Hans Lindblad, Avy Soffer
We prove Asymptotic Completeness of one dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity.
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The three-quasi-particle scattering problem: asymptotic completeness for short-range systems
本文针对广义N体系统发展散射理论方法，重点研究一类三准粒子系统，并证明了其渐近完备性。
Equivalence of classical and quantum completeness for real principal type operators on the circle
本文证明了圆上实主型算子的哈密顿流完备性与本质自伴性等价，基于散射理论中构造非实特征值特征函数的方法，并探讨了其谱性质及与散射理论的关系。
Modified Scattering and Asymptotic Completeness for the Derivative Nonlinear Schrödinger equation
本文首次在拟线性情形下证明导数非线性薛定谔方程的修正散射与渐近完备性，方法上结合波包检验、bootstrap 论证与 Klainerman Sobolev 向量场方法。
We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials $(V,A)$ where $V(x)=O(|x|^{-ρ}), A(x)= O(|x|^{-ρ}), |x| \to \infty, ρ>1,$ are dense in the set of all solutions to the Schrödinger equation that are in $L^2(K)$ where $K$ is any connected bounded open set in $\ere^n,n\geq 2,$ with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials $(V_1,A_1)$ and $(V_2,A_2)$ in $\ere^n, n\geq 3,$ have the same scattering matrix at a fixed positive energy and if the electric potentials $V_j$ and the…
The current paper is devoted to the scattering theory of a class of continuum Schrödinger operators with deterministic sparse potentials. We first establish the limiting absorption principle for both modified free resolvents and modified perturbed resolvents. This actually is a weak form of the classical limiting absorption principle. We then prove the existence and completeness of local wave operators, which, in particular, imply the existence of wave operators. Under additional assumptions on the sparse potential, we prove the completeness of wave operators. In the context of continuum Schrö…
A note on the non-$L^1$ asymptotic completeness of the Vlasov-Maxwell system
本文针对Vlasov-Maxwell系统，在电荷的一般渐近条件下，利用方程的洛伦兹不变性证明其小初值解不满足线性散射，即系统不具备非L^1渐近完备性。
We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.
We consider the NLS on Schwarzschild manifold.For radial solutions with sufficiently localized initial data,global existence,L^p estimates and asymptotic completeness of the wave operators is proved
We give a short description of the proof of asymptotic-completeness for NLS-type equations, including time dependent potential terms, with radial data in three dimensions. We also show how the method applies for the two-body Quantum Scattering case.
Long-range scattering
研究时间依赖的长程势散射问题，证明散射波算子的存在性和完备性，并发现解的弱局部非散射部分性质，方法基于短程系统的最新方法。
We study a time-dependent scattering theory for Schrödinger operators on a manifold with asymptotically polynomially growing ends. We use the Mourre theory to show the spectral properties of self-adjoint second-order elliptic operators. We prove the existence and the asymptotic completeness of wave operators using the smooth perturbation theory of Kato. We also consider a two-space scattering with a simple reference system.
We show asymptotic completeness for the charged Klein-Gordon equation in the exterior De Sitter-Reissner-Nordström spacetime when the product of the charge of the black hole with the charge of Klein-Gordon field is small enough. We then interpret scattering as asymptotic transports along principal null geodesics in a Kaluza-Klein extension of the original spacetime.
We show asymptotic completeness for linear massive Dirac fields on the Schwarzschild-Anti-de Sitter spacetime. The proof is based on a Mourre estimate. We also construct an asymptotic velocity for this field.
Scattering of Dirac Fields in the Interior of Kerr-Newman(-Anti)-de Sitter Black Holes via Comparison and Symmetry Operators
本文针对亚极端Kerr-Newman(-anti)-de Sitter黑洞内部的有质量带电Dirac场，构建了从事件视界到柯西视界的散射理论，证明了散射数据的存在性、唯一性与渐近完备性，并采用比较算符与对称算符两种方法实现渐近完备性的证明。
We study the theory of scattering in the energy space for the Hartree equation in space dimension n>2. Using the method of Morawetz and Strauss, we prove in particular asymptotic completeness for radial nonnegative nonincreasing potentials satisfying suitable regularity properties at the origin and suitable decay properties at infinity. The results cover in particular the case of the potential |x|^(- gamma) for 2 < gamma < Min(4,n).
The purpose of this paper is to study the evolution of moving interacting particles on the mesoscopic scale. We will introduce an uncertainty principle and a new priori bound for the evolution of particles subject to a general mesoscopic interaction and use this setting to demonstrate evidence for the universal dispersion of particles in the absence of the conventional methods like velocity averaging lemmas or Strichartz-type estimates. We will develop a scattering theory in generality and use its frame work to show the existence and uniqueness of a class scattering solutions to the Boltzmann …
We consider the defocusing nonlinear Schr{ö}dinger equation in several space dimensions, in the presence of an external potential depending on only one space vari-able. This potential is bounded from below, and may grow arbitrarily fast at infinity. We prove existence and uniqueness in the associated Cauchy problem, in a suitable functional framework, as well as the existence of wave operators when the power of the nonlinearity is sufficiently large. Asymptotic completeness then stems from at least two approaches, which are briefly recalled.
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 asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which kind of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavio…
We show asymptotic completeness for the massive charged Dirac equation between the black hole and Cauchy horizons of a sub-extremal ((Anti-) De Sitter) Reissner-Nordström black hole.
Asymptotic completeness for a scalar quasilinear wave equation satisfying the weak null condition
本文利用几何约化系统，首次证明了满足弱零条件的标量拟线性波动方程的渐近完备性，通过渐近变量恢复散射数据并构造匹配的精确解。
We consider the scattering theory for the Schrodinger equation with $-Δ-|x|^α$ as a reference Hamiltonian, for $0< α\leq 2$, in any space dimension. We prove that when this Hamiltonian is perturbed by a potential, the usual short range/long range condition is weakened: the limiting decay for the potential depends on the value of $α$, and is related to the growth of classical trajectories in the unperturbed case. The existence of wave operators and their asymptotic completeness are established thanks to Mourre estimates relying on new conjugate operators. We construct the asymptotic velocity an…
We prove small-data global existence to semi-linear wave equations on hyperbolic space of dimension greater than or equal to three, for nonlinearities that have the form of a sufficiently high integer power of the solution. We also prove the existence and asymptotic completeness of wave operators in this setting.
In this paper we consider the long time behavior of solutions to the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times\mathbb{T}^{d}$, $1\leq d\leq4$. For sufficiently small, smooth, decaying data we prove global existence and derive modified asymptotic dynamics. We also consider the asymptotic completeness problem.
On completeness of modified wave operators for defocusing NLS
本文研究临界规范不变非线性散焦Schrödinger方程的修正散射，通过波算子完备性刻画解的长时间渐近首项，并分别在n=1和n=2的sup范数控制条件下证明完备性。
We study the theory of scattering for a Schr"odinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schr"odinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We a…
On wave operators for Hardy operators with and without perturbations
本文证明了 Hardy 算子及其带短程扰动情形的波算子渐近完备性，并指出长程势下波算子不存在，通过三种方法（半群、Kato 光滑性、传播估计）实现。
Scattering for the fractional magnetic Schrodinger operators
本文通过构造带磁势的分数阶畸变傅里叶变换，证明了分数阶磁薛定谔算子散射算子的存在性，并得到波算子的存在性与渐近完备性，同时证明了薛定谔算子无正特征值。
Scattering theory of the nonlinear wave equations on lattices
本文综合离散波动方程的一致衰减估计与色散方程散射理论抽象框架，建立了DNLW的散射理论，并利用Strichartz估计给出另一证明。
We prove the optimal endpoint Strichartz estimates for Schrödinger equation with charge transfer potentials and a general source term in $\mathbb{R}^n$ for $n\geq3$. The proof is based on using the projection on the scattering states defined implicitly in Rodnianski, Schlag and Soffer \cite{RSS1}, and asymptotic completeness for such systems. The method works as well in the matrix non-selfadjoint case.
We study the scattering problem for the nonlinear Schrödinger equation $i\partial_t u + Δu = |u|^p u$ on $\mathbb{R}^d$, $d\geq 1$, with a mass-subcritical nonlinearity above the Strauss exponent. For this equation, it is known that asymptotic completeness in $L^2$ with initial data in $Σ$ holds and the wave operator is well-defined on $Σ$. We show that there exists $0<β<p$ such that the wave operator and the data-to-scattering-state map do not admit extensions to maps $L^2\to L^2$ of class $C^{1+β}$ near the origin. This constitutes a mild form of ill-posedness for the scattering problem in t…
We develop a scattering theory for the linear wave equation $\Box_g ψ= 0$ on the interior of Reissner-Nordström black holes, connecting the fixed frequency picture to the physical space picture. Our main result gives the existence, uniqueness and asymptotic completeness of finite energy scattering states. The past and future scattering states are represented as suitable traces of the solution $ψ$ on the bifurcate event and Cauchy horizons. The heart of the proof is to show that after separation of variables one has uniform boundedness of the reflection and transmission coefficients of the resu…
Nonlinear Scattering Theory for Asymptotically de Sitter Vacuum Solutions in All Even Spatial Dimensions
本文针对偶数空间维度下爱因斯坦真空方程的渐近德西特解，建立了包含散射态存在唯一性、渐近完备性及可逆散射映射的定量非线性散射理论，解决了偶数维的挑战并避免导数损失。
The well-posedness and scattering theory of nonlinear Schrödinger equations on lattice graphs
针对格图上的非线性 Schrödinger 方程，本文提出新型一阶导数，建立广义拟线性与半线性方程的适定性、解映射解析依赖性及小数据散射结果。
We study the scattering theory for charged Klein-Gordon equations: \[\{{array}{l} (\p_{t}- ıv(x))^{2}ϕ(t,x) ε^{2}(x, D_{x})ϕ(t,x)=0,[2mm] ϕ(0, x)= f_{0}, [2mm] ı^{-1} \p_{t}ϕ(0, x)= f_{1}, {array}. \] where: \[ε^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} ıb_{j}(x))A^{jk}(x)(\p_{x_{k}} ıb_{k}(x))+ m^{2}(x),\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \[ h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar…
Coulomb Sectors and Scattering for Maxwell-Higgs Fields on Schwarzschild and Slowly Rotating Kerr Backgrounds
针对 Schwarzschild 和慢旋转 Kerr 黑洞外部，发展小数据 Maxwell-Higgs 理论，通过传递原理实现全局存在性、辐射场和散射理论，并识别 Coulomb 扇区中的长程归一化。
We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We prove that if the energy is below or equal to the critical value, then the solution approaches a free Klein-Gordon solution at the time infinity. The interesting feature in the critical case is that the…
We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the …
In this paper we study the stability of the self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions $χ_a(t,x)$ form a family of evolving regular curves of $\mathbb R^3$ that develop a singularity in finite time, indexed by a parameter $a>0$. We consider curves that are small regular perturbations of $χ_a(t_0,x)$ for a fixed time $t_0$. In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence for the binormal flow. Nevertheless, w…
This article is concerned with the small data problem for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, and short range modifications of it. We provide a new, simpler approach in order to prove that global solutions exist for data which is small in $H^{0,1}$. In the same setting we also discuss the related problems of obtaining a modified scattering expansion for the solution, as well as asymptotic completeness.
Timelike asymptotics for global solutions to a scalar quasilinear wave equation satisfying the weak null condition
研究满足弱零条件的标量拟线性波动方程整体解的类时渐近性，推导光锥内部整体解的渐近公式（含特定散射数据），并证明违反零条件时在衰减假设下解必为零。
Scattering for the Wave Equation on de Sitter Space in All Even Spatial Dimensions
本文研究任意偶数空间维度下 de Sitter 空间上的线性波动方程，通过时间微分所得模型方程建立完整散射理论，并构造过去与未来无穷远数据间的散射同构。
In this paper, we study the long time dynamics of small solutions to Schrödinger map flows from $\Bbb R$ to Riemannian surfaces. The results are threefold. (i) We prove that for general Riemannian surface targets the points with some geometric condition can be completely divided into two categories according to the sectional curvature so that the long time dynamics of small solutions of 1D Schrödinger map flow near them are described by modified scattering and scattering respectively for the two categories. (ii) If the geometric condition fails, we prove that solutions with slow time growth in…
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