于Rollnik和整体Kato类的小时间无关势的色散估计。最后，我们解决的问题所提出的Quarne，Soffer，Sogge有关的潜在的衰减速度比$|x|^{-2}$的Quarhartz估计。
Smoothing And Dispersive Estimates For 1d Schrödinger Equations With BV Coefficients And Applications
我们证明了薛定谔方程$i\partial_t ϕ+\partial_x (a(x) \partial_x ϕ) =0$的光滑估计，其中$a(x)\in \mathrm{BV}$是具有有界全变差的函数空间，是实的，正的和下有界的。然后，我们自举这些估计，以获得最佳的Eschenhartz和最大函数的估计，所有这些原来是相同的常数系数的情况下。我们还提供了反例显示$a\in \mathrm{BV}$是一个最低要求。最后，我们提供了一个应用程序的广义Benjamin-Ono方程的尖锐适定性。
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。
Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations
考虑聚焦的能量临界波动方程的有界解，它不散射成线性解。我们证明了该解在某种弱意义下，沿着时间序列，直到尺度和空间平移，收敛到一个孤波的和。这个结果是一个新的一般紧凑性/刚性参数的基础上配置文件分解的后果。我们也给出了这种方法在能量临界薛定谔方程中的应用。
We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\|U^tP_c u_0\|_{l^\infty}\lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_c$ is the projection to the continuous spectrum. This is an analogous result for Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.
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 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.
Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy
我们调查$L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$色散估计薛定谔运营商$H=-Δ+V$时，有障碍物，共振或本征值，在零能量。特别是，我们表明，如果有一个共振或本征值在零能量，然后有一个不定常的，有限秩算子$F_t$满足$\|F_t\|_{L^1\to L^\infty} \lesssim 1/\log t$的$t>2$，使$$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for} t>2.$$我们还表明，运营商$F_t=0$，如果有一个本征值，但没有共振在零能量。然后，我们开发类似的色散估计的解决方案运营商的四维波动方程的潜力。
On the stability of ground states in 4D and 5D nonlinear Schrodinger equation including subcritical cases
考虑了四维和五维空间中一类具有吸引势的非线性薛定谔方程。的非线性是本地的，但相当普遍的第一次包括亚临界和超临界（在L2）的非线性。我们证明了由空间上的周期解（束缚态）形成的中心流形对所有具有小初值的解都是吸引子。的证明取决于分散的估计，我们获得的不定常，哈密顿，线性化动力学围绕一个参数家庭的束缚态，“阴影”的非线性演化系统。我们采用的方法是对更高维度的扩展，因此不同的线性色散行为，以及我们以前的结果[10，11，7]的粗糙非线性。
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。
Wave operator bounds for 1-dimensional Schrödinger operators with singular potentials and applications
本文证明了一类奇异势的薛定谔算子的波算子在一维空间中的有界性，其中允许有100多个Dirac δ分布.应用，例如，分散估计和换向器界。
The effect of geometric focusing on dispersive estimates for the Schrödinger and wave equations
本文分类了非捕获渐近锥流形和精确度量锥上薛定谔方程和波动方程色散估计的长期衰减率，揭示了共轭点导致衰减阶损失和正则性指数偏移，并提出了可容许条件避免π距离共轭点对的损失。
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。
Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations
本文分析线性与非线性薛定谔方程数值逼近格式的收敛速率，引入满足离散Strichartz估计的色散格式，证明其对L2(R)初值收敛，并对Hs(R)数据（0<s<1/2）获得多项式收敛速率，优于非色散格式的对数衰减。
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