Linear response theory for magnetic Schroedinger operators in disordered media

Wave Packets and Coherent Structures for Nonlinear Schroedinger equations in Variable Nonuniform Media

We determine conditions under which a generic gauge invariant nonautonomous and inhomogeneous nonlinear partial differential equation in the two-dimensional space-time continuum can be transform into standard autonomous forms. In addition to the nonlinear Schroedinger equation, important examples include the derivative nonlinear Schroedinger equation, the quintic complex Ginzburg-Landau equation, and the Gerdjikov-Ivanov equation. This approach provides a mathematical description of nonstationary media supporting unidimensional signal propagation and/or total field trapping. In particular, we …

Continuous integral kernels for unbounded Schroedinger semigroups and their spectral projections

By suitably extending a Feynman-Kac formula of Simon [Canadian Math. Soc. Conf. Proc, 28 (2000), 317-321], we study one-parameter semigroups generated by (the negative of) rather general Schroedinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of…

无序介质中的玻色-爱因斯坦凝聚体

On Bose-Einstein condensates in disordered media

该研究针对平均场区域内受无序势作用的相互作用玻色子量子动力学，通过安德森局域化单体动力学的相互作用绘景分析，证明凝聚体涨落因无序而慢传播，提供了任意维度下慢输运的无序多体系统实例。

Uniform N-particle Anderson localization and unimodal eigenstates in deterministic disordered media without induction on the number of particles

We present the first rigorous result on Anderson localization for interacting systems of quantum particles subject to a deterministic (e.g., almost periodic) disordered external potential. For a particular class of deterministic, fermionic, Anderson-type Hamiltonians on the lattice of an arbitrary dimension, and for a large class of underlying dynamical systems generating the external potential, we prove that the spectrum is pure point, all eigenstates are unimodal and feature a uniform exponential decay. In contrast to all prior mathematical works on multi-particle Anderson localization, we d…

Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with bi…

Random matrix theory of polarized light scattering in disordered media

In this work we present a method for generating random matrices describing electromagnetic scattering from disordered media containing dielectric particles with prescribed single particle scattering characteristics. Resulting scattering matrices automatically satisfy the physical constraints of unitarity, reciprocity and time reversal, whilst also incorporating the polarization properties of electromagnetic waves and scattering anisotropy. Our technique therefore enables statistical study of a variety of polarization phenomena, including depolarization rates and polarization-dependent scatteri…