K. Ito, E. Skibsted
We study spectral theory for the Schrodinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends.
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We study the stationary scattering theory for a perturbed 1-body Stark operator. We prove existence and completeness of the stationary wave operators, construct the associated generalized Fourier transforms, and characterize asymptotics of the generalized eigenfunctions of minimal growths in terms of the stationary scattering matrix.
In the present paper we discuss stationary scattering theory for repulsive Hamiltonians. We show the existence and completeness of stationary wave operators and unitarity of the scattering matrix. Moreover we completely characterize asymptotic behaviors of generalized eigenfunctions with minimal growth in terms of the scattering matrix. In our argument the radiation condition bounds for limiting resolvents play major roles. In fact, it is used to construct the stationary wave operators.
Stationary scattering theory on manifolds, II
在文献[IS]的基础上，我们发展了具有逃逸函数的流形上薛定谔算子的平稳散射理论。一类特殊的例子是具有欧几里得和/或双曲端点的流形，可能有无界和非光滑的障碍。我们发展的理论主要沿着经典的路线[SA，Co]，并特别推导出WKB-渐近适当的广义本征函数。作为一个应用，我们解决了猜想[HPW]的跨端传输在其自然和强大的形式在我们的理论框架内。
A theoretical framework is developed for scattering of scalar radiation from stationary, three-dimensional media with correlation functions of scattering potentials obeying $\mathcal{PT}$-symmetry. It is illustrated that unlike in scattering from deterministic $\mathcal{PT}$-symmetric media, its stationary generalization involves two mechanisms leading to symmetry breaking in the statistics of scattered radiation, one stemming from the complex-valued medium realizations and the other - from the complex-valued degree of medium's correlation.
We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\cal H}_0,{\cal H}$ and for an identification operator $J:{\cal H}_0\to{\cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation form…
Non-Stationary Quantum Systems Associated with Stationary Scattering in One Dimension
该研究将一维定态散射关联到非定态非幺正二能级量子系统，通过布洛赫球分析其演化，发现量子测量可获取反射振幅，无反射构型对应极点循环演化，谱奇点与相干完美吸收对应极点间的动力学自旋翻转。
We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in [AIIS1,AIIS2]. The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the metho…
On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II
建立了非线性Lamb系统双曲定态的渐近完备性。为了证明这一点，我们利用Banach空间中的反函数定理构造了一个约化方程（这是一个非线性非自治常微分方程）收敛到双曲驻点的轨迹。我们给出了反例，表明不存在这样的轨迹的非双曲型驻点。
Dirac equation: the stationary and dynamical scattering problems
证明了具有库仑型势的径向Dirac方程的广义动力学散射算子与相应的广义定态散射算子是一致的。这个事实是经典力学中遍历结果的量子力学模拟。
Transfer matrix formulation of stationary scattering in 2D and 3D: A concise review of recent developments
本文综述二维与三维稳态散射的转移矩阵表述，说明其可消除 δ 势 Lippman-Schwinger 方程中的紫外发散，并用于构造截止频率以下全向完美隐身的复杂散射势。
Stability estimates in stationary inverse transport
本文从$n\geq3$维的全隐式算子出发，研究了稳态线性输运方程中散射系数和吸收系数重构的稳定性。在紧致凸区域的边界上，将输入边界条件映射到输出输运解，定义了一种新的输运算子。文[M. Choulli-P. Stefanov，1996和1999]，并在[J. N. Wang，1999]，用于空间独立的散射系数。我们推广了这些结果，并证明了空间相关的散射系数的$L^1$稳定性估计。
Potential scatterings in $L^2$ space: (1) non-orthogonality of stationary states
本文研究有限宽度势中不同能量散射态的非正交性，指出其叠加态范数随时间变化，并提出用微扰法和变分法求严格跃迁概率。
Within the class of Derezi{ń}ski-Enss pair-potentials which includes Coulomb potentials and for which asymptotic completeness is known \cite{De}, we show that all entries of the $N$-body quantum scattering matrix have a well-defined meaning at any given non-threshold energy. As a function of the energy parameter the scattering matrix is weakly continuous. This result generalizes a similar one obtained previously by Yafaev for systems of particles interacting by short-range potentials \cite{Ya1}. As for Yafaev's paper we do not make any assumption on the decay of channel eigenstates. The main p…
Generalized wave operators: dynamical and stationary cases and divergence problem
进一步发展了动力学和定常情况下的广义波算子理论的思想和结果，并得到了广泛类微分方程广义散射算子的精确表达式。导出了广义散射算子结构的新结果。有趣的相互关系动态和固定的情况下，发现径向薛定谔方程和狄拉克方程，以及狄拉克型方程。对于一些重要的例子，我们解释了为什么众所周知的发散在高阶近似的散射矩阵不出现在“广义波算子”的方法。
Scattering through a straight quantum waveguide with combined boundary conditions
利用定态散射理论，研究了二维直量子波导Rx（0，d）中的散射问题，其中Dirichlet边界条件作用在（- \infty，0）x{y=0} \cup（0，\infty）x{y=d}上，Neumann边界条件作用在（-infty，0）x{y=d} \cup（0，\infty）x{y=0}上.证明了在x=0时匹配条件解的存在性。稳态散射理论的使用是合理的，表明它与波包运动的关系。作为说明，匹配条件也数值求解和转移概率显示。
The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D
考虑了含无限多粒子的相互作用气体的非线性Hartree方程，研究了描述均匀费米气体的$f(-Δ)$定态的大时间稳定性.在适当的相互作用势和动量分布$f$的假设下，我们证明了定态在2维空间是渐近稳定的。更准确地说，对于任何初始数据，这是一个小扰动的$f(-Δ)$在Schatten空间，系统弱收敛到平稳状态的大时间。
A constructive approach to stationary scattering theory
本文给出了Hilbert空间$\mathcal H$上满足下列条件的自伴算子对$H_0$和$H_1$的平稳散射理论的一种新的构造性方法：（i）对于$\mathbb R,$的任何开有界子集$Δ$，算子$F E_Δ^{H_0}$和$F E_Δ^{H_1}$是Hilbert-Schmidt，并且（ii）$V = H_1- H_0$是有界的，并允许分解$V = F^*JF,$，其中$F$是从$\mathcal H$到另一个Hilbert空间$\mathcal K$的具有平凡核的有界算子，$J$是有界自$\mathcal K.$上的伴随算子满足这些条件的一对算子的例子是作用在$L^2(\mathbb R^ν),$上的薛定谔算子$H_0 = -Δ+ V_0$，其中$V_0$是类$K_ν$的势（参见B. Simon，{ \it Schrödinger semigrops，} Bull. AMS 7，1982，447- 526）和$H_1 = H_0 + V_1,$，其中$V_1 \in L^\infty(\mathbb R^ν) \cap L^1(\mathbb R^ν).$本文的结果是波算子$W_\pm(H_1,H_0)$的存在性和完备性的一个新的证明以及散射矩阵的定态公式的一个新的构造性证明。这种散射理论的方法是基于显式对角化的自伴算子$H$的希尔伯特空间层$\EuScript S(H,F)…
Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends
本文综述了渐近圆柱端点流形的谱分析和散射理论的含时方法。对于谱分析，通过矩阵理论得到了度量的短程和长程行为以及无穷远处扰动的高阶预解估计。对于散射理论，在两个Hilbert空间中证明了波算子的存在性和渐近完备性。导出了散射算子的定态公式和映射性质。最后给出了时滞的存在性及其与Eisenbud-Wigner时滞的等价性。我们的分析主要不同于现有的文献上的选择一个更简单的比较动力学，以及互补使用的时间相关的和固定的散射理论。
Stationary scattering problem (when the distance $r$ tends to infinity) and dynamical scattering problem (when the time $t$ tends to infinity) are considered for the 3D Schrödinger equation. A simple interconnection between the scattering amplitude (stationary case) and scattering operator (dynamical case) is given in the paper. This result is a quantum mechanical analog of the ergodic formulas in the classical mechanics.
Stability of the gauge equivalent classes in stationary transport
对于各向异性衰减介质，Escherdo算子确定散射和衰减系数直到规范变换。我们表明，这样的决心是稳定的。
In this paper, we consider two types of the scattering problems (relativistic case), namely, the stationary scattering problem, where the distance $r$ tends to infinity, and the dynamical scattering problem, where the time $t$ tends to infinity. Using our results on Lippmann-Schwinger equation in the relativistic case, we found the connection between the stationary scattering problem (the scattering amplitude) and the dynamical scattering problem (the scattering operator). This result is the quantum mechanical analog of the ergodic formulas in the classical mechanics.
Stationary completeness: the $N$-body short-range case
本文针对N体短程薛定谔算子，证明所有非阈值能量均为稳态完备，解决了相关猜想，并由此得到散射量强连续依赖能量、散射矩阵酉性及渐近完备性的新稳态证明。
Stationary iteration methods for solving 3D electromagnetic scattering problems
本文研究广义Chebyshev迭代和广义简单迭代法求解非自伴算子线性方程，通过复平面谱定位条件保证收敛，并应用于三维介质体波衍射的体积奇异积分方程数值求解。
We study the stationary scattering theory for the matrix Schrödinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption principle, we construct the generalized Fourier maps, and we prove that they are partially isometric with initial space the subspace of absolute continuity of the matrix Schrödinger operator and final space $L^2((0, \infty))$. We prove the existence and the completeness of the wave operators and we establish that they are given by the stationary formulae. We als…
Scattering theory for $C^2$ long-range potentials
本文为$\mathbb R^d$上$C^2$长程势Schrödinger算子建立完整稳态散射理论，证明其与含时理论等价，并利用程函方程解和新交换子方案改进极限预解式估计。
Emergence of Reflectionless Scattering from Linearizations of Integrable PDEs around Solitons
我们提出了四个数学物理的可积偏微分方程（PDE）的例子-当线性化周围的一个固定的孤子-表现出散射无反射在{ \it所有}的能量。从最著名的和最相关的经验现象的透明性的一维明亮玻色孤子Bogoliubov激发，我们进行到正弦戈登，Korteweg-de Vries和刘维方程的定态孤子也支持我们的断言。可积性和无反射散射之间的联系似乎跨越了至少两种不同的可积性范式：前三种情况下的S-可积性和最后一种情况下的C-可积性。我们认为，线性化可积偏微分方程的透明性是必要的，以确保他们可以支持原来的可积偏微分方程中的固定孤子的透明性。作为对比，分析进一步扩展到两个不可可积系统：锯齿Gordon和$ϕ^4$模型。
Determinantal Representation of the Time-Dependent Stationary Correlation Function for the Totally Asymmetric Simple Exclusion Model
非平衡低维物理的基本模型--所谓的完全非对称排斥过程与$SU_q(2)$量子代数的“结晶极限”（$q\to\infty$）有关。利用量子逆散射方法，得到了一维晶格上周期边界条件下完全非对称简单排斥过程的含时定态关联函数的精确表达式.
Stationary phase analysis of ambient noise cross-correlations: Focusing on non-ballistic arrivals
本研究针对环境噪声互相关中非弹道波至理论分析不足的问题，推导其解析稳相解，明确区分互相关与格林函数中非弹道波至的条件，指出需排除源影响才能用非弹道波至获取介质信息，为高分辨率成像监测提供理论支撑。
Green functions and completeness; the $3$-body problem revisited
本文针对三体系统证明了所有非阈值能量处散射量的稳态完备性与强连续性，解决了N=3情形下的一个猜想，并给出长程对势下渐近完备性的独立稳态证明。
Scattering by a toroidal coil
在本文中，我们考虑薛定谔算子在${\mathbb R}^3$与长程磁势相关的磁场内支持的圆环${\mathbb{T}}$。利用光滑扰动方法构造了稳态修正波算子和相应的散射矩阵$S(λ)$。我们证明了$S(λ)$的本质谱是单位圆的一个区间，它只取决于穿过$\mathbb{T}$截面的磁通量$ϕ$.此外，我们表明，在对比的Aharonov-Bohm势在${\mathbb{R}}^2$，总散射截面总是有限的。我们还推测，这里处理的情况下是一个典型的例子在3维。
Low energy spectral and scattering theory for relativistic Schroedinger operators
研究了相对论薛定谔算符的低能谱和散射理论。一些惊人的性能，在阈值的操作员表现出，例如没有0-能量共振。研究了波算子和散射算子的低能行为，证明了波算子和散射算子的广义本征函数定态表达式。在稍强的扰动条件下，证明了谱在正半轴上的绝对连续性。最后，导出了自由演化群作用量的一个显式公式.这样一个公式，在通常的薛定谔例子中是众所周知的，显然在相对论环境中是不可用的。
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