non perturbative anderson localization in heavy tailed potentials via large deviations moment analysis we study a class of anderson hamiltonians with heavy tailed independent and identically distributed random potential on graphs with sub exponential growth of the balls and of the number of self avoiding paths connecting pairs of points we show that for a class of marginal distributions anderson localization occurs non perturbatively i e for any nonzero amplitude of the potential like in one dimensional systems the proof is based on the moment analysis of the green functions via large deviations estimates

localization in the ground state of the one dimensional x y model with a random transverse field we consider the ground state of the quantum spin model h j sigma langle i j rangle sigma x i sigma x j sigma y i sigma y j sigma i h i sigma z i in one dimension where hi iez are independent identically distributed random variables by means of a jordan wigner transformation the model is mapped into a free fermi gas in the presence of a random external potential we then use exponential localization of the one particle states to prove exponential decay for the spin spin correlation functions

moment analysis for localization in random schrodinger operators we study localization effects of disorder on the spectral and dynamical properties of schrodinger operators with random potentials the new results include exponentially decaying bounds on the transition amplitude and related projection kernels including in the mean these are derived through the analysis of fractional moments of the resolvent which are finite due to the resonance diffusing effects of the disorder the main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the lifshitz krein spectral shift associated with the local potential terms the difficulty is avoided here through the use of a weak l 1 estimate concerning the boundary value distribution of resolvents of maximally dissipative operators combined with standard tools of relative compactness theory

localized wavefunctions in quantum systems with multiwell potentials one dimensional systems with the multiwell potential v x 2 1cos x 2 2cos x are considered localized wavefunctions in the case when the energy is greater than the maximum value of the potential are presented wigner and weyl functions corresponding to the wavefunctions of these systems are also studied it is shown that they are very sensitive to the value of the parameter

localization phenomenon in gaps of the spectrum of random lattice operators we consider a class of random lattice operators including schrodinger operators of the formh w gv wherew x is a real valued periodic function g is a positive constant andv x x d are independent identically distributed real random variables we prove that if the operator w has gaps in the spectrum andg is sufficiently small then the operatorh develops pure point spectrum with exponentially decaying eigenfunctions in a vicinity of the gaps

localization criteria for anderson models on locally finite graphs we prove spectral and dynamical localization for anderson models on locally finite graphs using the fractional moment method our theorems extend earlier results on localization for the anderson model on d we establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

new characterizations of the region of complete localization for random schrodinger operators we study the region of complete localization in a class of random operators which includes random schrodinger operators with anderson type potentials and classical wave operators in random media as well as the anderson tight binding model we establish new characterizations or criteria for this region of complete localization given either by the decay of eigenfunction correlations or by the decay of fermi projections these are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity

an algorithm to study tunnelling in a wide class of one dimensional multiwell potentials i the author proves a theorem which gives an algorithmic solution to the problem of finding the logarithmic derivative of the ground state wavefunction of one dimensional systems by means of this quantity as is well known one can determine the lowest part of the spectrum of the hamiltonian by probabilistic methods the author shows that in some natural classes of potentials the complexity of the algorithm is less than n3 where n is the number of the absolute minima of the potential the author s approach allows a systematic treatment of cases of much greater complexity than those analysed so far in the literature and it can be useful in the study of physical systems like for example long molecular chains or superlattice structures

the localization properties of a random steady flow on a lattice we consider a random stationary vector field on a multidimensional lattice and investigate flow connected subsets of the lattice invariant under the action of the associated flow the subsets of primary interest are cycles and vortices each of which is the set of orbits terminating in the same cycle we prove that with probability 1 each vortex only involves a finite number of sites of the lattice under the assumption of independence of the vector field in different sites we find that with probability 1 the vortices exhaust all possible maximal flowconnected invariant subsets of the lattice if and only if the probability of existence of a cycle is positive thus if cycles exist a particle under the action of the flow only moves within a bounded region i e it is completely localized

a characterization of the anderson metal insulator transport transition we investigate the anderson metal insulator transition for random schrodinger operators we define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the hilbert schmidt norm we introduce a local transport exponent e and set the weak metallic transport region to be the part of the spectrum with nontrivial transport i e e 0 we prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent e provides a characterization of the metal insulator transport transition moreover we show that if there is such a transition then e has to be discontinuous at a transport mobility edge more precisely we show that if the transport is nontrivial then e 1 2d where d is the space dimension these results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors bootstrap multiscale analysis we also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis proving that the multiscale analysis is valid all the way up to a transport mobility edge

cells with many facets in a poisson hyperplane tessellation abstract let z be the typical cell of a stationary poisson hyperplane tessellation in r d the distribution of the number of facets f z of the typical cell is investigated it is shown that under a well spread condition on the directional distribution the quantity n 2 d 1 p f z n n is bounded from above and from below when f z is large the isoperimetric ratio of z is bounded away from zero with high probability these results rely on one hand on the complementary theorem which provides a precise decomposition of the distribution of z and on the other hand on several geometric estimates related to the approximation of polytopes by polytopes with fewer facets from the asymptotics of the distribution of f z tail estimates for the so called content of z are derived as well as results on the conditional distribution of z when its content is large

finite trotter approximation to the averaged mean square distance in the anderson model we prove that a finite trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice d exhibits at most a diffusive behavior in dimensions d 3 as long as the fourier transform of the single site probability hat mu is in l2

anderson localization for generic deterministic operators abstract we consider a class of ensembles of lattice schrodinger operators with deterministic potentials including quasi periodic potentials with diophantine frequencies depending upon an infinite number of parameters in an auxiliary measurable space using a variant of the multi scale analysis we prove anderson localization for generic ensembles in the strong disorder regime and establish an analog of minami type bounds for spectral spacings

localization for a class of one dimensional quasi periodic schrodinger operators we prove for small and satisfying a certain diophantine condition the operator h varepsilon 2 delta frac 1 2 pi cos 2 pi j alpha theta j in mathbb z has pure point spectrum for almost all a similar result is established at low energy for h frac d 2 dx 2 k 2 cos 2 pi x cos 2 pi alpha x theta providedk is sufficiently large

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

low lying eigenvalues of randomly curved quantum waveguides abstract we consider the negative dirichlet laplacian on an infinite waveguide embedded in r 2 and finite segments thereof the waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature we derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling i e weak disorder regime this allows us to estimate the probability of low lying eigenvalues a tool which is relevant in the context of anderson localization for random schrodinger operators

anderson localization for the almost mathieu equation a nonperturbative proof we prove that for any diophantine rotation angle and a e phase the almost mathieu operator h n n 1 n 1 cos 2 n n has pure point spectrum with exponentially decaying eigenfunctions for 15 we also prove the existence of some pure point spectrum for any 5 4

sur une conjecture de d g kendall concernant la cellule de crofton du plan et sur sa contrepartie brownienne in connection with a conjecture stated by d g kendall in the forties we describe the asymptotic behavior of the distribution function of the area of the planar crofton cell we deduce from this in support of his conjecture that expressed in terms of eigenvalues the large crofton cells are nearly circular we obtain also the asymptotic behavior of the laplace transform of the law of the perimeter of the convex hull of planar brownian motion run until time 1 this last result implies that the small convex hulls of brownian motion are nearly circular

localization in general one dimensional random systems i jacobi matrices we consider random discrete schrodinger operators in a strip with a potentialv n n a label in and a finite label across the strip andv an ergodic process we prove thath0 v has only point spectrum with probability one under two assumptions 1 theconditional distribution of v n n 0 1 all conditioned on v n 0 1 all has an absolutely continuous component with positive probability 2 for a e e no lyaponov exponent is zero

localization for random and quasiperiodic potentials a survey is made of some recent mathematical results and techniques for schrodinger operators with random and quasiperiodic potentials a new proof of localization for random potentials established in collaboration with h von dreifus is sketched

inverse scattering problem for a two dimensional random potential we study an inverse problem for the two dimensional random schrodinger equation q k 2 u 0 the potential q x is assumed to be a gaussian random function whose covariance operator is a classical pseudodifferential operator we show that the backscattered field obtained from a single realization of the random potential q determines uniquely the principal symbol of the covariance operator of q the analysis is carried out by combining harmonic and microlocal analysis with stochastic methods

dynamical localization for discrete and continuous random schrodinger operators we show for a large class of random schrodinger operators h on and on that dynamical localization holds i e that with probability one for a suitable energy interval i and for q a positive real here is a function of sufficiently rapid decrease and p i h is the spectral projector of h corresponding to the interval i the result is obtained through the control of the decay of the eigenfunctions of h and covers in the discrete case the anderson tight binding model with bernoulli potential dimension 1 or singular potential 1 and in the continuous case anderson as well as random landau hamiltonians

structures of polyhedra determined by submodular functions on crossing families the present paper shows that for any submodular functionf on a crossing family with if the polyhedron is nonempty then there exist a unique distributive lattice with and a unique submodular function with such thatb f coincides with the base polyhedron associated with the submodular system here iff is integer valued thenf 1 is also integer valued based on this fact we also show the relationship between the independent flow problem considered by the author and the minimum cost flow problem considered by j edmonds and r giles

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 unbounded random jacobi matrices there have been several recent proofs of one dimensional anderson localization based on positive lyapunov exponent that hold for bounded potentials we provide a lyapunov exponent based proof for unbounded potentials simultaneously treating the singular and unbounded jacobi case by extending the techniques in a recent work by jitomirskaya and zhu there have been several recent proofs of one dimensional anderson localization based on positive lyapunov exponent that hold for bounded potentials we provide a lyapunov exponent based proof for unbounded potentials simultaneously treating the singular and unbounded jacobi case by extending the techniques in a recent work by jitomirskaya and zhu

zariski closure of subgroups of the symplectic group and lyapunov exponents of the schrodinger operator on the strip we consider the schrodinger equation with a random potential y n 1 q n ei y n y n 1 0 whereqn is a sequence of independent identically distributed random symmetricm m matrices with real valued elements yn m n e is the real parameter andi is the identity matrix we show that if the smallest jordan algebra of matrices containing the support of the distribution of matricesqn coincides with jordan algebra of all real valued symmetric matrices then for all but maybe a finite number of values ofe all the lyapunov exponents of our schrodinger equation are different and thus the spectrum of the corresponding schrodinger operator is pure point

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

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

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

the quantum hall effect for electrons in a random potential we assume the existence of sufficiently localised states near the edges of each landau band we then prove that the hall conductivity is quantised and the parallel conductivity vanishes when the filling factor stays close to an integer the hall integer is a topological invariant given by the landau band index

kotani theory for one dimensional stochastic jacobi matrices we consider families of operators h on l2 given by h u n u n 1 u n 1 v n u n wherev is a stationary bounded ergodic sequence we prove analogs of kotani s results including that for a e ac h is the essential closure of the set ofe where e the lyaponov index vanishes and the result that ifv is non deterministic then ac is empty

notes on l m convex functions and the separation theorems the concepts of l convex function and m convex function have recently been introduced by murota as generalizations of submodular function and base polyhedron respectively and discrete separation theorems are established for l convex concave functions and for m convex concave functions as generalizations of frank s discrete separation theorem for submodular supermodular set functions and edmonds matroid intersection theorem this paper shows the equivalence between murota s l convex functions and favati and tardella s submodular integrally convex functions and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis

theory of submodular programs a fenchel type min max theorem and subgradients of submodular functions we consider submodular programs which are problems of minimizing submodular functions on distributive lattices with or without constraints we define a convex or concave conjugate function of a submodular or supermodular function and show a fenchel type min max theorem for submodular and supermodular functions we also define a subgradient of a submodular function and derive a necessary and sufficient condition for a feasible solution of a submodular program to be optimal which is a counterpart of the karush kuhn tucker condition for convex programs

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

exponential scaling limit of the single particle anderson model via adaptive feedback scaling we propose a twofold extension of the germinet klein bootstrap multi scale analysis bmsa for the anderson models on graphs first we show with the help of a single scaling algorithm that power law decay bounds at some initial scale imply an asymptotically exponential decay of eigenfunctions efs and of ef correlators efcs even on graphs of polynomial growth which do not fulfill the uniform scalability condition required for the existing bmsa techniques we also show that the exponential scaling limit of the efs and efcs holds true for a class of marginal distributions of the random potential with regularity lower than holder continuity of any positive order

the schrodinger equation with a quasi periodic potential we consider the schrodinger equation dx2 v e cosx cos cix 0 v eqi where e is small and a satisfies the diophantine inequality lp qcl c q2 forp qez q 0 we look for solutions of the form ikx ikx in x im ax 0 m x e q x e 2m ne inxne e if we try to solve for v v mn we are led to the schrodinger equation on the lattice z2 h k q ba v k q en where a is the discrete laplacian without diagonal terms and v k is some 2 potential on z we have two main results 4 for e sufficiently small h k has pure point spectrum for almost every k 2 for e sufficiently small the operator d dx 2 e cosx cos ax has no point spectrum to prove our results we must get decay estimates on the green s function e h 1 the decay of the eigenfunction follows from this in general we must keep track of small divisors which can make the green s function large this is accomplished by a kam kolmogorov arnold moser type of multiscale perturbation analysis

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

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

perturbation theory for periodic orbits in a class of infinite dimensional hamiltonian systems we consider a class of hamiltonian systems describing an infinite array of coupled anharmonic oscillators and we study the bifurcation of periodic orbits off the equilibrium point the family of orbits we construct can be parametrized by their periods which belong to cantor sets of large measure containing certain periods of the linearized problem as accumulation points the infinitely many holes forming a dense set on which the existence of a periodic orbit cannot be proven originate from a dense set of resonances that are present in the system we also have a result concerning the existence of solutions of arbitrarily large amplitude

parabolic problems for the anderson model i intermittency and related topics the objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics in particular in hydrodynamics on a qualitative level intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments the paper deals with the cauchy problem t u t x hu t x u 0 x t 0 x 0 t x d for the anderson hamiltonianh x x d where is a generally unbounded spatially homogeneous random potential this first part is devoted to some basic problems using percolation arguments a complete answer to the question of existence and uniqueness for the cauchy problem in the class of all nonnegative solutions is given in the case of i i d random variables necessary and sufficient conditions for intermittency of the fieldsu t ast are found in spectral terms ofh rough asymptotic formulas for the statistical moments and the almost sure behavior ofu t x ast are also derived

localization for random schrodinger operators with correlated potentials we prove localization at high disorder or low energy for lattice schrodinger operators with random potentials whose values at different lattice sites are correlated over large distances the class of admissible random potentials for our multiscale analysis includes potentials with a stationary gaussian distribution whose covariance functionc x y decays as x y where 0 can be arbitrarily small and potentials whose probability distribution is a completely analytical gibbs measure the result for gaussian potentials depends on a multivariable form of nelson s best possible hypercontractive estimate

metal insulator transition for the almost mathieu operator we prove that for diophantine and almost every the almost mathieu operator h n n 1 ni 1 cos 2 n n exhibits localization for 2 and purely absolutely continuous spectrum for 2

large faces in poisson hyperplane mosaics a generalized version of a well known problem of d g kendall states that the zero cell of a stationary poisson hyperplane tessellation in d under the condition that it has large volume approximates with high probability a certain definite shape which is determined by the directional distribution of the underlying hyperplane process this result is extended here to typical k faces of the tessellation for k 2 d 1 this requires the additional condition that the direction of the face be in a sufficiently small neighbourhood of a given direction

some rigorous results for random and quasi periodic potentials abstract we review some mathematical results and conjectures for random and quasi periodic schrodinger operators the role of localization in certain nonlinear systems is also discussed in addition we present some general arguments which show that the correlation length exponent v 2 d for a wide class of d dimensional random systems including the dilute ising ferro magnet

anderson localization for time quasi periodic random schrodinger and wave equations we prove that at large disorder with large probability and for a corresponding set of diophantine frequencies of large measure anderson localization in d is stable under localized time quasi periodic perturbations by proving that the associated quasi energy operator has pure point spectrum the main tools are the frohlich spencer mechanism for the random component and the bourgain goldstein schlag mechanism for the quasi periodic component this paper paves the way for the construction of time quasi periodic kam type of solutions of non linear random schrodinger equations in bw

the limit shape of the zero cell in a stationary poisson hyperplane tessellation in the early 1940s d g kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic poisson line process in the plane tends to circularity given that the area of the zero cell tends to a proof was given by i n kovalenko in 1997 this paper generalizes kovalenko s result in two directions to higher dimensions and to not necessarily isotropic stationary poisson hyperplane processes in the anisotropic case the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of minkowski s existence theorem for convex bodies with given area measures

anomalous thermostat and intraband discrete breathers abstract we investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice including quenched disorder the coupling between both parts is bilinear elimination of the harmonic degrees of freedom leads to a nonlinear langevin equation with memory kernels t and noise term t for the anharmonic coordinates q t q t for zero temperature i e for t 0 we prove that the support of the fourier transform of t and of the time averaged velocity velocity correlation functions k t of the anharmonic system cannot overlap as a consequence the asymptotic solutions can be constant periodic quasiperiodic or almost periodic and possibly weakly chaotic for a sinusoidal trajectory q t with frequency we find that the energy e t transferred to the harmonic system up to time t is proportional to t if equals one of the phonon frequencies it is 2 we prove that there is a zero measure set l such that for in its full measure complement r l it is 0 i e there is no energy dissipation under certain conditions l contains a subset l such that for l the dissipation rate is nonzero and may be subdissipative 0 1 or superdissipative 1 2 compared to ordinary dissipation 1 consequently the harmonic bath does act as an anomalous thermostat in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation forcing convergence to equilibrium intraband discrete breathers are such solutions which do not relax we prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a cantor set c k of finite lebesgue measure this is achieved by estimating the contribution of small denominators appearing for g t related to t for c k the small denominators do not lead to divergencies such that g t is a smooth and bounded function in t

holder continuity of the integrated density of states for quasi periodic schrodinger equations and averages of shifts of subharmonic functions in this paper we consider various regularity results for discrete quasiperiodic schr6dinger equations n l pn 1 v 9 nw on eon with analytic potential v we prove that on intervals of positivity for the lyapunov exponent the integrated density of states is holder continuous in the energy provided w has a typical continued fraction expansion the proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche principle the latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks in the multi frequency case the integrated density of states is shown to have a modulus of continuity of the form exp log tl for some 0 a 1 but currently we do not obtain holder continuity in the case of more than one frequency we also present a mechanism for proving the positivity of the lyapunov exponent for large disorders for a general class of equations the only requirement for this approach is some weak form of a large deviation theorem for the lyapunov exponents in particular we obtain an independent proof of the herman sorets spencer theorem in the multi frequency case the approach in this paper is related to the recent nonperturbative proof of anderson localization in the quasi periodic case by j bourgain and m goldstein

on the subdifferential of a submodular function the author recently introduced a concept of a subdifferential of a submodular function defined on a distributive lattice each subdifferential is an unbounded polyhedron in the present paper we determine the set of all the extreme points and rays of each subdifferential and show the relationship between subdifferentials of a submodular function and subdifferentials in an ordinary sense of convex analysis of lovasz s extension of the submodular function furthermore for a modular function on a distributive lattice we give an algorithm for determining which subdifferential contains a given vector and finding a nonnegative linear combination of extreme vectors of the subdifferential which expresses the given vector minus the unique extreme point of the subdifferential

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

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

the density of states in the anderson model at weak disorder a renormalization group analysis of the hierarchical model we study the density of states in a hierarchical approximation of the anderson tight binding model at weak disorder using a renormalization group approach since the laplacian term in our model is hierarchical the renormalization group transformations act essentially on the local potential distribution and the energy technically we use the supersymmetric replica trick and study the averaged green s function starting with a gaussian distribution with small variance we find that the density of states is analytic as soon as the variance of the potential is turned on except possibly near the band edge where we can show this only for 2 which corresponds tod 4 moreover it is perturbatively close to the free one except near the eigenvalues of the hierarchical laplacian where it is given up to perturbative corrections by the rescaled potential distribution

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

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

dynamical phase transitions in sampling complexity we make the case for studying the complexity of approximately simulating sampling quantum systems for reasons beyond that of quantum computational supremacy such as diagnosing phase transitions we consider the sampling complexity as a function of time t due to evolution generated by spatially local quadratic bosonic hamiltonians we obtain an upper bound on the scaling of t with the number of bosons n for which approximate sampling is classically efficient we also obtain a lower bound on the scaling of t with n for which any instance of the boson sampling problem reduces to this problem and hence implies that the problem is hard assuming the conjectures of aaronson and arkhipov proceedings of the forty third annual acm symposium on theory of computing acm press new york new york usa 2011 p 333 this establishes a dynamical phase transition in sampling complexity further we show that systems in the anderson localized phase are always easy to sample from at arbitrarily long times we view these results in light of classifying phases of physical systems based on parameters in the hamiltonian in doing so we combine ideas from mathematical physics and computational complexity to gain insight into the behavior of condensed matter atomic molecular and optical systems

upper bound on the energy gap in terms of the localization in tight binding systems we present an upper bound on the energy gap above the ground energy in terms of the variance of the particle position in one particle tight binding systems our inequality gives a complementary relationship between the energy gap and the localization of the particle that is the energy gap narrows as the ground state spreads spatially while the energy gap can broaden as the ground state becomes localized we first consider the ground state in one dimensional tight binding systems and relate the energy gap to the variance of the particle position in the ground state we then extend the result to higher dimensional systems we compare our upper bound with the exact energy gap as calculated numerically in one and two dimensions this comparison confirms that our upper bound places a tight restriction on the energy gap

on mott s formula for the ac conductivity in the anderson model we study the ac conductivity in linear response theory in the general framework of ergodic magnetic schrodinger operators for the anderson model if the fermi energy lies in the localization regime we prove that the ac conductivity is bounded from above by c 2 log 1 d 2 at small frequencies this is to be compared to mott s formula which predicts the leading term to be c 2 log 1 d 1

a localization principle for multiplicative perturbations abstract a certain resolvent estimate implies point spectrum for discrete linear wave equations with random local propagation speed the significance of point spectrum in this context is that scattering within a random medium produces only localized standing waves the implication is derived from a general result about multiplicative perturbations of a self adjoint operator the estimate implies point spectrum for almost every value of the perturbation parameter

fenchel type duality for matroid valuations the weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem given a pair of valuated matroidsmi v i i i 1 2 defined on a common ground setv find a common baseb 1 2 that maximizes 1 b 2 b this paper develops a fenchel type duality theory related to this problem with a view to establishing a convexity framework for nonlinear integer programming a fenchel type min max theorem and a discrete separation theorem are given furthermore the subdifferentials of matroid valuations are investigated 1998 the mathematical programming society inc published by elsevier science b v

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

simplicity of eigenvalues in the anderson model we give a transparent and intuitive proof that all eigenvalues of the anderson model in the region of localization are simple

disordered quantum walks in one lattice dimension we study a spin 12 particle moving on a one dimensional lattice subject to disorder induced by a random space dependent quantum coin the discrete time evolution is given by a family of random unitary quantum walk operators where the shift operation is assumed to be deterministic each coin is an independent identically distributed random variable with values in the group of two dimensional unitary matrices we derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization put differently the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance our findings imply that this effect takes place if the coin is chosen at random from the haar measure or some measure continuous with respect to it but also for a class of discrete probability measures which support consists of two coins one of them being the hadamard coin

directed submodularity ditroids and directed submodular flows set relations and operations such as inclusion union and intersection are generalized to directed subsets whose elements are distinguished between forward and backward elements the concepts of submodular functions matroids and polymatroidal network flows are extended to the concepts of directed submodular functions ditroids and directed submodular flows on directed subsets two unrelated matroids submodular functions can be embedded in one ditroid directed submodular function total dual integrality is preserved in these generalizations and proved for very general set function class directed odd submodular functions

odd submodular functions dilworth functions and discrete convex functions three set function classes more general than submodular ones are discussed an odd submodular function defines a box totally dual integral system an integral odd submodular function and its dilworth truncation i e a dilworth function give rise to the same polyhedron with integral vertices the minimum of two submodular functions is an odd submodular function the convolution of two submodular functions is a dilworth function a dilworth function is a discrete convex function a discrete convex function can be characterized by its convex hull subgradients epigraph and general subadditivity discrete convexity is preserved under many natural operations

almost periodic one dimensional systems the schrodinger equation with potential v x 2 1cos x 2 2cos x is considered its solution reduces to the problem of finding the eigenvectors of a matrix the eigenvalues of this matrix show a band structure which is very sensitive to the value of the parameter the solutions of the schrodinger equation are presented and their physical meaning is discussed the potential v x has a complex multiwell structure and quantum tunneling occurs the accuracy of all approximations is carefully studied

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

eigenfunctions in a two particle anderson tight binding model we establish the phenomenon of anderson localisation for a quantum two particle system on a lattice mathbb z d with short range interaction and in presence of an iid external potential with sufficiently regular marginal distribution

analyticity of the density of states and replica method for random schrodinger operators on a lattice we analyze the density of states and some aspects of the replica method for anderson s tight binding model on a lattice of arbitrary dimension with diagonal disorder we give heuristic arguments for the conjectures that the classical value of the exponent of the localization length is 1 2 and that the upper critical dimension d c loc is bounded by 4 d c loc 6

extremal theory for spectrum of random discrete schrodinger operator ii distributions with heavy tails we study the asymptotic structure of the first k largest eigenvalues k v and the corresponding eigenfunctions k v of a finite volume anderson model discrete schrodinger operator mathcal h v kappa delta v xi cdot on the multidimensional lattice torus v increasing to the whole of lattice provided the distribution function f of i i d potential satisfies condition log 1 f t o t3 and some additional regularity conditions as t for z v denote by 0 z the principal eigenvalue of the single peak hamiltonian v z z in l2 v and let lambda 0 k v be the kth largest value of the sample 0 in v we first show that the eigenvalues k v are asymptotically close to lambda 0 k v we then prove extremal type limit theorems i e poisson statistics for the normalized eigenvalues k v bv av where the normalizing constants av 0 and bv are chosen the same as in the corresponding limit theorems for lambda 0 k v the eigenfunction k v is shown to be asymptotically completely localized as v at the sites zk v v defined by lambda 0 z k v lambda 0 k v proofs are based on the finite rank in particular rank one perturbation arguments for discrete schrodinger operator when potential peaks are sparse

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

almost periodic schrodinger operators iii the absolutely continuous spectrum in one dimension we studyh d2 dx2 v x withv x limit periodic e g v x ancos x 2n with an we prove that for a genericv and for generican in the explicit example h is a cantor nowhere dense perfect set for a dense set the spectrum is both cantor and purely absolutely continuous and therefore purely recurrent absolutely continuous

instability of dense point spectrum under finite rank perturbations we prove that if a is a self adjoint operator and r a non negative finite rank operator whose range is cyclic for a then for a generic in the baire sense t in each eigenvalue of the operator has a neighborhood containing no other eigenvalues

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies