Torben Fattler, Martin Grothaus
By using Dirichlet form techniques we construct the dynamics of a tagged particle in an infinite particle environment of interacting particles for a large class of interaction potentials. In particular, we can treat interaction potentials having a singularity at the origin, non-trivial negative part and infinite range, as e.g., the Lennard-Jones potential.
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We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in [FG11], using closures of pre-Dirichlet forms which were already proposed in [GP87] and [Osa98]. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from [FG11], we show…
Singular diffusion limit of a tagged particle in zero range processes with Sinai-type random environment
该研究针对一维环面Sinai型随机环境下的零程过程，通过两步法推导标记粒子位置的奇异扩散极限，先引入正则化得到非平衡极限方程，再证明正则化参数趋于零时解依分布收敛于对应奇异扩散过程。
Consider a complex energy $z$ for a $N$-particle Hamiltonian $H$ and let $χ$ be any wave packet accounting for any channel flux. The time independent mean field (TIMF) approximation of the inhomogeneous, linear equation $(z-H)|Ψ>=|χ>$ consists in replacing $Ψ$ by a product or Slater determinant $ϕ$ of single particle states $ϕ_i.$ This results, under the Schwinger variational principle, into self consistent TIMF equations $(η_i-h_i)|ϕ_i>=|χ_i>$ in single particle space. The method is a generalization of the Hartree-Fock (HF) replacement of the $N$-body homogeneous linear equation $(E-H)|Ψ>=0$ …
Bound States in Continuum via Singular Transfer Matrices
本研究针对当前非均匀薄图案化薄层器件的BICs研究缺乏传输矩阵精确解析解与数值数据结合方法的问题，利用此前提出的解析公式转化为方程，探究BICs的起源、特性及实现，推导器件几何参数与工作频段。
We review recent developments in the spectral theory of continuum one-dimensional quasicystals, yielding purely singular continuous spectrum for these Schrödinger operators. Allowing measures as potentials we can generalize some results to very singular potentials, including Kronig-Penney type models.
In this paper we construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Furthermore, we show that almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman.
We apply the factorization technique developed by Kuru et. al. [Ann. Phys. {\bf 323} (2008) 413] to obtain the exact analytical classical trajectories and momenta of the continuum states of the non Hermitian but ${\cal{PT}}$ symmetric Scarf II potential. In particular, we observe that the strange behaviour of the quantum version at the spectral singularity has an interesting classical analogue.
The introduction of a lattice converts a singular boundary-layer problem in the continuum into a regular perturbation problem. However, the continuum limit of the discrete problem is extremely nontrivial and is not completely understood. This paper examines two singular boundary-layer problems taken from mathematical physics, the instanton problem and the Blasius equation, and in each case examines two strategies, Pade resummation and variational perturbation theory, to recover the solution to the continuum problem from the solution to the associated discrete problem. Both resummation procedur…
Continuum limit of resonances for discrete Schrödinger operators
本文证明离散 Schrödinger 算子的复共振在连续极限下收敛到连续算子的共振，并借助推广的范数预解收敛与 Fourier 空间复畸变方法，为奇异势情形提供近似离散算子构造方案。
We consider the radial Schroedinger equation with an attractive potential singular in the origin. The additional continuum of states caused by the singularity, that usually remain nontreatable, are shown to correspond to particles, asymptotically free near the singularity (in the inner channel). Depending on kinematics, they are either confined by the centre or may escape to infinity (to the outer channel). The orthonormality within the continuum of confined states is established and the scattering phase of the particle emitted by the centre and then reflected back to it is found. For the deco…
We prove the Wegner bounds for the one-dimensional interacting multi-particle Anderson models in the continuum. The results apply to singular probability distribution functions such as the Bernoulli's measures. The proofs need the amplitude of the inter-particle interaction potential to be sufficiently weak. As a consequence, the results imply the Anderson localization via the multi-scale analysis.
A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current $T$ and the associated defects are given by its boundary. For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown…
We consider Schrödinger operators on the real line with limit-periodic potentials and show that, generically, the spectrum is a Cantor set of zero Lebesgue measure and all spectral measures are purely singular continuous. Moreover, we show that for a dense set of limit-periodic potentials, the spectrum of the associated Schrödinger operator has Hausdorff dimension zero. In both results one can introduce a coupling constant $λ\in (0,\infty)$, and the respective statement then holds simultaneously for all values of the coupling constant.
We establish necessary and sufficient conditions for complex potentials in the Schrödinger equation to enable spectral singularities (SSs) and show that such potentials have the universal form $U(x) = -w^2(x) - iw_x(x) + k_0^2$, where $w(x)$ is a differentiable function, such that $\lim_{x\to \pm \infty}w(x)=\mp k_0$, and $k_0$ is a nonzero real. We also find that when $k_0$ is a complex number, then the eigenvalue of the corresponding Shrödinger operator has an exact solution which, depending on $k_0$, represents a coherent perfect absorber (CPA), laser, a localized bound state, a quasi bound…
We study the existence, regularity and so-called `strict physicality' of weak solutions of a coupled Navier-Stokes Q-tensor system which is proposed as a model for the incompressible flow of nematic liquid crystal materials. An important contribution to the dynamics comes from a singular potential introduced by John Ball and Apala Majumdar (Nematic Liquid Crystals: From Maier-Saupe to a Continuum Theory, Mol. Cryst. Liq. Cryst., Vol. 525: pp. 1-11, 2010) which replaces the commonly employed Landau-de Gennes bulk potential. This is built into our model to ensure that a natural physical constrai…
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