hamiltonians on discrete structures jumps of the integrated density of states and uniform convergence we study equivariant families of discrete hamiltonians on amenable geometries and their integrated density of states ids we prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support the size of a jump of the ids is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions from this we deduce uniform convergence w r t the spectral parameter of the finite volume approximants of the ids our framework includes quasiperiodic operators on delone sets periodic and random operators on quasi transitive graphs and operators on percolation graphs

generalized polar coordinates on lie groups and numerical integrators motivated by developments in numerical lie group integrators we introduce a family of local coordinates on lie groups denoted generalized polar coordinates fast algorithms are derived for the computation of the coordinate maps their tangent maps and the inverse tangent maps in particular we discuss algorithms for all the classical matrix lie groups and optimal complexity integrators for n spheres

holder continuity of the spectral measures for one dimensional schrodinger operator in exponential regime avila and jitomirskaya prove that the spectral measure v xf of quasi periodic schrodinger operator is 1 2 holder continuous with appropriate initial vector f if satisfies diophantine condition and is small in the present paper the conclusion is extended to that for all with the spectral measure v xf is 1 2 holder continuous with small if v is real analytic in a neighbor of x c where c is a large absolute constant in particular the spectral measure xf of almost mathieu operator is 1 2 holder continuous if e c with c a large absolute constant

spectral concentration and rapidly decaying potentials abstract we consider the spectral function 0 for the sturm liouville equation y q y 0 on 0 with a boundary condition y 0 cos y 0 sin 0 and where q l 0 a point 0 is called a point of spectral concentration if has a local maximum there we give a new formula for in terms of the prufer transformation angle and we identify a rapid transitional property of in the neighbourhood of points of spectral concentration this property provides a sensitive test of the existence of such points and lends itself to a detailed computational investigation which reveals such points in greater profusion than shown by for example sledge our approach also leads to the possibility of a theoretical investigation

eigenvalue and eigenfunction asymptotics for regular sturm liouville problems abstract in this paper we obtain asymptotic formulas for eigenvalues eigenfunctions and the reciprocals of the eigenfunction norms for eigenvalue problems associated with the general sturm liouville equation pu r q u 0 having regular endpoints the method is based on a conversion to liouville normal form and an iterative procedure of solving the associated volterra integral equation producing an asymptotic expansion of the solution in higher powers of 1 1 2 as

boundary value method for inverse sturm liouville problems in this paper we present a method to recover symmetric and non symmetric potential functions of inverse sturm liouville problems from the knowledge of eigenvalues the linear multistep method coupled with suitable boundary conditions known as boundary value method bvm is the main tool to approximate the eigenvalues in each iteration step of the used newton method the bvm was extended to work for neumann neumann boundary conditions moreover a suitable approximation for the asymptotic correction of the eigenvalues is given numerical results demonstrate that the method is able to give good results for both symmetric and non symmetric potentials

numerical approximation of high dimensional fokker planck equations with polynomial coefficients this paper is concerned with the numerical solution of high dimensional fokker planck equations related to multi dimensional diffusion with polynomial coefficients or pearson diffusions classification of multi dimensional pearson diffusion follows from the classification of one dimensional pearson diffusion there are six important classes of pearson diffusion three of them possess an infinite system of moments gaussian gamma beta while the other three possess a finite number of moments inverted gamma student and fisher snedecor numerical approximations to the solution of the fokker planck equation are generated using the spectral method the use of an adaptive reduced basis technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition svd the basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite dimensional subspace of the solution space this is achieved through basis enrichment and projection stages numerical results with different boundary conditions are presented to demonstrate the accuracy and efficiency of the numerical scheme

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

sharp holder continuity of the lyapunov exponent of finitely differentiable quasi periodic cocycles we show that if the base frequency is diophantine then the lyapunov exponent of a c k quasi periodic sl 2 mathbb r cocycle is 1 2 holder continuous in the almost reducible regime as a consequence we show that if the frequency is diophantine and the potential is small then the integrated density of states of the corresponding quasi periodic schrodinger operator is 1 2 holder continuous

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

conservation properties of a time fe method part i time stepping schemes forn body problems in the present paper one step implicit integration algorithms for the n body problem are developed the time stepping schemes are based on a petrov galerkin finite element method applied to the hamiltonian formulation of the n body problem the approach furnishes algorithmic energy conservation in a natural way the proposed time finite element method facilitates a systematic implementation of a family of time stepping schemes a particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals the influence of various standard as well as non standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements copyright 2000 john wiley sons ltd

random schrodinger operators arising from lattice gauge fields i existence and examples we consider new models of ergodic schrodinger operators whose existence relies on a cohomological theorem of feldman and moore in ergodic theory these operators generalize the harper operator which describes the case of a constant magnetic field an example is the case when the magnetic field is given by independent random variables attached to the lattice plaquettes a generalization of the feldman moore theorem by lind to non abelian groups also allows us to consider schrodinger operators obtained from non abelian lattice gauge fields the existence result extends to more general graphs like to operators on tilings and to higher dimensions we compute some moment expansions for the density of states for example for independent identically and uniformly distributed magnetic fields a model which has been studied at least since 1970 and whose existence can also be seen without involving the above mentioned existence theorem we show that the nth moment is the number of closed paths in the two dimensional lattice starting at the origin for which the winding number vanishes at each plaquette point this goes beyond the brinkman rice self retracing path approximation other examples are a higher dimensional example a one dimensional anderson model which can be treated in this framework as well as the hofstadter model with constant magnetic field where one averages over all possible magnetic fields we also reprove a result of jitomirskaya mandelshtam stating that the deterministic aharonov bohm model is a compact perturbation of the free laplacian

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

efficient calculation of spectral density functions for specific classes of singular sturm liouville problems new families of approximations to sturm liouville spectral density functions are derived for cases where the potential function has one of several specific forms this particular form dictates the type of expansion functions used in the approximation error bounds for the residuals are established for each case in the case of power potentials the approximate solutions of an associated terminal value problem at are shown to be asymptotic power series expansions of the exact solution numerical algorithms have been implemented and several examples are given demonstrating the utility of the approach

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

stability of periodic plane waves nonlinear partial differential equations with explicit periodic plane wave solutions are considered for a class of such equations the variational equations governing linear stability are solved explicitly also an explicit equation for the growth rate of a perturbation is obtained the analysis is applied to nonlinear forms of the schrodinger and klein gordon equations and to a generalized korteweg de vries equation

stability of periodic plane wave solutions of wave equations abstract nonlinear wave equations with explicit periodic plane wave solutions are considered the variational equations governing linear stability are derived and solved the analysis is applied to the special cases of 4 and sine gordon wave equations

computing the spectral function for singular sturm liouville problems algorithms for computing sturm liouville spectral density functions are developed based on several mathematical characterizations convergence and error bounds are derived and methods are tested on several examples the results are compared with those from the existing software package sledge

a spectral transform method for singular sturm liouville problems with applications to energy diffusion in plasma physics we develop a spectrally accurate numerical method to compute solutions of a model pde used in plasma physics to describe diffusion in velocity space due to fokker planck collisions the solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular sturm liouville operator we present a new algorithm for computing the spectral density function of the operator that uses chebyshev polynomials to extrapolate the value of the titchmarsh weyl m function from the complex upper half plane to the real axis the eigenfunctions and density function are rescaled and a new formula for the limiting value of the m function is derived to avoid amplification of roundoff errors when the solution is reconstructed the complexity of the algorithm is also analyzed showing that the cost of computing the spectral density function at a point grows less rapidly than any fractional inverse power of the desired acc

universality of general beta ensembles we prove the universality of the ensembles with convex analytic potentials and for any 0 i e we show that the spacing distributions of log gases at any inverse temperature coincide with those of the gaussian ensembles

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

motion of vortex lines in the ginzburg landau model equations of motion for vortex lines in the ginzburg landau theory are derived we construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field using the method of matched asymptotic expansions we show that to leading order the line moves in the binormal direction with a curvature dependent velocity we also consider the contribution of remote parts of the line interaction between several vortex lines and interaction with external fields

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

a general framework for conservative single step time integration schemes with higher order accuracy for a central force system a general framework for algorithms that conserve angular momentum for single body central force problems is presented it is shown that any family of momentum conserving algorithms can have at most three free parameters one of which may be used to ensure energy conservation and hence will be configuration dependent further restrictions can be made that enable the algorithms to recover the orbits of relative equilibria of the underlying physical problem in addition the algorithms can be made time reversible whilst still leaving two parameters unspecified the order of accuracy of a general momentum conserving family is analysed and it is shown that energy momentum algorithms that preserve the underlying physical relative equilibria can have unlimited accuracy if the two remaining parameters are appropriately chosen functions of the configuration and the time step this does not require any additional degrees of freedom extra stages of calculation or information from past solutions numerical examples are given that show the performance of some representative higher order schemes when applied to stiff and non stiff problems and the issue of newton raphson convergence is discussed

completely conservative covariant numerical methodology for a distance dependent potential o newtonian dynamics is characterized by conservation of energy linear momentum and angular momentum in addition the basic dynamical equations are covariant that is invariant under fundamental coordinate transformations in this paper it is shown that related n body problems can be solved numerically in such a fashion that independently of the time step the identical conservation laws continue to be valid for the approximating difference equations in addition the numerical method is covariant

holder continuity of absolutely continuous spectral measure for multi frequency schrodinger operators abstract we establish sharp results on the modulus of continuity of the distribution of the spectral measure for multi frequency schrodinger operators with diphantine frequencies and small analytic potentials

new characterizations of spectral density functions for singular sturm liouville problems a generalization is given for a characterization of the spectral density function of weyl and titchmarsh for a singular sturm liouville problem having absolutely continuous spectrum in 0 a recurrent formulation is derived that generates a family of approximations based on this scheme proofs of convergence for these new approximations are supplied and a numerical method is implemented the computational results show more rapid rates of convergence which are in accord with the theoretical rates