lifschitz tails for random schrodinger operator in bernoulli distributed potentials this paper presents an elementary proof of lifschitz tail behavior for random discrete schrodinger operators with a bernoulli distributed potential the proof approximates the low eigenvalues by eigenvalues of sine waves supported where the potential takes its lower value this is motivated by the idea that the eigenvectors associated to the low eigenvalues react to the jump in the values of the potential as if the jump were infinite

enlargement of obstacles for the simple random walk we consider a continuous time simple random walk moving among obstacles which are sites resp bonds of the lattice z d we derive in this context a version of the technique of enlargement of obstacles developed by sznitman in the brownian case this method gives controls on exponential moments of certain death times as well as good lower bounds for certain principal eigenvalues we give an application to recover an asymptotic result of donsker and varadhan on the number of sites visited by the random walk and another application to the number of bonds visited by the random walk

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

quantum control with piecewise constant control functions one of the long standing problems in the field of computational quantum control is to devise algorithms that converge to high yield controls with a desired structure particularly controls that can be reproduced by laboratory laser pulse shapers in this paper we propose an approach that is a considerable step forward toward the solution of this problem our treatment employs piecewise constant controls for these the concept of desired structure is addressed by including appropriate quadratic forms in the performance functionals thereafter correct extremal control equations are derived whereas the equations appearing in the literature are correct only to first or second order we illustrate numerically that our approach can give extremely high yield controls with a desired structure our results are valid not only in the context of quantum control but also for similiar bilinear systems arising in different areas in particular they are suitable for controlling linear flows with drift on unitary matrix groups

a robust geostatistical approach to contaminant source identification 1 estimation under model uncertainty remains a practical concern in many scientific and engineering fields a commonly encountered example in groundwater remediation is the contaminant source identification problem like many other inverse problems contaminant source identification is inherently ill posed and is sensitive to both data and model uncertainties model uncertainties which may be introduced at virtually any stage of a model building process can adversely affect estimator performance if they are not accounted for properly in this paper a robust geostatistical approach rgs which is extended from the geostatistical inversion approach gs is used to solve linear estimation problems the uncertainties in both model and covariance matrices are taken into account in the rgs formulation the nominal correlation structural parameters are estimated using a structural analysis procedure the resulting minimax optimization problem is solved using semidefinite programming techniques the rgs is generic and can be applied to any problem for which the gs is suitable and the upper bound of uncertainty can be quantified the rgs is illustrated for source release history identification in a two dimensional aquifer where the model uncertainty is caused by variability in hydraulic conductivity it is shown that when the model is perfectly known the rgs solution coincides with that of the gs when the model has uncertainty the rgs is robust against unknown variations from a nominal model and its overall performance is better than that of the gs using the same nominal model

recovering the history of a groundwater contaminant plume method of quasi reversibility the method of quasi reversibility qr lattes and lions 1969 has been used previously to solve the diffusion equation with reversed time we develop a quasi reversible solution to a convection dispersion equation by solving the qr diffusion operator in a moving coordinate system the solution procedure is applied to the problem of recovering the history of a groundwater contaminant plume from observations of its present conditions this approach to the plume history problem is potentially superior to the tikhonov regularization approach used by skaggs and kabala 1994 because it is easier to implement and readily allows for space and time dependent transport parameters however our results for a few example problems suggest that the qr procedure is less accurate than the regularization technique thus the easy implementation and improved generality of the qr procedure come at the expense of accuracy this trade off will have to be weighed if the qr technique is to be used

local fluctuation of the spectrum of a multidimensional anderson tight binding model we consider the anderson tight binding modelh v acting inl 2 z d and its restrictionh to finite hypercubes z d herev v x x z d is a random potential consisting of independent identically distributed random variables let e j j be the eigenvalues ofh and let j e e j e j 1 be its rescaled eigenvalues then assuming that the exponential decay of the fractional moment of the green function holds for complex energies neare and that the density of statesn e exists ate we shall prove that the random sequence j e j considered as a point process onr 1 converges weakly to the stationary poisson point process with intensity measuren e dx as gets large thus extending the result of molchanov proved for a one dimensional continuum random schrodinger operator on the other hand the exponential decay of the fractional moment of the green function was established recently by aizenman molchanov and graf as a technical lemma for proving anderson localization at large disorder or at extreme energy thus our result in this paper can be summarized as follows near the energye where anderson localization is expected there is no correlation between eigenvalues ofh if is large

an introduction to lie group integrators basics new developments and applications we give a short and elementary introduction to lie group methods a selection of applications of lie group integrators are discussed finally a family of symplectic integrators on cotangent bundles of lie groups is presented and the notion of discrete gradient methods is generalised to lie groups

ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential in this paper we show the that the ground state energy of the one dimensional discrete random schrodinger operator with bernoulli potential is controlled asymptotically as the system size n goes to infinity by the random variable l n the length the longest consecutive sequence of sites on the lattice with potential equal to zero specifically we will show that for almost every realization of the potential the ground state energy behaves asymptotically as frac pi 2 ell n 1 2 in the sense that the ratio of the quantities goes to one

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

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

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

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

comparison of inverse methods for reconstructing the release history of a groundwater contamination source inverse methods can be used to reconstruct the release history of a known source of groundwater contamination from concentration data describing the present day spatial distribution of the contaminant plume using hypothetical release history functions and contaminant plumes we evaluate the relative effectiveness of two proposed inverse methods tikhonov regularization tr and minimum relative entropy mre inversion in reconstructing the release history of a conservative contaminant in a one dimensional domain skaggs and kabala 1994 woodbury and ulrych 1996 we also address issues of reproducibility of the solution and the appropriateness of models for simulating random measurement error the results show that if error free plume concentration data are available both methods perform well in reconstructing a smooth source history function with error free data the mre method is more robust than tr in reconstructing a nonsmooth source history function however the tr method is more robust if the data contain measurement error two error models were evaluated in this study and we found that the particular error model does not affect the reliability of the solutions the results for the tr method have somewhat greater reproducibility because in some cases its input parameters are less subjective than those of the mre method however the mre solution can identify regions where the data give little or no information about the source history function while the tr solution cannot

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

quantum particle in a hierarchical potential with tunnelling over arbitrarily large scales the authors introduce a notion of hierarchical potential by iterating a basic elementary geometric construction over an increasing sequence of length scales they then show that a quantum particle in such a potential exhibits a subdiffusive behaviour characterised by r2 or c ln t beta if the initial state is a normalised wavepacket superposition of states of sufficiently low energy

a constrained robust least squares approach for contaminant release history identification 1 contaminant source identification is an important type of inverse problem in groundwater modeling and is subject to both data and model uncertainty model uncertainty was rarely considered in the previous studies in this work a robust framework for solving contaminant source recovery problems is introduced the contaminant source identification problem is first cast into one of solving uncertain linear equations where the response matrix is constructed using a superposition technique the formulation presented here is general and is applicable to any porous media flow and transport solvers the robust least squares rls estimator which originated in the field of robust identification directly accounts for errors arising from model uncertainty and has been shown to significantly reduce the sensitivity of the optimal solution to perturbations in model and data in this work a new variant of rls the constrained robust least squares crls is formulated for solving uncertain linear equations crls allows for additional constraints such as nonnegativity to be imposed the performance of crls is demonstrated through one and two dimensional test problems when the system is ill conditioned and uncertain it is found that crls gave much better performance than its classical counterpart the nonnegative least squares the source identification framework developed in this work thus constitutes a reliable tool for recovering source release histories in real applications

extremal theory for spectrum of random discrete schrodinger operator iii localization properties in this paper we study the asymptotic localization properties with high probability of the kth eigenfunction associated with the kth largest eigenvalue k 1 fixed of the multidimensional anderson hamiltonian in torus v increasing to the whole of lattice denote by z k v v the site at which the kth largest value of potential is attained it is well known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity then the kth eigenfunction is asymptotically delta function at the site z k v localization centre for some random k v k 1 we study the asymptotic behavior of the index v k by distinguishing between three cases of the tails of potential distribution i for the heavy tails including gaussian v k is asymptotically bounded ii for the light tails but heavier than the double exponential the index v k unboundedly increases like v o 1 iii finally for the double exponential tails with high disorder the index v k behaves like a power of v for weibull s and fractional double exponential types distributions associated with the case ii we obtain the first order expansion formulas for log v k

a supersymmetric transfer matrix and differentiability of the density of states in the one dimensional anderson model let h v on 2 z where v x 9 xez are d r v s with common probability distribution v let h t e itvdv v and let k e be the integrated density of states it is proven i if h is n times differentiable with h j t 0 l t for some 0 7 0 l n then k e is a cn function in particular if v has compact support and h t 0 l h 1 with 0 then k e is c this allows v to be singular continuous ii lfh t 0 e a tl for some 0 then k e is analytic in a strip about the real axis the proof uses the supersymmetric replica trick to rewrite the averaged green s function as a two point function of a one dimensional supersymmetric field theory which is studied by the transfer matrix method

on a backward problem for the kirchhoff s model of parabolic type abstract we study for the first time the backward problem for nonlocal nonlinear boundary value problem of kirchhoff s model of parabolic type first we show that the problem is severely ill posed in the sense of hadamard we propose two methods the fourier truncation method for stabilizing the problem with homogeneous source and the quasi reversibility method for regularizing the problem with nonlinear source under some priori assumptions on the exact solution we establish some stability estimates in the h 0 1 norm

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

a rigorous replica trick approach to anderson localization in one dimension leth v onl2 wherev x x are i i d r v s and letg l x y e i x hl e i 1 y whereh l denotes the operatorh restricted to l l 1 l with dirichlet boundary conditions we use a supersymmetric replica trick to prove that e g l 0 x e i eta 2 leqq k eta 2 exp m log eta sigma x for somem 0 0 k uniformly inl ande this estimate together with the usual necessary estimate on the density of states implies zero conductivity and gives exponential localization by the frohlich martinelli scoppola and spencer method

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

lifschitz tails for the anderson model we present in an expository way an elementary rigorous proof patterned after an argument of kirsch martinelli that the anderson model has lifschitz tails in very great generality

optimal quantum control by an adapted coordinate ascent algorithm this paper explores an adaption of the coordinate ascent approach to quantum control problems it was motivated by the observation that several of the existing monotone converging schemes for quantum control may be viewed as approximations of the well known coordinate ascent method our implementation employs line searches in coordinate directions in control space using only evaluations of the performance functional without invoking its derivatives it is based on recasting the performance functional as a local tracking function which gives the future quality of a control at each moment in time back propagation of a basis of the target space enables these performance functional i e tracking function evaluations to be done efficiently the performance functional may include or not include regularization terms as needed convergence of the resulting algorithm and its relation to previous schemes are discussed and numerical examples are provided in our tests the coordinate ascent algorithm exhib

a method for enforcing parameter nonnegativity in bayesian inverse problems with an application to contaminant source identification received 28 may 2002 revised 18 september 2002 accepted 24 september 2002 published 14 february 2003 1 when an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site one constraint is that the unknown function is known to be everywhere nonnegative in this work we develop a statistically rigorous method for enforcing function nonnegativity in bayesian inverse problems the proposed method behaves similarly to a gaussian process with a linear variogram i e unrestricted brownian motion for parameter values significantly greater than zero the method uses the method of images to define a prior probability density function based on reflected brownian motion that implicitly enforces nonnegativity this work focuses on problems where the unknown is a function of a single variable e g time a markov chain monte carlo mcmc method specifically a highly efficient gibbs sampler is implemented to generate conditional realizations of the unknown function the new method is applied to the estimation of the trichloroethylene tce and perchloroethylene pce contamination history in an aquifer at dover air force base delaware based on concentration profiles obtained from an underlying aquitard index terms 1831 hydrology groundwater quality 1869 hydrology stochastic processes 3260 mathematical geophysics inverse theory keywords stochastic inverse modeling contaminant source identification inference under constraints markov chain monte carlo mcmc gibbs sampling bayesian inference

estimation of historical groundwater contaminant distribution using the adjoint state method applied to geostatistical inverse modeling 1 as the incidence of groundwater contamination continues to grow a number of inverse modeling methods have been developed to address forensic groundwater problems in this work the geostatistical approach to inverse modeling is extended to allow for the recovery of the antecedent distribution of a contaminant at a given point back in time which is critical to the assessment of historical exposure to contamination such problems are typically strongly underdetermined with a large number of points at which the distribution is to be estimated to address this challenge the computational efficiency of the new method is increased through the application of the adjoint state method in addition the adjoint problem is presented in a format that allows for the reuse of existing groundwater flow and transport codes as modules in the inverse modeling algorithm as demonstrated in the presented applications the geostatistical approach combined with the adjoint state method allow for a historical multidimensional contaminant distribution to be recovered even in heterogeneous media where a numerical solution is required for the forward problem