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Register a new userOn the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential
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It is known that a one-dimensional quantum particle is localized when subjected to an arbitrarily weak random potential. It is conjectured that localization also occurs for an arbitrarily weak potential generated from the nonlinear skew-shift dynamics: $v_n=2cosleft(binom{n}{2}omega +ny+xright)$ with $omega$ an irrational number. Recently, Han, Schlag, and the second author derived a finite-size criterion in the case when $omega$ is the golden mean, which allows to derive the positivity of the infinite-volume Lyapunov exponent from three conditions imposed at a fixed, finite scale. Here we numerically verify the two conditions among these that are amenable to computer calculations.
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We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $binom{j}{2} omega+jy+x mod 1$ for irrational $omega$. We prove that the eigenvalue distribution of these matrices converges to the corresponding distribution from random matrix theory on the global scale, namely, the Wigner semicircle law for square matrices and the Marchenko-Pastur law for rectangular matrices. The results evidence the quasi-random nature of the skew-shift dynamics which was observed in other contexts by Bourgain-Goldstein-Schlag and Rudnick-Sarnak-Zaharescu.
We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the Lyapunov exponent of this model. Making use of a connection between quantum walks and CMV matrices our result implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients as well as for quasi-periodic unitary band matrices.
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We study the behavior of the soliton solutions of the equation i((partial{psi})/(partialt))=-(1/(2m)){Delta}{psi}+(1/2)W_{{epsilon}}({psi})+V(x){psi} where W_{{epsilon}} is a suitable nonlinear term which is singular for {epsilon}=0. We use the strong nonlinearity to obtain results on existence, shape, stability and dynamics of the soliton. The main result of this paper (Theorem 1) shows that for {epsilon}to0 the orbit of our soliton approaches the orbit of a classical particle in a potential V(x).
A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary differential equation. In this representation it can be shown that solitons and solutions with compact support only exist in systems with linear or quadratic dispersion, respectively. In particular, this article deals with so the called K(n,m) equations. It is shown that these equations can be classified according to a simple point transformation. As a result, all equations that allow for soliton solutions join the same equivalence class with the Korteweg-deVries equation being its representative.
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