Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣ ^σ ) for some 0 < σ < 1.     

Energy transfer in scattering by rotating potentials

Quantum mechanical scattering theory is studied for time-dependent Schrödinger operators, in particular for particles in a rotating potential. Under various assumptions about the decay rate at infinity we show uniform boundedness in time for the kinetic energy of scattering states, existence and completeness of wave operators, and existence of a conserved quantity under scattering. In a simple model we determine the energy transferred to a particle by collision with a rotating blade.     

The Wegner estimate and the integrated density of states for some random operators

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local H?lder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on theL ^p-theory of the spectral shift function (SSF), forp ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace idealI _p, for 0p ≤ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local H?lder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.     

On some Schrödinger and wave equations with time dependent potentials

The existence and uniqueness of solutions in the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate.     

A remark on the Lifshitz tail for Schrödinger operator with random magnetic field

In this note, we consider the Lifshitz singularity for Schrödinger operator with ergodic random magnetic field. A key estimate is an energy bound for magnetic Schrödinger operators as discussed in Nakamura [8]. Here we remove a technical assumption in [8], namely, the uniform boundedness of the magnetic field.     

Solutions of mKdV in Classes of Functions Unbounded at Infinity

Using P. Lax’s concept of a Lax pair we prove global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrödinger operator under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.     

Asymptotics of eigenelements of boundary value problems for the Schrödinger operator with a large potential localized on a small set

Asymptotics of eigenelements of a singularly perturbed boundary value problem for the three-dimensional Schrödinger operator is constructed in a bounded domain with the Dirichlet and Neumann boundary condition. The perturbation is described by a large potential whose support contracts into a point. In the case of the Dirichlet boundary conditions, this problem corresponds to a potential well with infinitely high walls and a narrow finite peak at the bottom.     

Magnetic bottles for the Neumann problem: The case of dimension 3

The main object of this paper is to analyze the recent results obtained on the Neumann realization of the Schrödinger operator in the case of dimension 3 by Lu and Pan. After presenting a short treatment of their spectral analysis of keymodels, we show briefly how to implement the techniques of Helffer-Morame in order to give some localization of the ground state. This leaves open the question of the localization by curvature effect which was solved in the case of dimension 2 in our previous work and will be analysed in the case of dimension 3 in a future paper.     

Lifshitz tails for random perturbations of periodic Schrödinger operators

The present paper is a non-exhaustive review of Lifshitz tails for random perturbations of periodic Schrödinger operators. It is not our goal to review the whole literature on Lifshitz tails; we will concentrate on a single model, the continuous Anderson model.     

Wegner estimate for sparse and other generalized alloy type potentials

We prove a Wegner estimate for generalized alloy type models at negative energies (Theorems 8 and 13). The single site potential is assumed to be non-positive. The random potential does not need to be stationary with respect to translations from a lattice. Actually, the set of points to which the individual single site potentials are attached, needs only to satisfy a certain density condition. The distribution of the coupling constants is assumed to have a bounded density only in the energy region where we prove the Wegner estimate.     

Explicit examples of nonsolvable weakly hyperbolic operators with real coefficients

We give in this paper two explicit examples of nonsolvable weakly hyperbolic operators with real coefficients in two-space-dimensions.     

Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents

We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant.     

Krylov subspace spectral methods for the time-dependent Schrödinger equation with non-smooth potentials

This paper presents modifications of Krylov Subspace Spectral (KSS) Methods, which build on the work of Gene Golub and others pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to the time-dependent Schrödinger equation in the case where either the potential energy or the initial data is not a smooth function. These modifications consist of using various symmetric perturbations to compute off-diagonal elements of functions of matrices. It is demonstrated through analytical and numerical results that KSS methods, with these modifications, achieve the same high-order accuracy and possess the same stability properties as they do when applied to parabolic problems, even though the solutions to the Schrödinger equation do not possess the same smoothness.     

One-dimensional Schrödinger operator on the half-line: The differential equation for eigenfunctions with respect to the spectral parameter and an analog of the Freud equation

It is shown that the eigenfunctions of the Schrödinger operator on the half-line satisfy an explicitly constructed differential equation with respect to the spectral parameter. Such an equation was earlier obtained for orthogonal polynomials. An analog of the Freud equation is found.     

Continuity properties of the integrated density of states on manifolds

We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS. Subsequently, alloy-type perturbations of the periodic operator are considered. The randomness may enter both via the potential and the metric. A Wegner estimate is proven which implies the continuity of the corresponding IDS. This gives an example of a discontinuous “periodic” IDS which is regularized by a random perturbation.     

带跳的线性SDE的上Lyapunov指数的逼近

在假设线性随机微分方程的Lyapunov上指数q存在的条件下，我们将线性随机微分方程离散化，获得了几种逼近线性随机微分方程解的Markov链，并且证明了这些Markov链存在Lyapunov指数q^h。当离散化步长h很小时，我们给出了误差|q—q^h|阶的理论估计，这是Talay[9]中相应结果的推广。     

Atomic characterization of the Hardy space of one-dimensional Schrödinger operators with nonnegative potentials

Given a Schrödinger operator on with nonnegative potential , we present an atomic characterization of the associated Hardy space .

     

Evolution of solitons of nonlinear Schrödinger equation with variable parameters

The nonlinear Schrödinger equation with variable parameters is solved by means of variational technique. A set of evolution equations for the solitary-wave solution is derived. The propagation properties of the solitons in an adiabatic amplification system and in a dispersion-decreasing fiber are analyzed. An explicit analytical approximate soliton solution in the exponentially dispersion-decreasing fiber is obtained using the derived dynamical equations.     

Existence and mapping properties of the wave operator for the Schrödinger equation with singular potential

We consider the Schrödinger equation in three-dimensional space with small potential in the Lorentz space and we prove Strichartz-type estimates for the solution to this equation. Moreover, using Cook's method, we prove the existence of the wave operator. In the last section we prove the equivalence between the homogeneous Sobolev spaces and in the case .