maximum tsallis entropy with generalized gini and gini mean difference indices constraints abstract using the maximum entropy principle with tsallis entropy some distribution families for modeling income distribution are obtained by considering income inequality measures maximum tsallis entropy distributions under the constraint on generalized gini and gini mean difference indices are derived it is shown that the tsallis entropy maximizers with the considered constraints belong to generalized pareto family

bound states and the szegő condition for jacobi matrices and schrodinger operators for jacobi matrices with a n 1 1 n n b n 1 n n we study bound states and the szeg condition we provide a new proof of nevai s result that if 12 the szeg condition holds which works also if one replaces 1 n by cos n we show that if 0 0 and 12 the szeg condition fails we also show that if 1 and are small enough 2 8 2 1 24 will do then the jacobi matrix has finitely many bound states for 0 large it has infinitely many

preservation of a c spectrum for random decaying perturbations of square summable high order variation we consider random self adjoint jacobi matrices of the form j u n an u n 1 bn u n an 1 u n 1 on l2 n where an 0 and bn r are sequences of random variables on a probability space dp such that there exists q n such that for any l n 2l al al q and 2l 1 bl bl q are independent random variables of zero mean satisfying n 1 n2 dp let jp be the deterministic periodic of period q jacobi matrix whose coefficients are the mean values of the corresponding entries in j we prove that for a e the a c spectrum of the operator j equals to and fills the spectrum of jp if moreover n 1 n4 dp then for a e the spectrum of j is purely absolutely continuous on the interior of the bands that make up the spectrum of jp

zeroes of the spectral density of the schroedinger operator with the slowly decaying wigner von neumann potential we consider the schrodinger operator mathcal l alpha on the half line with a periodic background potential and a perturbation which consists of two parts a summable potential and the slowly decaying wigner von neumann potential frac c sin 2 omega x delta x gamma where gamma in frac 1 2 1 the continuous spectrum of this operator has the same band gap structure as the continuous spectrum of the unperturbed periodic operator in every band there exist two points called critical where the eigenfunction equation has square summable solutions every critical point nu cr is an eigenvalue of the operator mathcal l alpha for some value of the boundary parameter alpha alpha cr specific to that particular point we prove that for alpha ne alpha cr the spectral density of the operator mathcal l alpha has a zero of the exponential type at nu cr

on the absolutely continuous and negative discrete spectra of schrodinger operators on the line with locally integrable globally square summable potentials for one dimensional schrodinger operators with potentials q subject to n nn 1 q x dx 2 we prove that the absolutely continuous spectrum is 0 extending the 1999 result due to dieft killip as a by product we show that under the same condition the sequence of the negative eigenvalues is 3 2 summable improving the relevant result by lieb thirring

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

schrodinger operators with sparse potentials asymptotics of the fourier transform of the spectral measure we study the pointwise behavior of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with sparse potentials we find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model we also present an improved version of known results on the spectrum of such operators

stability of spectral types for jacobi matrices under decaying random perturbations abstract we study stability of spectral types for semi infinite self adjoint tridiagonal matrices under random decaying perturbations we show that absolutely continuous spectrum associated with bounded eigenfunctions is stable under hilbert schmidt random perturbations we also obtain some results for singular spectral types

absolutely continuous spectrum of dirac operators with square integrable potentials we show that the absolutely continuous part of the spectral function of the one dimensional dirac operator on a half line with a constant mass term and a real square integrable potential is strictly increasing throughout the essential spectrum 1 1 the proof is based on estimates for the transmission coefficient for the full line scattering problem with a truncated potential and a subsequent limiting procedure for the spectral function furthermore we show that the absolutely continuous spectrum persists when an angular momentum term is added thus also establishing the result for spherically symmetric dirac operators in higher dimensions

asymptotics of the fourier transform of the spectral measure for schrodinger operators with bounded and unbounded sparse potentials we study the pointwise behaviour of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with unbounded sparse potentials particularly with the potentials of the special type which give rise to the spectra with hausdorff dimensionality between 1 2 and 1 operators with bounded sparse potentials are also considered

on local and global measures of separation in fuzzy topological spaces this paper introduces as far as separation properties are concerned a new way of studying a fuzzy topological space we define separation functions which measure the degree to which points in a given space can be separated in a t 0 t 1 or t 2 like fashion both local and global measures are introduced and their interrelation their relation with previously defined properties their behaviour with regard to subspaces and products and their behaviour in the special classes of fuzzy neighborhood spaces and fuzzy uniform spaces are studied

spectral analysis of selfadjoint jacobi matrices with periodically modulated entries this paper deals with the spectral analysis of a class of selfadjoint unbounded jacobi matrices j with modulated entries the entries have the form of smooth sequences that increase to infinity multiplied by proper periodic sequences for this class criteria for pure absolute continuity of the spectrum or its discreteness and the asymptotics of generalized eigenvectors of j are given some examples illustrating the stability zones of spectral structure are presented

maximization of statistical heterogeneity from shannon s entropy to gini s index abstract different fields of science apply different quantitative gauges to measure statistical heterogeneity statistical physics and information theory commonly use shannon s entropy which measures the randomness of probability laws whereas economics and the social sciences commonly use gini s index which measures the evenness of probability laws motivated by the principle of maximal entropy we explore the maximization of statistical heterogeneity for probability laws with a given mean in the four following scenarios i shannon entropy maximization subject to a given dispersion level ii gini index maximization subject to a given dispersion level iii shannon entropy maximization subject to a given gini index iv gini index maximization subject to a given shannon entropy analysis of these four scenarios results in four different classes of heterogeneity maximizing probability laws yielding an in depth description of both the marked differences and the interplay between the physical randomness based and the economic evenness based approaches to the maximization of statistical heterogeneity

on the application of some of m g krein s results to the spectral analysis of sturm liouville operators abstract in one of his works m g krein discovered an analogy between polynomials orthogonal on the unit circle and generalized eigenfunctions of certain differential systems we used some ideas of this paper to obtain new results in spectral analysis of sturm liouville operators

central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients we study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles we study those biorthogonal ensembles for which the underlying biorthogonal family s

absence of singular continuous spectrum for perturbed discrete schrodinger operators abstract we show that the spectral measure of discrete schrodinger operators h u n u n 1 u n 1 v n u n does not have singular continuous component if the potential v n o n 1

orthogonal polynomials with recursion coefficients of generalized bounded variation we consider probability measures on the real line or unit circle with jacobi or verblunsky coefficients satisfying an l p condition and a generalized bounded variation condition this latter condition requires that a sequence can be expressed as a sum of sequences l each of which has rotated bounded variation i e sum n 0 infty vert e i phi l beta n 1 l beta n l vert infty for some phi l this includes a large class of discrete schrodinger operators with almost periodic potentials modulated by l p decay i e linear combinations of lambda n rm cos 2 pi alpha n phi with lambda in ell p of bounded variation and any

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

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

solutions spectrum and dynamics for schrodinger operators on infinite domains 1 introduction and main results in this paper we investigate the relations between the rate of decay of solutions of schrodinger equations continuity properties of spectral measures of the corresponding operators and dynamical properties of the corresponding quantum systems the first main result of this paper shows that in great generality certain upper bounds on the rate of growth of l 2 norms of generalized eigenfunctions over expanding balls imply certain minimal singularity of the spectral measures consider an operator h v defined by the differential expression h v v x

positive lyapunov exponents for a class of deterministic potentials letv be a smooth non constant function on the torus and lett be a hyperbolic toral automorphism consider a discrete one dimensional schrodinger operatorh whose potential at sitej is given bygv j gv t j we prove that wheng 0 is small andg 1 2 e 2 g 1 2 the lyapunov exponent for the cocycle generated byh e is proportional tog 2 the proof relies on a formula of pastur and figotin and on symbolic dynamics

schrodinger operators with many bound states consider the schrodinger operators h d2 dx2 v x we present a method for estimating the potential in terms of the negative eigenvalues of these operators among the applications are inverse lieb thirring inequalities and several sharp results concerning the spectral properties of h

de localization in the prime schrodinger operator we report a combined numerical approach to study the localization properties of the one dimensional tight binding model with potential modulated along the prime numbers a localization delocalization transition was found as a function of the potential intensity it is also argued that there are delocalized states for any value of the potential intensity

measuring statistical heterogeneity the pietra index abstract there are various ways of quantifying the statistical heterogeneity of a given probability law statistics uses variance which measures the law s dispersion around its mean physics and information theory use entropy which measures the law s randomness economics uses the gini index which measures the law s egalitarianism in this research we explore an alternative to the gini index the pietra index which is a counterpart of the kolmogorov smirnov statistic the pietra index is shown to be a natural and elemental measure of statistical heterogeneity which is especially useful in the case of asymmetric and skewed probability laws and in the case of asymptotically paretian laws with finite mean and infinite variance moreover the pietra index is shown to have immediate and fundamental interpretations within the following applications renewal processes and continuous time random walks infinite server queueing systems and shot noise processes financial derivatives the interpretation of the pietra index within the context of financial derivatives implies that derivative markets in effect use the pietra index as their benchmark measure of statistical heterogeneity

finite gap potentials and wkb asymptotics for one dimensional schrodinger operators consider the schrodinger operator h d2 dx2 v x with power decaying potential v x o x we prove that a previously obtained dimensional bound on exceptional sets of the wkb method is sharp in its whole range of validity the construction relies on pointwise bounds on finite gap potentials these bounds are obtained by an analysis of the jacobi inversion problem on hyperelliptic riemann surfaces

maximum entropy estimation of density and regression functions abstract this paper presents an information efficient technique to determine the functional forms of density and regression functions by maximizing entropy subject to given side conditions a flexible functional form approach is established as special cases of this approach exponential polynomial cobb douglas translog generalized cobb douglas generalized leontief fourier flexible form and minflex laurent series etc can be derived in a monte carlo experiment the maximum entropy regression method shows good performance in terms of goodness of fit boxplots and isoquants produced by the maximum entropy method are compared with those produced by other well known functions

pure point spectrum under 1 parameter perturbations and instability of anderson localization we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic fora we study the set of all eigenvalues of the operatora t a tp t that belong to its essential spectrum which does not depend on the parametert we prove that this set is empty for a dense set of values oft then we apply this result or its idea to questions of anderson localization for 1 dimensional schrodinger operators discrete and continuous

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

orthogonal polynomials measures and recurrences on the unit circle on donne de nouvelles caracterisations des polynomes orthogonaux sur le cercle unite et les mesures associees en termes de coefficients de reflexion dans l equation de recurrence satisfaite par le polynome

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

spectral and dynamical properties of certain random jacobi matrices with growing parameters in this paper a family of random jacobi matrices with off diagonal terms that exhibit power law growth is studied since the growth of the randomness is slower than that of these terms it is possible to use methods applied in the study of schr odinger operators with random decaying potentials a particular result of the analysis is the existence of operators with arbitrarily fast transport whose spectral measure is zero dimensional the results are applied to the infinite dumitriu edelman model 2002 and its spectral properties are analyzed

asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients abstract we study solutions of three term recurrence relations whose n step transfer matrices belong to the uniform stolz class in particular we derive the first order of their uniform asymptotics for orthonormal polynomials we show more namely we find the constructive formula for the density of their orthogonality measure in terms of turan determinants and we determine their exact asymptotic behavior we treat both bounded and unbounded cases in a unified way

on the existence of embedded eigenvalues i relate the existence of eigenfunctions of decaying perturbations of the free laplacian to the semi classical behavior of the spectrum of quasi periodic schrodinger operators in particular i construct embedded eigenvalues in the spectrum of the discrete laplacian perturbed by a potential of generalized bounded variation

spectral stability for compact perturbations of toeplitz matrices abstract let f be a regular real valued non constant symbol defined on the one dimensional torus t denote respectively by and t its set of critical points and the associated toeplitz matrix on l 2 n if v is a suitable compact perturbation we prove that the operator t v has no singular continuous spectrum and only finite point spectrum away from the set of thresholds f we also obtain some propagation estimates and apply these results to concrete examples

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

wkb and spectral analysis of one dimensional schrodinger operators with slowly varying potentials consider a schrodinger operator on l 2 of the line or of a half line with appropriate boundary conditions if the potential tends to zero and is a finite sum of terms each of which has a derivative of some order in l 1 l p for some exponent p 2 then an essential support of the the absolutely continuous spectrum equals almost every generalized eigenfunction is bounded and satisfies certain wkb type asymptotics at infinity if moreover these derivatives belong to l p with respect to a weight x with 0 then the hausdorff dimension of the singular component of the spectral measure is strictly less than one

wave propagation through sparse potential barriers we prove that the three dimensional schrodinger operator with slowly decaying sparse potential has an a c spectrum that fills a new kind of wkb asymptotics for green s function is obtained the absence of positive eigenvalues is established as well 2007 wiley periodicals inc

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

some jacobi matrices with decaying potential and dense point spectrum we discuss doubly infinite matrices of the formm ij i j 1 i j 1 v i ij as operators on l2 we present for each 0 examples of potentialsv n with v n o n 1 2 and wherem has only point spectrum our discussion should be viewed as a remark on the recent work of delyon kunz and souillard

szegő difference equations transfer matrices and orthogonal polynomials on the unit circle we develop the theory of orthogonal polynomials on the unit circle based on the szeg recurrence relations written in matrix form the orthogonality measure and c function arise in exactly the same way as weyl s function in the weyl approach to second order linear differential equations on the half line the main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one dimensional schrodinger operators discrete and continuous and the notion of subordinacy from the gilbert pearson theory we study the relations between transfer matrices and the structure of orthogonality measures the theory is illustrated by the szeg equations with reflection coefficients having bounded variation

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

new classes of lorenz curves by maximizing tsallis entropy under mean and gini equality and inequality constraints abstract in this paper by using the entropy maximization principle with tsallis entropy new distribution families for modeling the income distribution are derived also new classes of lorenz curves are obtained by applying the entropy maximization principle with tsallis entropy under mean and gini index equality and inequality constraints

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

the limiting absorption principle for the discrete wigner von neumann operator abstract we apply weighted mourre commutator theory to prove the limiting absorption principle for the discrete schrodinger operator perturbed by the sum of a wigner von neumann and long range type potential in particular this implies a new result concerning the absolutely continuous spectrum for these operators even for the one dimensional operator we show that methods of classical mourre theory based on differential inequalities and on the generator of dilation cannot apply to the aforementioned schrodinger operators

sparse potentials with fractional hausdorff dimension abstract we construct non random bounded discrete half line schrodinger operators which have purely singular continuous spectral measures with fractional hausdorff dimension in some interval of energies to do this we use suitable sparse potentials our results also apply to whole line operators as well as to certain random operators in the latter case we prove and compute an exact dimension of the spectral measures

spectral theory of one channel operators and application to absolutely continuous spectrum for anderson type models abstract a one channel operator is a self adjoint operator on l 2 g for some countable set g with a rank 1 transition structure along the sets of a quasi spherical partition of g jacobi operators are a very special case in essence there is only one channel through which waves can travel across the shells to infinity this channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells not all of the transfer matrices are defined for some countable set of energies still many theorems from the world of jacobi operators are translated to this setup the results are then used to show absolutely continuous spectrum for the anderson model on certain finite dimensional graphs with a one channel structure this result generalizes some previously obtained results on antitrees

new functional forms of lorenz curves by maximizing tsallis entropy of income share function under the constraint on generalized gini index abstract the lorenz curve is one of the most powerful tools in the analysis of the size distribution of income and wealth in the past decades many authors have proposed different functional forms for estimating lorenz curves using a variety of approaches in this paper new functional forms are derived by maximizing tsallis entropy of income share function subject to a given generalized gini index the obtained lorenz curves are fitted to the income data sets of three asian countries in 1988 and their relative performances with respect to some well known parametric models of lorenz curves are compared using two types of goodness of fit measures

wkb asymptotic behavior of almost all generalized eigenfunctions for one dimensional schrodinger operators with slowly decaying potentials we prove the wkb asymptotic behavior of solutions of the differential equation d2u dx2 v x u eu for a e e a where v v1 v2 v1 lp r and v2 is bounded from above with a lim supx v x while v 2 x lp r 1 p 2 these results imply that schrodinger operators with such potentials have absolutely continuous spectrum on a we also establish wkb asymptotic behavior of solutions for some energy dependent potentials

on the preservation of absolutely continuous spectrum for schrodinger operators abstract we present general principles for the preservation of a c spectrum under weak perturbations the schrodinger operators on the strip and on the caley tree bethe lattice are considered

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

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

criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed stark type operators abstract in this paper we consider the perturbed stark operator h u h 0 u q u u x u q u where q is the power decaying perturbation the criteria for q such that h h 0 q has at most one eigenvalue finitely many infinitely many eigenvalues are obtained all the results are quantitative and are generalized to the perturbed stark type operator

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

entropy maximization under the constraints on the generalized gini index and its application in modeling income distributions abstract in economics and social sciences the inequality measures such as gini index pietra index etc are commonly used to measure the statistical dispersion there is a generalization of gini index which includes it as special case in this paper we use principle of maximum entropy to approximate the model of income distribution with a given mean and generalized gini index many distributions have been used as descriptive models for the distribution of income the most widely known of these models are the generalized beta of second kind and its subclass distributions the obtained maximum entropy distributions are fitted to the us family total money income in 2009 2011 and 2013 and their relative performances with respect to generalized beta of second kind family are compared

absolutely continuous spectrum of one dimensional schrodinger operators and jacobi matrices with slowly decreasing potentials we prove that for any one dimensional schrodinger operator with potentialv x satisfying decay condition v x cx 3 4 e the absolutely continuous spectrum fills the whole positive semi axis the description of the set in on which the singular part of the spectral measure might be supported is also given analogous results hold for jacobi matrices

power law bounds on transfer matrices and quantum dynamics in one dimension ii abstract we establish quantum dynamical lower bounds for a number of discrete one dimensional schrodinger operators these dynamical bounds are derived from power law upper bounds on the norms of transfer matrices we develop further the approach from part i and study many examples particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power law bounds on transfer matrices the models discussed in this paper include substitution models sturmian models a hierarchical model the prime model and a class of moderately sparse potentials

half line schrodinger operators with no bound states acting in l2 0 c with the boundary condition r for convenience we require that the potential v be uniformly locally square integrable we write l l 2 for the banach space of such functions the free operators that is when v 0 can be diagonalized by the fourier transform this shows that they have spectra 2 2 and 0 co respectively and that in both cases the spectrum is purely absolutely continuous for a general discrete operator the mere fact that the spectrum is contained in 2 2 forces it to be purely absolutely continuous this is our first main result

value distribution and spectral theory of schrodinger operators with l 2 sparse potentials we apply the methods of value distribution theory to the spectral asymptotics of schrodinger operators with l2 sparse potentials

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

sparse one dimensional discrete dirac operators ii spectral properties we study spectral properties of some discrete dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions as observed in the corresponding schrodinger operators we determine the hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous

generalized prufer variables for perturbations of jacobi and cmv matrices abstract prufer variables are a standard tool in spectral theory developed originally for perturbations of the free schrodinger operator we adapt the generalized prufer variables introduced by kiselev remling simon for perturbations of an arbitrary schrodinger operator to the setting of jacobi and szeg recursions we present an application to random l 2 perturbations of jacobi and cmv matrices and an application to decaying oscillatory perturbations of periodic jacobi and cmv matrices

maximum entropy lorenz curves abstract for describing the distribution of income a family of maximum entropy density functions is derived under optional side conditions on the distance of the mean income from the minimum income and the absolute gini index of concentration many of the resulting lorenz curves obtained from the reciprocal of the density function by repeated quadrature turn out to be well known models

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

some schrodinger operators with power decaying potentials and pure point spectrum we construct deterministic potentials such that the schrodinger equation on has dense pure point spectrum in for almost all boundary conditions at as a by product we also obtain power decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions