singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization this paper introduces an algorithm for minimizing a single variable locally lipschitz function subject to a like function being nonpositive the method combines polyhedral and quadratic approximation a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point the convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases the algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms

the fibonacci hamiltonian we consider the fibonacci hamiltonian the central model in the study of electronic properties of one dimensional quasicrystals and establish relations between its spectrum and spectral characteristics namely the optimal holder exponent of the integrated density of states the dimension of the density of states measure the dimension of the spectrum and the upper transport exponent and the dynamical properties of the fibonacci trace map such as dimensional characteristics of the non wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures topological entropy multipliers of periodic orbits we also exhibit a connection between the spectral quantities and the thermodynamic pressure function as a result a detailed description of the spectral properties for all values of the coupling constant is obtained in contrast to all previous quantitative results which could be established only in the regime of small or large coupling in particular we show that the spectrum of this operator is a dynamically defined cantor set and that the density of states measure is exact dimensional this implies that all standard fractal dimensions coincide in each case we show that all the gaps of the spectrum allowed by the gap labeling theorem are open for all values of the coupling constant also we establish strict inequalities between the four spectral characteristics in question and provide the exact large coupling asymptotics of the dimension of the density of states measure for the other three quantities the large coupling asymptotics were known before

recurrence of the eigenstates of a schrodinger operator with automatic potential we consider the schrodinger eigenvalue problem in the discrete case with a potential assuming two values distributed according to the automatic sequence of prouhet thue morse we show that there are no localized states and that the generalized eigenvectors are recurrent on a geometrical set stemming from the hierarchical nature of the potential

the spectrum of a quasiperiodic schrodinger operator the spectrum h of the tight binding fibonacci hamiltonian hmn m n 1 m 1 n m n v n v n chi omega 3 omega 2 n 1 1 is the golden number is shown to coincide with the dynamical spectrum the set on which an infinite subsequence of traces of transfer matrices is bounded the point spectrum is absent for any and h is a cantor set for 4 combining this with casdagli s earlier result one finds that the spectrum is singular continuous for 16

power law subordinacy and singular spectra i half line operators we study hausdorff dimensional spectral properties of certain whole line quasiperiodic discrete schrodinger operators by using the extension of the gilbert pearson subordinacy theory that we previously developed in 19

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

on subordinacy and analysis of the spectrum of schrodinger operators with two singular endpoints the theory of subordinacy is extended to all one dimensional schrodinger operatorsfor which the corresponding differential expression l d 2 dr 2 v r is in the limit point case at both ends of an interval a b with v r locally integrable this enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of lu xu x e as r a and r b the result provides a rigorous but straightforward method of direct spectral analysis which has very general application and somefurther properties of the spectrum are deduced from the underlying theory

on the hausdorff dimension of the spectrum of the thue morse hamiltonian we show that the hausdorff dimension of the spectrum of the thue morse hamiltonian has a common positive lower bound for all coupling

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

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

on subordinacy and analysis of the spectrum of one dimensional schrodinger operators abstract by considering the behaviour as n of the ratio of l2 0 n norms of solutions of d 2 dr 2 v r u xu 0 r x r a characterisation of the absolutely continuous and singular spectra of one dimensional schrodinger operators is deduced the analysis is applicable to all operators for which l d 2 dr 2 v r is regular at 0 and in the limit point case at infinity with v r locally integrable

spectrum and extended states in a harmonic chain with controlled disorder effects of the thue morse symmetry along the lines of previous work we give the general framework together with a detailed and rigorous study of the spectrum and born von karman eigenstates of a 1d harmonic chain with controlled disorder determined by the thue morse sequence the spectrum is a cantor like set we prove numerically that its measure is zero and calculate its bouligand minkowski dimension box dimension we prove that the value of the ids on each of the gaps is 2k 1 3 2 p withk andp integers finally we also prove that points in a dense subset of the spectrum give rise to extended states an exceptional property due to the symmetry of the thue morse substitution which can have important applications to multilayered structures and we illustrate this situation

power law growth for the resistance in the fibonacci model many one dimensional quasiperiodic systems based on the fibonacci rule such as the tight binding hamiltonianh n n 1 n 1 v n n ne el2 e wherev n n 1 n x denoting the integer part ofx and the golden mean sqrt 5 1 2 give rise to the same recursion relation for the transfer matrices it is proved that the wave functions and the norm of transfer matrices are polynomially bounded critical regime if and only if the energy is in the spectrum of the hamiltonian this solves a conjecture of kohmoto and sutherland on the power law growth of the resistance in a one dimensional quasicrystal

measure zero spectrum of a class of schrodinger operators we study the measure of the spectrum of a class of one dimensional discrete schrodinger operators hv with potential v generated by any primitive substitutions it is well known that the spectrum of hv is singular continuous 1 we will give a more exact result that the spectrum of hv is a set of lebesgue measure zero by removing two hypotheses the semi primitive of a certain induced substitution and the existence of square word from a theorem due to bovier and ghez 2

power law bounds on transfer matrices and quantum dynamics in one dimension we present an approach to quantum dynamical lower bounds for discrete one dimensional schrodinger operators which is based on power law bounds on transfer matrices it suffices to have such bounds for a nonempty set of energies we apply this result to various models including the fibonacci hamiltonian

uniform spectral properties of one dimensional quasicrystals iii α continuity we study the spectral properties of one dimensional whole line schrodinger operators especially those with sturmian potentials building upon the jitomirskaya last extension of the gilbert pearson theory of subordinacy we demonstrate how to establish continuity of a whole line operator from power law bounds on the solutions on a half line however we require that these bounds hold uniformly in the boundary condition

singular spectrum of lebesgue measure zero for one dimensional quasicrystals the spectrum of one dimensional discrete schrodinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the lyapunov exponent if and only if the lyapunov exponent exists uniformly this is used to obtain the cantor spectrum of zero lebesgue measure for all aperiodic subshifts with uniform positive weights this covers in particular all aperiodic subshifts arising from primitive substitutions including new examples such as e g the rudin shapiro substitution our investigation is not based on trace maps instead it relies on an oseledec type theorem due to a furman and a uniform ergodic theorem due to the author

unbounded trace orbits of thue morse hamiltonian it is well known that an energy is in the spectrum of fibonacci hamiltonian if and only if the corresponding trace orbit is bounded however it is not known whether the same result holds for the thue morse hamiltonian in this paper we give a negative answer to this question more precisely we construct two subsets sigma ii and sigma iii of the spectrum of the thue morse hamiltonian both of which are dense and uncountable such that each energy in sigma ii cup sigma iii corresponds to an unbounded trace orbit exact estimates on the norm of the transfer matrices are also obtained for these energies for e in sigma ii cup sigma iii the norms of the transfer matrices behave like begin aligned e c 1 gamma sqrt n le vert t n e vert le e c 2 gamma sqrt n end aligned however two types of energies are quite different in the sense that each energy in sigma ii is associated with a two sided pseudo localized state while each energy in sigma iii is associated with a one sided pseudo localized state the difference is also reflected by the local dimensions of the spectral measure the local dimension is 0 for energies in sigma ii and is larger than 1 for energies in sigma iii as a comparison we mention another known countable dense subset sigma i each energy in sigma i corresponds to an eventually constant trace map and the associated eigenvector is an extended state in summary the thue morse hamiltonian exhibits mixed spectral nature

spectral properties of one dimensional schrodinger operators with potentials generated by substitutions we investigate one dimensional discrete schrodinger operators whose potentials are invariant under a substitution rule the spectral properties of these operators can be obtained from the analysis of a dynamical system called the trace map we give a careful derivation of these maps in the general case and exhibit some specific properties under an additional easily verifiable ypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying schrodinger operator is singular and supported on a set of zero lebesgue measure a condition allowing to exclude point spectrum is also given the application of our theorems is explained on a series of examples

an aggregate subgradient method for nonsmooth and nonconvex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

solution of linear differential equations with fuzzy boundary values we investigate linear differential equations with boundary values expressed by fuzzy numbers in contrast to most approaches which search for a fuzzy valued function as the solution we search for a fuzzy set of real functions as the solution we define a real function as an element of the solution set if it satisfies the differential equation and its boundary values are in intervals determined by the corresponding fuzzy numbers the membership degree of the real function is defined as the lowest value among membership degrees of its boundary values in the corresponding fuzzy sets to find the fuzzy solution we use a method based on the properties of linear transformations we show that the fuzzy problem has a unique solution if the corresponding crisp problem has a unique solution we prove that if the boundary values are triangular fuzzy numbers then the value of the solution at a given time is also a triangular fuzzy number the defined solution is the same as one of the solutions obtained by zadeh s extension principle for a second order differential equation with constant coefficients the solution is expressed in analytical form examples are given to describe the proposed approach and to compare it to a method that uses the generalized hukuhara derivative which demonstrates the advantages of our method

resistance of one dimensional quasicrystals abstract one dimensional quasiperiodic systems are often based on an inflation rule giving rise to a recursion formula on the transfer matrices it is shown in a general setting that the norm growth of these matrices can be evaluated a consequence is for instance a purely critical regime namely polynomial bounds in function of the sample size for the electrical resistance when the energy is in the spectrum this solves a conjecture by sutherland and kohmoto