Uniform Convergence of Schr?dinger Cocycles over Bounded Toeplitz Subshift

For locally constant cocycle defined on an aperiodic subshift, Damanik and Lenz proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. For any simple Toeplitz subshift, we proved that the corresponding Schr?dinger cocycle is uniform, although it does not satisfy condition (B) in general. In this paper, we study bounded Toeplitz subshift. In general, it does not satisfy condition (B); and it contains non-simple case, which make us cannot use Chebishev polynomial. By a combination of trace formula and avalanche principle, we prove that for any bounded Toeplitz subshift, the corresponding Schr?dinger cocycle is also uniform.     

Simple Harish-Chandra supermodules over the super Schrdinger algebra

We study the N=1 super Schrdinger algebra S in(1+1)-dimensional spacetime. The first part of this paper determines the necessary and sufficient conditions for highest weight supermodules over S to be simple. The paper also describes the structure of all Verma supermodules and determines all simple Harish-Chandra supermodules over S.     

Joint Continuity of Lyapunov Exponent for Finitely Smooth Quasi-periodic Schr?dinger Cocycles

We prove the joint continuity of Lyapunov exponent in the energy and the Diophantine frequency for quasi-periodic Schr?dinger cocycles with the C~2 cos-type potentials. In particular, the Lyapunov exponent is log-H?lder continuous at each Diophantine frequency.     

Two Theorems on Convergence of Schrödinger Means

Journal of Fourier Analysis and Applications - Localization and convergence almost everywhere of Schrödinger means are studied.     

Problems on Pointwise Convergence of Solutions to the Schr?dinger Equation

In this paper we consider several variants of the pointwise convergence problem for the Schr?dinger equation, which generalize the previously known results.     

Pointwise Convergence and Uniform Convergence of Wavelet Frame Series

Pointwise convergence and uniform convergence for wavelet frame series is a new topic. With the help of band-limited dual wavelet frames, this topic is first researched.     

Lyapunov exponents of continuous Schrödinger cocycles over irrational rotations

We consider the Lyapunov exponent of those continuous SL $(2,\mathbb{R})We consider the Lyapunov exponent of those continuous SL-valued cocycles over irrational rotations that appear in the study of Schr?dinger operators and prove generic results related to large coupling asymptotics and uniform convergence.     

Matrix-valued Schrödinger operators over local fields

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to ameasure on the Skorokhod space of paths,D[0,∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman-Kac formula in the local field setting as a consequence of the Hille-Yosida theory of semi-groups. The text was submitted by the authors in English.     

Convergence of ground state solutions for nonlinear Schrdinger equations on graphs

We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|~(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|~(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.     

A Note on the Convergence of the Schrödinger Operator along Curve

In this paper we set up a convergence property for the fractional Schödinger operator for $0相似文献   

Convergence of a semiclassical wavepacket based time-splitting for the Schrödinger equation

We propose a new algorithm for solving the semiclassical time-dependent Schrödinger equation. The algorithm is based on semiclassical wavepackets. The focus of the analysis is only on the time discretization: convergence is proved to be quadratic in the time step and linear in the semiclassical parameter $\varepsilon $ .     

Convergence of Nonlinear Schrödinger–Poisson Systems to the Compressible Euler Equations

Abstract

The combined semi-classical and quasineutral limit in the bipolar defocusing nonlinear Schrödinger–Poisson system in the whole space is proven. The electron and current densities, defined by the solution of the Schrödinger–Poisson system, converge to the solution of the compressible Euler equation with nonlinear pressure. The corresponding Wigner function of the Schrödinger–Poisson system converges to a solution of a nonlinear Vlasov equation. The proof of these results is based on estimates of a modulated energy functional and on the Wigner measure method.     

Riesz Transforms of Schr?dinger Operators on Manifolds

We consider Schr?dinger operators A=???+V on L ^p (M) where M is a complete Riemannian manifold of homogeneous type and V=V ⁺?V ^? is a signed potential. We study boundedness of Riesz transform type operators $\nabla A^{-\frac{1}{2}}$ and $|V|^{\frac{1}{2}}A^{-\frac{1}{2}}$ on L ^p (M). When V ^? is strongly subcritical with constant ????(0,1) we prove that such operators are bounded on L ^p (M) for $p\in(p_{0}', 2]$ where $p_{0}'=1$ if N??2, and $p_{0}'=(\frac{2N}{(N-2)(1-\sqrt{1-\alpha })})' \in (1, 2)$ if N>2. We also study the case p>2. With additional conditions on V and M we obtain boundedness of ?A ^?1/2 and |V|^1/2 A ^?1/2 on L ^p (M) for p??(1,inf?(q ₁,N)) where q ₁ is such that $\nabla(-\Delta)^{-\frac{1}{2}}$ is bounded on L ^r (M) for r??[2,q ₁).     

Small Time Uniform Controllability of the Linear One-Dimensional Schrödinger Equation with Vanishing Viscosity

This paper addresses the exact controllability problem of the linear one-dimensional Schrödinger equation perturbed by a vanishing viscosity term depending on a strictly positive parameter. It is shown that, for any time and for each initial datum in a suitable space, there exists a uniformly bounded family of boundary controls. Any weak limit of this family is a control for the linear Schrödinger equation.     

Commutators of Riesz Transforms Related to?Schr?dinger Operators

In this work we obtain boundedness on L ^p , for 1<p<??, of commutators T _b f=bTf?T(bf) where T is any of the Riesz transforms or their conjugates associated to the Schr?dinger operator ???+V with V satisfying an appropriate reverse H?lder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p=??, under a slightly stronger condition on?b.