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

on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

on the differential geometry of the euler lagrange equations and the inverse problem of lagrangian dynamics the conditions for a system of second order differential equations to be derivable from a lagrangian the conditions of self adjointness in the terminology of santilli 1978 and others are related in the time independent case to the differential geometry of the tangent bundle of configuration space these conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second order differential equations necessary and sufficient conditions for such a vector field to be derivable from a lagrangian may be stated as the existence of a two form with certain properties it is interesting that it is a deduction not an assumption that this two form is closed and thus defines a symplectic structure some other differential geometric properties of euler lagrange second order differential equations are described

similarity vs unitary similarity and perturbation analysis of sign characteristics complex and real indefinite inner products for selfadjoint and unitary matrices with respect to indefinite inner products on finite dimensional complex or real vector spaces the unitary similarity relation is studied in contrast to the similarity relation the number of unitary similarity classes in a similarity class of such matrices is identified it is proved that under sufficiently small perturbations similarity of selfadjoint or unitary with respect to an indefinite inner product matrices implies unitary similarity estimates are given for the sizes of perturbation that guarantee this property canonical forms and sign characteristics play a key role in the proofs a perturbation analysis is developed of the sign characteristics in the real case under sufficiently small structure preserving perturbations of both selfadjoint or unitary matrices and of the indefinite inner product

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

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

modified equations for variational integrators it is well known that if a symplectic integrator is applied to a hamiltonian system then the modified equation whose solutions interpolate the numerical solutions is again hamiltonian we investigate this property from the variational side we present a technique to construct a lagrangian for the modified equation from the discrete lagrangian of a variational integrator

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

singularities in spectra of disordered systems abstract this paper starts with a review of work on singularities in the spectral density of random harmonic chains detailed descriptions of singular behavior near the maximal frequency near special frequencies and near island frequencies are discussed next the site disordered anderson model is considered here the discussion is limited to the singularity in the density of states near the low energy band edge for systems with arbitrary disorder in dimensions d 1 2 3 in a specific example it is shown that renormalization improves the prediction for the band tail quantitatively

on the asymptotic behaviour of correlators of multi cut matrix models we consider invariant ensembles of n n hermitian random matrices known also as the matrix models in the case where the support of the limiting density of states dos consists of several intervals by using recent results on the asymptotics of orthogonal polynomials we find first that the amplitudes of the leading terms of the correlator of the normalized traces of resolvent of random matrices and of their densities of states are quasi periodic functions of n whose frequencies are the integrals of the limiting dos over the intervals of the support and whose form is uniquely determined by the edges of the support this suggest a certain parametrization of the universality classes of the correlator second we show that the leading terms of these correlators can be expressed correspondingly via the matrix elements of the resolvent and of the spectral kernel of a certain quasi periodic jacobi matrix whose coefficients are determined by the same frequencies

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

the distribution of the reflection phase of disordered conductors the reflection coefficients of one dimensional disordered conductors is asymptotically a pure phase when the sample length is much larger than the localisation length the authors study the stationary distribution of this reflection phase for tight binding chains with diagonal disorder i e random site potentials they derive the dyson integral equation obeyed by this distribution and relate it to the lyapunov exponent and the integrated density of states this general approach allows them to consider then more specifically two types of disorder in the case of a binary alloy where the potentials take two values the distribution of the reflection phase is shown to have generically an infinity of power law singularities related to the occurrence of periodic patterns of potentials this phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy dependent threshold they also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials this non trivial solvable model allows them to examine analytically various properties of the phase distribution such as the anomalies which occur at weak disorder

the inverse problem of the calculus of variations for discrete systems fil barbero linan maria instituto de ciencias matematicas espana universidad politecnica de madrid espana

random operator approach for word enumeration in braid groups we investigate analytically the problem of enumeration of nonequivalent primitive words in the locally free and braid groups for by analysing the random word statistics and target spaces on these groups we develop a symbolic dynamics method for exact word enumeration in locally free groups and give arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations we touch briefly the connection of these problems with conventional random operator theory localization phenomena and statistics of systems with quenched disorder we also discuss the relation of particular problems of random operator theory to the theory of modular functions

basic classes of matrices with respect to quaternionic indefinite inner product spaces basic classes of matrices or linear transformations in finite dimensional quaternionic vector spaces with a regular indefinite inner product are introduced and studied the classes include plus matrices selfadjoint skew adjoint and unitary matrices in particular results are proved concerning extensions of invariant semidefinite subspaces canonical form for unitary matrices is developed and subsequently applied to stability of periodic hamiltonian systems

singular behavior of the density of states and the lyapunov coefficient in binary random harmonic chains we study the integrated density of statesh 2 of a chain of harmonic oscillators with a binary random distribution of the masses we show in particular that there is a dense set of values of the squared frequency for which the differenceh 2 h 2 has a singularity of the type 2 multiplied by a periodic function of ln where the exponent and the period depend continuously on 2 in the region where 1 2 h is not differentiate on a dense set of points the same type of singularities is also present in the lyapunov coefficient

on the large coupling constant behavior of the liapunov exponent in a binary alloy we consider the usual one dimensional tight binding anderson model with the random potential taking only two values 0 and with probabilityp and 1 p 0 p 1 we show that the liapunov exponent e e e r diverges as uniformly in the energye using a result of carmona klein and martinelli this proves that for large enough the integrated density of states is singular continuous we also compute explicitly the exact asymptotics for a dense set of energies and we compare the results with numerical simulations

special energies and special frequencies special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators our investigation includes both this model and the random one dimensional schrodinger operator describing an alloy or its discrete analogue using the phase method we exactly determine a bilateral lifsic asymptotic of the integrated density of statesk e at special energiese s which is not only of the classical type exp c e e s 1 2 but also exp c e e s is a typical behaviour in addition other asymptotics occur e g e e s c which show thatk e need not bek

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

diffusion in a one dimensional lattice with random asymmetric transition rates the authors study one dimensional continuous time random walks for which the pairs wn wn 1 of nearest neighbour transition rates are assumed to be independent equally distributed random variables the long time asymptotic behaviour of the mean displacement x t is determined exactly for a specific model system in which diodes u 0 and two way bonds lambda v v occur with probabilities p and 1 p respectively for lambda 1 p we find that x t approximately tnu f beta 1ln t where nu ln 1 p ln lambda and beta ln lambda and where f is a periodic function with period 1 the mean displacement thus not only increases slower than linearly in time but exhibits superimposed non decaying oscillations

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

a new approach to the problem of disordered harmonic chains abstract in the problem of harmonic chains with random masses the exponential growth rate and the spectral distribution function are shown to be the real and imaginary part of dyson s characteristic function along its cut in the complex plane a new function satisfying a certain integral equation is introduced from which the characteristic function follows immediately the functions introduced by dyson schmidt and matsuda and ishii are recovered as special cases and it is shown that schmidt s distributions w n converge as n relations between dyson s and schmidt s functions are obtained finally the same method is applied to the problem of harmonic chains with random variables k n m n and k n 1 m n and to the tight binding electron model with random site energies

bounds for the smallest singular value of a jordan block with an application to eigenvalue perturbation abstract let j m be an m m jordan block with on its diagonal and 1 m be its smallest singular value as 1 m 1 m denoting we obtain the following inequalities m 1 m 1 2 cos m 1 2 2 m 1 sin 2 m 1 1 2 when m m 1 1 m 1 m 1 when m m 1 m 1 m m m m 1 1 m m when 0 m m 1 as a direct result of the inequalities a lower bound on the perturbation of eigenvalues in the case of a more general matrix is given

several observations on symplectic hamiltonian and skew hamiltonian matrices abstract we prove a hamiltonian skew hamiltonian version of the classical theorem relating strict equivalence and t congruence between pencils of complex symmetric or skew symmetric matrices then we give a pure symplectic variant of the recent result of xu concerning the singular value decomposition of a conjugate symplectic matrix finally we discuss implications that can be derived from veselic s result on definite pairs of hermitian matrices for the skew hamiltonian situation

simple forms and invariant subspaces of h expansive matrices in this paper we consider a simple form for pairs of matrices a h where h is a real symmetric invertible matrix and a is a real h expansive matrix that is atha h is positive semidefinite for such pairs we use the simple form to give an explicit construction of real a invariant maximal h semidefinite subspaces

low frequency expansion and specific heat for harmonic chains with random masses in the problem of harmonic chains with random masses the characteristic function is the analytic continuation into the complex frequency plane of the accumulated density of states and the exponential growth rate a scheme is developed for the calculation of its asymptotic expansion in powers of the frequency it is found that it changes sign under the unusual transformation to omega mk to 1 k 1 mk its first nine taylor coefficients are presented in a table with the first twelve of these coefficients a two point pade approximant for a related function is used for the calculation of the derivative of the specific heat without making use of the spectral density these calculations are carried out for several families of mass distributions

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

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

a geometrical version of the helmholtz conditions in time dependent lagrangian dynamics appropriate geometrical machinery for the study of time dependent lagrangian dynamics is developed it is applied to the inverse problem of the calculus of variations and a set of necessary and sufficient conditions for the existence of a lagrangian are given in terms of the existence of a 2 form with suitable properties which are exactly equivalent to the helmholtz conditions

reflectionless modes in chains with large size homogeneous impurities the authors study the spectra and localization properties of tight binding chains with randomly distributed binary blocks of arbitrary size m one having site energies epsilon a and the other epsilon b respectively they demonstrate that for block size m greater than one and delta mod epsilon a epsilon b less than a critical value delta c m perfect transmission resonance modes exist in the band their number is proportional to m and they occur via dominant 1 e2 divergencies of the localization length no transmission is found for delta delta c m the results are understood by solving exactly the scattering problem from a single homogeneous impurity block of arbitrary size m in the limit of hard impurities m to infinity transmission stops only when delta delta c infinity 4v v being the intersite matrix element that is when the pure a and b bands become detached

an alternative canonical form for quaternionic h unitary matrices abstract the field of linear algebra over the quaternions is a research area which is still in development in this paper we continue our research on canonical forms for a matrix pair a h where the matrix a is h unitary h is invertible and with a as well as h quaternionic matrices we seek an invertible matrix s such that the transformations from a h to s 1 a s s h s brings the matrix a in jordan form and simultaneously brings h into a canonical form canonical forms for such pairs of matrices already exist in the literature the goal of the present paper is to add one more canonical form which specifically keeps a in jordan form in contrast to the existing canonical forms

error analysis of variational integrators of unconstrained lagrangian systems an error analysis of variational integrators is obtained by blowing up the discrete variational principles all of which have a singularity at zero time step divisions by the time step lead to an order that is one less than observed in simulations a deficit that is repaired with the help of a new past future symmetry