large deviations of the lyapunov exponent and localization for the 1d anderson model the proof of anderson localization for the 1d anderson model with arbitrary e g bernoulli disorder originally given by carmona klein martinelli in 1987 is based in part on the multi scale analysis later in the 90s it was realized that for one dimensional models with positive lyapunov exponents some parts of multi scale analysis can be replaced by considerations involving subharmonicity and large deviation estimates for the corresponding cocycle leading to nonperturbative proofs for 1d quasiperiodic models in this paper we present a short proof along these lines for the anderson model to prove dynamical localization we also develop a uniform version of craig simon s bound that works in high generality and may be of independent interest

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

the spectrum of a magnetic schrodinger operator with randomly located delta impurities we consider a single band approximation to the random schrodinger operator in an external magnetic field the spectrum of such an operator has been characterized in the case where delta impurities are located on the sites of a lattice in this paper we generalize these results by letting the delta impurities have random positions as well as strengths they are located in squares of a lattice with a general bounded distribution we characterize the entire spectrum of this operator when the magnetic field is sufficiently high we show that the whole spectrum is pure point the energy coinciding with the first landau level is infinitely degenerate and that the eigenfunctions corresponding to other landau band energies are exponentially localized

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

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

painleve classification of coupled korteweg de vries systems in this work we give a classification of coupled korteweg de vries equations we found new systems of equations that are completely integrable in the sense of painleve

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 for the anderson model on a strip with singular potentials abstract we prove exponential localization in the anderson model on a one dimensional strip for any potential whose probability distribution is not concentrated on a single point and possesses some finite moment in particular we obtain localization for bernoulli potentials on a strip

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

two component integrable generalizations of burgers equations with nondiagonal linearity two component second and third order burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries new integrable systems are obtained master symmetries of the obtained symmetry integrable systems and bi poisson structures of those that also possess conservation laws are given

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

sasa satsuma complex modified korteweg de vries ii and the complex sine gordon ii equation revisited recursion operators nonlocal symmetries and more we present a new symplectic structure and a hereditary recursion operator for the sasa satsuma equation which is widely used in nonlinear optics using an integrodifferential substitution relating this equation to a third order symmetry flow of the complex sine gordon ii equation enabled us to find a hereditary recursion operator and higher hamiltonian structures for the latter equation we also show that both the sasa satsuma equation and the third order symmetry flow for the complex sine gordon ii equation are bi hamiltonian systems and we construct several hierarchies of local and nonlocal symmetries for these systems

on the integrability of systems of second order evolution equations with two components this paper is devoted to classifying second order evolution equations with two components combining the symbolic method and number theory we give the complete list of such homogeneous polynomial symmetry integrable systems with non zero diagonal linear terms the technique is applicable for more general systems

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

localization bounds for multiparticle systems we consider the spectral and dynamical properties of quantum systems of n particles on the lattice mathbb z d of arbitrary dimension with a hamiltonian which in addition to the kinetic term includes a random potential with iid values at the lattice sites and a finite range interaction two basic parameters of the model are the strength of the disorder and the strength of the interparticle interaction it is established here that for all n there are regimes of high disorder and or weak enough interactions for which the system exhibits spectral and dynamical localization the localization is expressed through bounds on the transition amplitudes which are uniform in time and decay exponentially in the hausdorff distance in the configuration space the results are derived through the analysis of fractional moments of the n particle green function and related bounds on the eigenfunction correlators

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

existence of the density of states for multi dimensional continuum schrodinger operators with gaussian random potentials a wegner estimate is proved for quantum systems in multi dimensional euclidean space which are characterized by one particle schrodinger operators with random potentials that admit a certain one parameter decomposition in particular the wegner estimate applies to systems with rather general gaussian random potentials as a consequence these systems possess an absolutely continuous integrated density of states whose derivative the density of states is locally bounded an explicit upper bound is derived

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

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

localization and universality of poisson statistics for the multidimensional anderson model at weak disorder we prove anderson localization with the mean field lyapunov exponent and poisson statistics for eigenvalue spacing for the multi dimensional anderson model at weak disorder these results are obtained by developing the supersymmetric formalism initiated in w1 see also sjw rid

small universal turing machines abstract let utm m n be the class of universal turing machine with m states and n symbols universal turing machines are proved to exist in the following classes utm 24 2 utm 10 3 utm 7 4 utm 5 5 utm 4 6 utm 3 10 and utm 2 18

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

wegner type bounds for a multi particle continuous anderson model with an alloy type external potential we consider an n particle quantum systems in d with interaction and in presence of a random external alloy type potential a continuous n particle anderson model we establish wegner type bounds inequalities for such models giving upper bounds for the probability that random spectra of hamiltonians in finite volumes intersect a given set

the determination of the value of rado s noncomputable function σ for four state turing machines the well defined but noncomputable functions e k and s k given by t rado as the score and shift number for the k state turing machine busy beaver game were previously known only for k 13 and s 4 a 107 reported by this author supported the conjecture that these lower bounds are the actual particular values of the functions for k 4 the four state case has previously been reduced to solving the blank input tape halting problem of only 5 820 individual machines in this final stage of the k 4 case one appears to move into a heuristic level of higher order where it is necessary to treat each machine as representing a distinct theorem the remaining set consists of two primary classes in which a machine and its tape are viewed as the representation of a growing string of cellular automata the proof techniques embodied in programs are entirely heuristic while the inductive proofs once established by the computer are completely rigorous and become the key to the proof of the new and original mathematical results l 4 13 and s 4 107 in any case even though skilled mathematicians and experienced programmers attempted to evaluate e 3 and s 3 there is no evidence that any known approach will yield the answer even if we avail ourselves of high speed computers and elaborate programs as regards l 4 s 4 the situation seems to be entirely hopeless at present tibor rado 1963 1 background and introduction the busy beaver game was devised by tibor rado 8 for the purpose of illustrating the notion of noncomputability given the set of turing machines of exactly k states which operate with the minimum alphabet of two symbols a space and a mark or 0 and 1 one considers the problem of behavior of these machines on a tape which is initially all blank all o s this is a finite set of machines there being exactly 4k 1 2k distinct machines where with two table entries per state each table entry may consist of printing 0 or 1 moving right or left branching to state 1 2 k or simply an entry to declare a halt since the input tape is blank each machine faces one of two possible fates either it eventually halts or else it continues running forever after a particular machine received july 22 1981 revised june 28 1982 and september 7 1982 1980 mathematics subject classification primary 03d10 03b35 secondary 68c30 68d20

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

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

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

landau hamiltonians with random potentials localization and the density of states we prove the existence of localized states at the edges of the bands for the two dimensional landau hamiltonian with a random potential of arbitrary disorder provided that the magnetic field is sufficiently large the corresponding eigenfunctions decay exponentially with the magnetic field and distance we also prove that the integrated density of states is lipschitz continuous away from the landau energies the proof relies on a wegner estimate for the finite area magnetic hamiltonians with random potentials and exponential decay estimates for the finitearea green s functions the proof of the decay estimates for the green s functions uses fundamental results from two dimensional bond percolation theory

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

spectral gap and decay of correlations in u 1 symmetric lattice systems in dimensions d 2 we consider many body systems with a global u 1 symmetry on a class of lattices with the fractal dimension d 2 and their zero temperature correlations whose observables behave as a vector under the u 1 rotation for a wide class of the models we prove that if there exists a spectral gap above the ground state then the correlation functions have a stretched exponentially decaying upper bound this is an extension of the mcbryan spencer method at finite temperatures to zero temperature the class includes quantum spin and electron models on the lattices and our method also allows finite or infinite quasi degeneracy of the ground state the resulting bounds rule out the possibility of the corresponding magnetic and electric long range orders

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

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

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

localization for one dimensional continuum bernoulli anderson models we use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional continuum anderson type models with singular distributions in particular the case of a bernoulli dis tribution is covered the operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi scale analysis we show that non reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs

2 dimensional localization of acoustic waves in random perturbation of periodic media abstract in this work we study internal band edges localization of acoustic waves in 2 dimensional space obtained by random perturbation of some periodic media our results rely on the study of lifshitz tails for the integrated density of states for random acoustic operators of the form a 1 localization is then deduced by the standard multiscale argument

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

dynamical localization ii with an application to the almost mathieu operator several recent works have established dynamical localization for schrodinger operators starting from control on the localization length of their eigenfunctions in terms of their centers of localization we provide an alternative way to obtain dynamical localization without resorting to such a strong condition on the exponential decay of the eigenfunctions furthermore we illustrate our purpose with the almost mathieu operator h cos 2 x 15 and with good diophantine properties more precisely for almost all for all q 0 and for all functions l2 mathbb z of compact support we show that mathop sup limits t left langle e ith theta lambda omega psi left x right q e ith theta lambda omega psi right rangle c psi the proof applies equally well to discrete and continuous random hamiltonians in all cases it uses as input a repulsion principle of singular boxes supplied in the random case by the multi scale analysis

on the swendsen wang dynamics i exponential convergence to equilibrium we present rigorous results on the exponential convergence to equilibrium for the swendsen wang stochastic dynamics for thed dimensional ising ferromagnet with external magnetic fieldh in the thermodynamic limit we consider various situations mainly in the low temperature regime in which boundary conditions are homogeneous and parallel or opposite to the external field in the latter case we relate directly the tunneling from the metastable phase to the stable one with the exponential convergence to equilibrium

localization in the ground state of a disordered array of quantum rotators we consider the zero temperature behavior of a disordered array of quantum rotators given by the finite volume hamiltonian h lambda mathop sigma limits x in lambda frac h x 2 frac partial 2 partial varphi x 2 j mathop sigma limits left langle x y right rangle in lambda cos varphi x varphi y wherex y z d denotes nearest neighbors inz d j 0 andh h x 0 x z d are independent identically distributed random variables with common distributiond h satisfying h d h 0 we prove that for anym 0 it is possible to choosej m sufficiently small such that if 0 j j m for almost every choice ofh and everyx z d the ground state correlation function satisfies left langle cos varphi x varphi y right rangle leqq c x h j e m left x y right for ally z d withc x h j

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

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

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 for discrete one dimensional random word models we consider schrodinger operators in l2 z whose potentials are obtained by randomly concatenating words from an underlying set w according to some probability measure on w our assumptions allow us to consider models with local correlations such as the random dimer model or more generally random polymer models we prove spectral localization and away from a finite set of exceptional energies dynamical localization for such models these results are obtained by employing scattering theoretic methods together with furstenberg s theorem to verify the necessary input to perform a multiscale analysis

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

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

the repulsion between localization centers in the anderson model in this note we show that a simple combination of deep results in the theory of random schrodinger operators yields a quantitative estimate of the fact that the localization centers become far apart as corresponding energies are close together

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

making almost commuting matrices commute suppose two hermitian matrices a b almost commute vert a b vert leq delta are they close to a commuting pair of hermitian matrices a b with vert a a vert vert b b vert leq epsilon a theorem of h lin 3 shows that this is uniformly true in that for every epsilon 0 there exists a 0 independent of the size n of the matrices for which almost commuting implies being close to a commuting pair however this theorem does not specify how depends on epsilon we give uniform bounds relating and epsilon the proof is constructive giving an explicit algorithm to construct a and b we provide tighter bounds in the case of block tridiagonal and tridiagonal matrices within the context of quantum measurement this implies an algorithm to construct a basis in which we can make a projective measurement that approximately measures two approximately commuting operators simultaneously finally we comment briefly on the case of approximately measuring three or more approximately commuting operators using povms positive operator valued measures instead of projective measurements

many body localization in the droplet spectrum of the random xxz quantum spin chain abstract we study many body localization properties of the disordered xxz spin chain in the ising phase disorder is introduced via a random magnetic field in the z direction we prove a strong form of dynamical exponential clustering for eigenstates in the droplet spectrum for any pair of local observables separated by a distance l the sum of the associated correlators over these states decays exponentially in l in expectation this exponential clustering persists under the time evolution in the droplet spectrum our result applies to the large disorder regime as well as to the strong ising phase at fixed disorder with bounds independent of the support of the observables

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

classification of polynomial integrable systems of mixed scalar and vector evolution equations i we perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries we consider polynomial systems that are homogeneous under a suitable weighting of variables this paper deals with the kdv weighting the burgers or potential kdv or modified kdv weighting the ibragimov shabat weighting and two unfamiliar weightings the case of other weightings will be studied in a subsequent paper making an ansatz for undetermined coefficients and using a computer package for solving bilinear algebraic systems we give the complete lists of second order systems with a third order or a fourth order symmetry and third order systems with a fifth order symmetry for all but a few systems in the lists we show that the system or at least a subsystem of it admits either a lax representation or a linearizing transformation a thorough comparison with recent work of foursov and olver is made

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

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

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

new reductions of integrable matrix partial differential equations sp m invariant systems we propose a new type of reduction for integrable systems of coupled matrix partial differential equations this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix via this reduction we obtain a new integrable system of coupled derivative modified kdv equations and a new integrable variant of the massive thirring model in addition to the already known systems we also discuss integrable semidiscretizations of the obtained systems and present new soliton solutions to both continuous and semidiscrete systems as a by product a new integrable semidiscretization of the manakov model self focusing vector nonlinear schrodinger equation is obtained

localization of classical waves i acoustic waves we consider classical acoustic waves in a medium described by a position dependent mass density x we assume that x is a reandom perturbation of a periodic function 0 x and that the periodic acoustic operator a 0 nabla cdot tfrac 1 varrho 0 x nabla has a gap in the spectrum we prove the existence of localized waves i e finite energy solutions of the acoustic equations with the property that almost all of the wave s energy remains in a fixed bounded region of space at all times with probability one localization of acoustic waves is a consequence of anderson localization for the self adjoint operators a nabla cdot tfrac 1 varrho 0 x nabla onl2 d we prove that in the random medium described by x the random operatora exhibits anderson localization inside the gap in the spectrum ofa0 this is shown even in situations when the gap is totally filled by the spectrum of the random opertor we can prescribe random environments that ensure localization in almost the whole gap

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

extinction of contact and percolation processes in a random environment we consider the inhomogeneous percolation process on z d r defined as follows along each vertical line x r we put cuts at times given by a poisson point process with intensity x and between each pair of adjacent vertical lines x r and y r we place bridges at times given by a poisson point process with intensity x y we say that x t and y s are connected or in the same cluster if there is a path from x t to y s made out of uncut segments of vertical lines and bridges if we consider only oriented percolation we have the graphical representation of the inhomogeneous d dimensional contact process we consider these percolation and contact processes in a random environment by taking x x z d and x y x y z d x y 2 1 to be independent families of independent identically distributed strictly positive random variables we use and for representative random variables we prove extinction i e no percolation of these percolation and contact processes for almost every and if and satisfy e log 1 2d 2 1 1 1 d 1 2d and if e log 1 is sufficiently small

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

the complex behavior of simple machines abstract this paper interprets work on understanding the actions of turing machines operating on an initially blank tape while this is impossible for arbitrary machines complete characterizations of behavior are possible if the number of states is sufficiently constrained the approach combines normalization to drastically reduce the number of machines considered human generated classification schemes and computer generated proofs of behavior this approach can be applied to other computational systems giving complete characterizations in sufficiently small domains this is of interest in the area of emergent systems since the properties of such systems are often difficult to determine by using computers to eliminate multitudes of machines with well understood behavior some unanticipated exotic machines with complex behavior were discovered these exotic machines show that it is quite difficult to estimate the number of states needed to produce a given behavior and hence subjective estimates of complexity may be poor approximations of the true complexity

anderson localization and the space time characteristic of continuum states a proof of anderson localization is obtained by ruling out any continuous spectrum on the basis of the space time characteristic of its states

spectral gap and exponential decay of correlations we study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short range interactions on a wide class of lattices we prove that if two observables anticommute with each other at large distance then the nonvanishing spectral gap implies exponential decay of the corresponding correlation when two observables commute with each other at large distance the connected correlation function decays exponentially under the gap assumption if the observables behave as a vector under the u 1 rotation of a global symmetry of the system we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself rather than just the connected correlation function decays exponentially under the gap assumption on a lattice with a certain self similarity in fractal dimensions d 2 in particular if the system is translationally invariant in one of the spatial directions then this self similarity condition is automatically satisfied we also treat systems with long range power law decaying interactions

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

localization for some continuous random schrodinger operators we study the spectrum of random schrodinger operators acting onl2 r d of the following type h delta w sum x in mathbb z d t x v x the t x x in mathbb z d are i i d random variables under weak assumptions onv we prove exponential localization forh at the lower edge of its spectrum in order to do this we give a new proof of the wegner estimate that works without sign assumptions onv

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

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

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

bootstrap multiscale analysis and localization in random media we introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events for quantum and classical waves in random media we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side l with probability higher than 1 e l for any 0 1 the starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent at some sufficiently large scale with probability bigger than 1 d is the dimension note that from the same starting hypothesis we get conclusions that are valid for any 0 1 this is achieved by the repeated use of a bootstrap argument as an application we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the hilbert schmidt norm and better estimates on the behavior of the eigenfunctions

small turing machines and generalized busy beaver competition let tm k l be the set of one tape turing machines with k states and l symbols it is known that the halting problem is decidable for machines in tm 2 3 and tm 3 2 we prove that the decidability of machines in tm 2 4 and tm 3 3 will be difficult to settle by giving machines in these sets for which the halting problem depends on an open problem in number theory a machine in tm 5 2 with the same result is already known and moreover this machine is the record holder for the busy beaver competitions this is the machine in tm 5 2 which halts when starting from a blank tape making the greatest number of steps and leaving the greatest number of non blank symbols we give potential winners for similar generalized busy beaver competitions in tm 2 3 tm 2 4 and tm 3 3

diffusion in the mean for an ergodic schrodinger equation perturbed by a fluctuating potential diffusive scaling of position moments and a central limit theorem are obtained for the mean position of a quantum particle hopping on a cubic lattice and subject to a random potential consisting of a large static part and a small part that fluctuates stochastically in time if the static random potential is strong enough to induce complete localization in the absence of time dependent noise then the diffusion constant is shown to go to zero proportional to the square of the strength of the time dependent part

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