on the stability of time harmonic localized states in a disordered nonlinear medium we study the problem of localization in a disordered one dimensional nonlinear medium modeled by the nonlinear schrodinger equation devillard and souillard have shown that almost every time harmonic solution of this random pde exhibits localization we consider the temporal stability of such time harmonic solutions and derive bounds on the location of any unstable eigenvalues by direct numerical determination of the eigenvalues we show that these time harmonic solutions are typically unstable and find the distribution of eigenvalues in the complex plane the distributions are distinctly different for focusing and defocusing nonlinearities we argue further that these instabilities are connected with resonances in a schrodinger problem and interpret the earlier numerical simulations of caputo newell and shelley and of shelley in terms of these instabilities finally in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time harmonic state observed in the simulations of shelley

decay rate of the green function in a random potential on the bethe lattice and a criterion for localization it is shown that in the tight binding anderson model on the bethe lattice the exponential decay rate of the green function can be obtained for arbitrary energies and arbitrary disorder analytical results in the case of the lorentzian distribution of site energies are presented as an application of these results a criterion for the localized region on the corresponding regular lattice on the level of the bethe peierls approximation is proposed our criterion yields exact results in the one dimensional limit and yields correct band edges for the hypercubic lattice in the vanishing limit of disorder the mobility edge trajectory obtained by our criterion is given by an elliptic curve in the case of the lorentzian distribution and its shape is found to be in qualitatively good agreement with that obtained by the finite size scaling method in the three dimensional system

a note on the use of vector barrier parameters for interior point methods a key feature to ensure desirable convergence properties in an interior point method is the appropriate choice of an updating rule for the barrier parameter in this work we analyze and describe updating rules based on the use of a vector of barrier parameters we show that these updating rules are well defined and satisfy sufficient conditions to ensure convergence to the correct limit points we also present some numerical results that illustrate the improved performance of these strategies compared to the use of a scalar barrier parameter

multiple positive solutions of singularly perturbed differential systems with different orders in this paper we consider the multiplicity of positive solutions for a class of singular higher order perturbed differential systems with different orders by employing a well known fixed point theorem some new existence results are given under the case where nonlinearity can be sign changing

uniqueness of positive solutions for a class of semilinear elliptic systems we prove existence and uniqueness of positive solutions for the system u f v v g u in b u v 0 on b where b is the open unit ball in rn f g r r and f x xp g x xq at for some p q 0 with pq 1

a new proof of localization in the anderson tight binding model we give a new proof of exponential localization in the anderson tight binding model which uses many ideas of the frohlich martinelli scoppola and spencer proof but is technically simpler particularly the probabilistic estimates

generalized eigenfunctions for waves in inhomogeneous media abstract many wave propagation phenomena in classical physics are governed by equations that can be recast in schrodinger form in this approach the classical wave equation e g maxwell s equations acoustic equation elastic equation is rewritten in schrodinger form leading to the study of the spectral theory of its classical wave operator a self adjoint partial differential operator on a hilbert space of vector valued square integrable functions physically interesting inhomogeneous media give rise to nonsmooth coefficients we construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients our construction yields polynomially bounded generalized eigenfunctions the set of generalized eigenvalues forming a subset of the operator s spectrum with full spectral measure

positive solutions of second order systems of boundary value problems abstract the existence of nonnegative solutions of boundary value problems of the form x t a t x t y t 0 y t b t g x t y t 0 x 0 x 1 y 0 y 1 0 is investigated where and g are functions from r p r q into r p and r q respectively and a b are functions defined on 0 1 with values in the spaces of diagonal matrices of order p and q respectively all functions are assumed to be continuous and nonnegativity as well as monotonicity and growth conditions are imposed on and g the existence results hold for a bounded interval of values of if and g are assumed to grow fast and have only nonzero values while if 0 0 0 g 0 0 0 it is shown that under reasonable conditions there exists a nontrivial nonnegative solution for every 0

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

anderson localization for bernoulli and other singular potentials we prove exponential localization in the anderson model under very weak assumptions on the potential distribution in one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment in particular this solves the longstanding problem of localization for bernoulli potentials i e potentials that take only two values in dimensions greater than one we prove localization at high disorder for potentials with holder continuous distributions and for bounded potentials whose distribution is a convex combination of a holder continuous distribution with high disorder and an arbitrary distribution these include potentials with singular distributions we also show that for certain bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

positive solutions for 2p order and 2q order nonlinear ordinary differential systems abstract in this paper by using krasnosel skii fixed point theorem and under suitable conditions we present the existence of single and multiple positive solutions to the following systems 1 p u 2 p a t f u t v t t 0 1 1 q v 2 q b t g u t v t t 0 1 u 2 i 0 u 2 i 1 0 0 i p 1 v 2 j 0 v 2 j 1 0 0 j q 1 where 0 0 p q n we derive two explicit intervals of and such that for any and in the two intervals respectively the existence of at least one solution to the systems is guaranteed and the existence of at least two solutions for and in appropriate intervals is also discussed

anderson localization for multi dimensional systems at large disorder or large energy we prove that discrete schrodinger operators on d with a random potential have almost surely only pure point spectrum and exponentially decaying eigenfunctions for large disorder or large energy this is the first proof of localization for multi dimensional anderson models

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

from fixed energy localization analysis to dynamical localization an elementary path we review several techniques initiated by a remarkable work written by spencer j stat phys 51 1009 1019 1988 a quarter of century ago used and further developed in numerous subsequent researches we also describe a fairly general elementary derivation of spectral and strong dynamical anderson localization from the fixed energy analysis of the green s functions on locally finite graphs of polynomial growth obtained either by the multi scale analysis or by the fractional moment method this derivation goes in the same direction as the simon wolf method commun pure appl math 39 75 90 1986 but provides more quantitative estimates can be adapted to multi particle models and combined with a simplified variant of the germinet klein argument commun math phys 222 415 448 2001 results in an elementary proof of strong dynamical localization on arbitrary graphs of polynomial growth

on absence of diffusion near the bottom of the spectrum for a random schrodinger operator on l 2 bf r nu we consider a random schrodinger operator onl2 v of the form h omega delta v omega v omega x sigma chi c i x q i omega ci being a covering of v with unit cubes around the sites of v and qi i i d random variables with values in 0 1 we assume that theqi s are continuously distributed with bounded densityf q and that 0 p q0 1 2 1 then we show that an ergodic mean of the quantity dx x 2 exp ith x 2 t 1 vanishes provided ge h where is well localized around the origin andge is a positivec function with support in 0 e e e f our estimate ofe f is such that the set x v v x e f may contain with probability one an infinite cluster of cubes ci which are nearest neighbours the proof is based on the technique introduced by frohlich and spencer for the analysis of the anderson model

remark on the absence of absolutely continuous spectrum for d dimensional schrodinger operators with random potential for large disorder or low energy we show that there is no absolutely continuous part in the spectrum of the anderson tight binding model for large disorder or low energy the proof is based on the exponential decay of the green s function proved by frohlich and spencer the extension of this result to the continuous case is also discussed

multiple solutions of singular boundary value problems for differential systems by constructing a special cone and using fixed point index theory in cone this paper investigates the existence of multiple solutions of singular boundary value problems for differential systems