某些Landau方程的粒子近似

Convergence to the Landau equation from the truncated BBGKY hierarchy in the weak-coupling limit

We consider the evolution of the one-particle function in the weak-coupling limit in three space dimensions, obtained by truncating the BBGKY hierarchy under a propagation of chaos approximation. For this dynamics, we rigorously show the convergence to a solution of the Landau equation, keeping the full singularity of the Landau kernel. This resolves the issue arising from [10], where the singular region has been removed artificially. Since the singularity appears in the Landau equation due to the geometry of particle interactions, it is an intrinsic physical property of the weak-coupling limi…

具有空间1:3共振的超图案形成系统中随机朗道近似的有效性

Validity of the stochastic Landau approximation for super-pattern forming systems with a spatial 1:3 resonance

本文针对带加性白噪声和空间1:3共振的Kuramoto-Shivashinsky类方程，推导耦合随机朗道方程，并证明其近似解与原始系统真实解之间的误差估计。

库仑势中辐射带电粒子Landau-Lifshitz方程的精确解

Exact Solution of the Landau-Lifshitz Equations for a Radiating Charged Particle in The Coulomb Potential

我们精确地解决了经典的非相对论性朗道-Lifshitz运动方程的带电粒子运动在库仑势，包括辐射阻尼。一般的解决方案涉及Painleve第二类超越。它证实了我们的物理直觉，即带负电荷的经典粒子将螺旋进入核，支持Landau-Lifshitz方程的有效性。

Derivation of the Landau-Pekar equations in a many-body mean-field limit

We consider the Fröhlich Hamiltonian in a mean-field limit where many bosonic particles weakly couple to the quantized phonon field. For large particle number and suitably small coupling, we show that the dynamics of the system is approximately described by the Landau-Pekar equations. These describe a Bose-Einstein condensate interacting with a classical polarization field, whose dynamics is effected by the condensate, i.e., the back-reaction of the phonons that are created by the particles during the time evolution is of leading order.

关于量子力学粒子在平方反比势作用下运动的Landau-Lifshitz猜想

Landau-Lifshitz's conjecture about the motion of a quantum mechanical particle under the inverse square potential

Landau和Lifshitz [4，Section 35]证明了对于任意的$k\in \mathbb{R}$，存在量子力学粒子在平方反比势$k|x|^{-2}$下的运动，即$x \in \mathbb{R}^3$ .当$k$为负且$| k |$非常大时，平方反比势变得非常深，并产生非常强的吸引力，因此量子力学粒子很可能落到原点（平方反比势的中心）。因此，这个猜想（Landau-Lifshitz猜想）乍一看似乎是错误的。然而，我们证明了朗道-Lifshitz的猜想，证明了存在一个自伴扩张的薛定谔算子的平方倒数潜在的$-Δ+k|x|^{-2}$在$\mathbb{R}^N\ (N\geq 2)$和频谱的自伴扩张是有界的任意$k\in \mathbb{R}$以下。从而在$\mathbb{R}^N\ (N\geq 2)$中对Landau-Lifshitz猜想给出了肯定的、完整的回答。

Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

We consider the Fröhlich Hamiltonian with large coupling constant $α$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $α^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account. This allows us to show that the Landau-Pekar equations approximately describe the evolution of the electron- and one-phonon reduced densi…

从粒子系统到朗道方程：一个相容性结果

From particle systems to the Landau equation: a consistency result

我们考虑一个N个经典粒子的系统，通过一个光滑的，短程势相互作用，在弱耦合制度。这意味着当相互作用被适当地重新缩放时，N趋于无穷大。服从通常BBGKY层次的j粒子边缘被分解为两个贡献：一个小但强烈振荡，另一个希望光滑。消除第一个，我们到达建立一个新的层次（平滑部分）涉及的记忆项的动力学问题。我们表明，一阶修正的自由流收敛，作为N \rightarrow \infty，相应的长期相关的朗道方程。我们还展示了相关的混沌传播。

Position-space representation of charged particles' propagators in a constant magnetic field as an expansion over Landau levels

We have obtained propagators in the position space as an expansion over Landau levels for the charged scalar particle, fermion, and massive vector boson in a constant external magnetic field. The summation terms in the resulting expressions consisted of two factors, one being rotationally invariant in the 2-dimensional Euclidean space perpendicular to the direction of the field, and the other being Lorentz-invariant in the 1+1-dimensional space-time. The obtained representations are unique in the sense that they allow for the simultaneous study of the propagator from both space-time and energe…

Multiple Superadiabatic Transitions and Landau-Zener Formulas

We consider nonadiabatic systems in which the classical Born-Oppenheimer approximation breaks down. We present a general theory that accurately captures the full transmitted wavepacket after multiple transitions through either a single or distinct avoided crossings, including phase information and associated interference effects. Under suitable approximations, we recover both the celebrated Landau-Zener formula and standard surface-hopping algorithms. Our algorithm shows excellent agreement with the full quantum dynamics for a range of avoided crossing systems, and can also be applied to singl…

Coordinate-space representation of a charged scalar particle propagator in a constant magnetic field expanded as a sum over the Landau levels

A coordinate-space representation for a charged scalar particle propagator in a constant magnetic field was obtained as a series over the Landau levels. Using the recently developed modified Fock-Schwinger method, an intermediate expression was constructed and symmetrized, thus, allowing for factorization of the series terms into two factors. The first one, a sum of Bessel functions, depends on time and $z$-coordinate, where the $z$-axis is chosen to be a direction of the magnetic field, and has a structure similar to the propagator of a free field. The second one, a product of a Laguerre poly…

宇宙弦时空中Dirac中性粒子的非惯性效应引起的类似Landau量子化

Noninertial effects on a Dirac neutral particle inducing an analogue of the Landau quantization in the cosmic string spacetime

我们讨论的行为与狄拉克中性粒子的永久电偶极矩，以实现相对论束缚态的解决方案，在非惯性系和拓扑缺陷时空的存在下，相互作用的外场。我们证明了费米-沃克参考系的非惯性效应即使在没有磁荷的情况下也会引起径向磁场，这受到宇宙弦时空拓扑结构的影响。然后，我们讨论了诱导场必须满足的条件，以产生对应于宇宙弦时空中的Landau-He-McKellar-Wilkens量子化的相对论束缚态。最后我们得到了正能解的Dirac旋量和Dirac几率流的Gordon分解。

球形胶体粒子的Landau-de Gennes能量极小化

Minimizers of the Landau-de Gennes energy around a spherical colloid particle

我们考虑能量最小化配置的液晶周围的球形胶体粒子，在Landau-de Gennes模型的上下文中。假设胶体粒子占据半径为r_0的球的外部，在胶体表面满足垂面弱锚定，并在无穷远处接近均匀单轴态。我们研究了在两种不同的极限状态下的极小元：对于与特征长度尺度r_0<<L相比较小的球，以及对于较大的球，r_0>>L。半径和锚固强度W之间的关系也是相关的。对于小球，我们得到一个限制四极配置，与“土星环”缺陷相对较强的锚定，对应于交换的Q-张量的本征值。在非常大的球的限制下，我们得到了Oseen-Frank能量的轴对称极小值，并得到了一个偶极配置，正好有一个点缺陷。

Landau-Lifshitz模型的S-矩阵分解

On S-Matrix factorization of the Landau-Lifshitz model

我们考虑Landau-Lifshitz模型的三粒子散射S-矩阵，直接计算Feynman图的集合直到二阶。通过对非线性薛定谔模型的类似计算，我们证明了三粒子S矩阵在第一非平凡阶是可因子分解的。