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

a kinetic theory for quantum transport in aperiodic media transport theory is reinvestigated in a one particle model including dissipation mechanisms through a phenomenological time dependent part using the description of an aperiodic crystal in terms of noncommutative geometry it is then possible to rigorously derive kubo formulas for the electric conductivity within the framework of the relaxation time approximation a fractal analysis applied to the spectral measures entering in the kubo formula allows us to examine anomalous transport owing to quantum interferences this leads to anomalies in drude s formula

quantization of edge currents for continuous magnetic operators abstract for a magnetic hamiltonian on a half plane given by the sum of the landau operator with dirichlet boundary conditions and a random potential a quantization theorem for the edge currents is proven this shows that the concept of edge channels also makes sense in presence of disorder moreover gaussian bounds on the heat kernel and its covariant derivatives are obtained

the quantum state diffusion model applied to open systems a model of a quantum system interacting with its environment is proposed in which the system is represented by a state vector that satisfies a stochastic differential equation derived from a density operator equation such as the bloch equation and consistent with it the advantages of the numerical solution of these equations over the direct numerical solution of the density operator equations are described the method is applied to the nonlinear absorber cascades of quantum transitions second harmonic generation and a measurement reduction process the model provides graphic illustrations of these processes with statistical fluctuations that mimic those of experiments the stochastic differential equations originated from studies of the measurement problem in the foundations of quantum mechanics the model is compared with the quantum jump model of dalibard 1992 carmichael and others which originated among experimenters looking for intuitive pictures and rules of computation

the condition number of real vandermonde krylov and positive definite hankel matrices we show that the euclidean condition number of any positive definite hankel matrix of order n geq 3 may be bounded from below by gamma n 1 16n with gamma exp 4 cdot it catalan pi approx 3 210 and that this bound may be improved at most by a factor 8 gamma n similar estimates are given for the class of real vandermonde matrices the class of row scaled real vandermonde matrices and the class of krylov matrices with hermitian argument improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval our findings confirm that all such matrices including for instance the famous hilbert matrix are ill conditioned already for moderate order as application we describe implications of our results for the numerical condition of various tasks in numerical analysis such as polynomial and rational i nterpolation at real nodes determination of real roots of polynomials computation of coefficients of orthogonal polynomials or the iterative solution of linear systems of equations

spectra and transport in almost periodic dimers we study spectral properties of discrete schrodinger operators with potentials obtained via dimerization of a class of aperiodic sequences it is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic full probability singular continuous spectrum in the hull of primitive and palindromic four block substitution potentials are robust under dimerization generic results also hold for circle potentials we illustrate these results with numerical studies of the quantum mean square displacement as a function of time the numerical techniques provide a very fast algorithm for the time evolution of wave packets

fractal wavelet dimensions and localization in this paper we want to give a new definition of fractal dimensions as small scale behavior of theq energy of wavelet transforms this is a generalization of previous multi fractal approaches with this particular definition we will show that the 2 dimension correlation dimension of the spectral measure determines the long time behavior of the time evolution generated by a bounded self adjoint operator acting in some hilbert space it will be proved that for we have mathop lim inf limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 and that mathop lim sup limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 wherek 2 are the upper and lower correlation dimensions of the spectral measure associated with and a quantitative version of the rage theorem shall also be given

computational procedure for sturm liouville problems the approaches of datzeff c r acad bulg sci 12 1959 113 285 and wittrick and williams quart j mech appl math xxiv 1971 263 int j mech sci 16 1974 209 have been incorporated to develop a procedure for the automatic computation of the eigen values and the eigen functions of one dimensional linear sturm liouville boundary value eigenproblems both singular and nonsingular the continuous coefficients of a regular sturm liouville problem have been approximated by a finite number of step functions in each step the resulting boundary value problem has been integrated exactly and the solutions have then been matched to construct the continuously differentiable solution of the original problem and the corresponding eigencondition step sizes have been chosen automatically so that the local error has been held in a predetermined interval representative test examples have been computed to illustrate the accuracy reliability and efficiency of the algorithm proposed

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

quantum return probability for substitution potentials we propose an effective exponent ruling the algebraic decay of the average quantum return probability for discrete schrodinger operators we compute it for some non periodic substitution potentials with different degrees of randomness and do not find a complete qualitative agreement with the spectral type of the substitution sequences themselves i e the more random the sequence the smaller such an exponent

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

minimal condition number for positive definite hankel matrices using semidefinite programming abstract we present a semidefinite programming approach for computing optimally conditioned positive definite hankel matrices of order n unlike previous approaches our method is guaranteed to find an optimally conditioned positive definite hankel matrix within any desired tolerance since the condition number of such matrices grows exponentially with n this is a very good test problem for checking the numerical accuracy of semidefinite programming solvers our tests show that semidefinite programming solvers using fixed double precision arithmetic are not able to solve problems with n 30 moreover the accuracy of the results for 24 n 30 is questionable in order to accurately compute minimal condition number positive definite hankel matrices of higher order we use a mathematica 6 0 implementation of the sdpha solver that performs the numerical calculations in arbitrary precision arithmetic by using this code we have validated the results obtained by standard codes for n 24 and we have found optimally conditioned positive definite hankel matrices up to n 100

perturbation theory for the random dimer model the authors study a one dimensional tight binding model with a random potential taking two values or nu with the restriction that on pairs of two neighbouring sites the potentials take the same value this random dimer model proposed by dunlap phillips and wu 1990 has a vanishing lyapunov exponent for the energy e or nu they compute the lyapunov exponent and the density of states perturbatively in the vicinity of this energy using the invariant measure formalism

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

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

uniform cantor singular continuous spectrum for nonprimitive schrodinger operators it is shown that some schrodinger operators with nonprimitive substitution potentials have pure singular continuous cantor spectrum with null lebesgue measure for all elements in the respective hulls

conditional entropy of some automatic sequences burrows and sulston 1991 have introduced conditional block entropies hn from information theory in order to give a quantitative measure of disorder for sequences and if possible a characterization of quasicrystalline sequences we give here some properties concerning these block entropies and give an explicit formula for the sequences hn n in n corresponding to the thue morse sequence the rudin shapiro sequence and the paperfolding sequence we deduce from these computations that this measure of disorder does not enable us to distinguish between deterministic sequences even if they have different spectral properties

how bad are hankel matrices considered are hankel vandermonde and krylov basis matrices it is proved that for any real positive definite hankel matrix of order n its spectral condition number is bounded from below by 3 cdot 2 n 6 also proved is that the spectral condition number of a krylov basis matrix is bounded from below by 3 frac 1 2 cdot 2 frac n 2 3 for v v x 1 ldots x n a vandermonde matrix with arbitrary but pairwise distinct nodes x 1 ldots x n we show that mbox rm cond 2 v geq 2 n 2 n frac 1 2 if either x j leq 1 or x j geq 1 for all j then mbox rm cond 2 v geq 2 n 2

remarks on the relation between quantum dynamics and fractal spectra abstract we investigate the large time behaviour of various solutions of the schrodinger equation in terms of some local and global dimensions of the spectral measure we emphasize in particular the role of the hausdorff and correlation dimensions for the growth exponents of position moments we also discuss the stability of such exponents under local perturbations

singular continuous spectrum on a cantor set of zero lebesgue measure for the fibonacci hamiltonian it is rigorously proven that the spectrum of the tight binding fibonacci hamiltonian h mn m n 1 m n 1 m n n 1 n where 5 1 2 and means integer part is a cantor set of zero lebesgue measure for all real nonzero and the spectral measures are purely singular continuous this follows from a recent result by kotani coupled with the vanishing of the lyapunov exponent in the spectrum

trace maps for arbitrary substitution sequences the authors prove the existence and show how to construct trace maps for products of 2 2 matrices generated by arbitrary substitution sequences the dimension of the underlying space of the trace map is lower than the one suggested recently by other authors

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

localization at large disorder and at extreme energies an elementary derivation the work presents a short proof of localization under the conditions of either strong disorder 0 or extreme energies for a wide class of self adjoint operators with random matrix elements acting inl 2 spaces a prototypical example is the discrete schrodinger operatorh u 0 x v x onz d d 1 withu 0 x a specified background potential and v x generated as random variables the general results apply to operators with replaced by a non local self adjoint operatort whose matrix elements satisfy y t x y s const uniformly inx for somes 1 localization means here that within a specified energy range the spectrum ofh is of the pure point type or equivalently the wave functions do not spread indefinitely under the unitary time evolution generated byh the effect is produced by strong disorder in either the potential or in the off diagonal matrix elementst x y under rapid decay oft x y the corresponding eigenfunctions are also proven to decay exponentially the method is based on resolvent techniques the central technical ideas include the use of low moments of the resolvent kernel i e g e x y s withs small enough 1 to avoid the divergence caused by the distribution s cauchy tails and an effective use of the simple form of the dependence ofg e x y on the individual matrix elements ofh in elucidating the implications of the fundamental equation h e g e x x 0 x x0 this approach simplifies previous derivations of localization results avoiding the small denominator difficulties which have been hitherto encountered in the subject it also yields some new results which include localization under the following sets of conditions i potentials with an inhomogeneous non random partu 0 x ii the bethe lattice iii operators with very slow decay in the off diagonal terms t x y 1 x y d e and iv localization produced by disordered boundary conditions

symmetric matrix methods for schrodinger eigenvectors the one dimensional single particle schrodinger eigenvalue equation is represented by a generalized matrix eigenvalue equation of the form a psi ea psi where a is a symmetric matrix a is a symmetric positive definite matrix and psi is a column vector the local error in making the representation is proportional to delta 4 where delta is the step size several methods of solution which make use of the sparsity of the matrices are investigated

a kinetic model of quantum jumps a new class of models describing the dissipative dynamics of an open quantum system s by means of random time evolutions of pure states in its hilbert space is considered the random evolutions are linear and defined by poisson processes at the random poissonian times the wavefunction experiences discontinuous changes quantum jumps these changes are implemented by some non unitary linear operators satisfying a locality condition if the hilbert space of s is infinite dimensional the models involve an infinite number of independent poisson processes and the total frequency of jumps may be infinite we show that the random evolutions in are then given by some almost surely defined unbounded random evolution operators obtained by a limit procedure the average evolution of the observables of s is given by a quantum dynamical semigroup its generator having the lindblad form 1 the relevance of the models in the field of electronic transport in anderson insulators is emphasised

on rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions we consider different definitions of the correlation dimension and find some relationships between them and other characteristics of dimension type such as hausdorff dimension box dimension etc we also introduce different ways to define and study the generalized spectrum for dimensions a one parameter family of characteristics of dimension type

wave propagation in almost periodic structures abstract the unusual phenomena occurring in the wave propagation in almost periodic structures both classical and quantum are studied in this paper focusing our attention on structures with spectral measures in the family of disconnected iterated function systems we describe and characterize the concept of quantum intermittency this theory shows that the nontrivial renormalization properties of the set of orthogonal polynomials associated with these systems are the origin of such intermittency and leads to a new determination of the exponents of the asymptotic growth of the moments of the position operator

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

fractal measures and mean p variations abstract recently strichartz proved that if is locally uniformly dimensional on r d then where 0 d and bt denotes the ball of radius t center at 0 if is self similar and satisfies a certain open set condition he also obtained a formula for the so that 0 lim sup t 1 t d b t f s 2 the can serve in some sense as the dimensional index of the measure by using the mean p variation and the tauberian theorems we extend the first inequality and its variants to p q forms and give necessary and sufficient conditions on for such inequalities to hold we then use the mean quadratic variation to study some self similar measures on r which do not satisfy the open set condition the s that are constructed from s 1 x x s 2 x x 1 1 2 with weights 1 2 each the index for corresponding to 5 1 2 is calculated the expression for such is significantly different from the one obtained by strichartz

a stable numerical method for inverting shape from moments we derive a stable technique based upon matrix pencils for the reconstruction of or approximation by polygonal shapes from moments we point out that this problem can be considered the dual of 2 d numerical quadrature over polygonal domains an analysis of the sensitivity of the problem is presented along with some numerical examples illustrating the relevant points finally an application to the problem of gravimetry is explored where the shape of a gravitationally anomalous region is to be recovered from measurements of its exterior gravitational field

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

limits of infinite interaction radius dimensionality and the number of components for random operators with off diagonal randomness we consider random one body operators that are analogs of the statistical mechanics hamiltonians with a varying interaction radiusr the dimensionality of spaced and the number of the field components orbitals n we prove that all the moments of the green functions for nonreal energies of these operators converge asr d n to the products of the average green functions just as in the mean field approximation of statistical mechanics we find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity which is nonzero on the whole support of the integrated density of states

schrodinger operators with rudin shapiro potentials are not palindromic we prove a conjecture of a hof o knill and b simon commun math phys 174 149 159 1995 by showing that the rudin shapiro sequence is not palindromic i e does not contain arbitrarily long palindromes we prove actually this property for all paperfolding sequences and all rudin shapiro sequences deduced from paperfolding sequences as a consequence and as guessed by the above authors their method cannot be used for establishing that discrete schrodinger operators with rudin shapiro potentials have a purely singular continuous spectrum

on mott s formula for the ac conductivity in the anderson model we study the ac conductivity in linear response theory in the general framework of ergodic magnetic schrodinger operators for the anderson model if the fermi energy lies in the localization regime we prove that the ac conductivity is bounded from above by c 2 log 1 d 2 at small frequencies this is to be compared to mott s formula which predicts the leading term to be c 2 log 1 d 1

spectral properties of a tight binding hamiltonian with period doubling potential we study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence we prove that its spectrum is purely singular continuous and supported on a cantor set of zero lebesgue measure for all nonzero values of the potential strength moreover we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength

quantum intermittency in almost periodic lattice systems derived from their spectral properties abstract hamiltonian tridiagonal matrices characterized by multi fractal spectral measures in the family of iterated function systems can be constructed by a recursive technique described here we prove that these hamiltonians are almost periodic they are suited to describe quantum lattice systems with nearest neighbours coupling as well as chains of linear classical oscillators and electrical transmission lines we investigate numerically and theoretically the time dynamics of the systems so constructed we derive a relation linking the long time power law behaviour of the moments of the position operator expressed by a scaling function of the moment order and spectral multi fractal dimensions dq via d1 we show cases in which this relation is exact and cases where it is only approximate unveiling the reasons for the discrepancies

positive definite hankel matrices of minimal condition abstract real positive definite hankel matrices h n have spectral condition numbers which are exponentially increasing with n this paper attempts to characterize those matrices h n in this class which are minimally conditioned and to describe some of their properties we also compute h n explicitly for n 16

singular continuous electron spectrum for a class of circle sequences we derive substitution rules for a class of binary quasiperiodic sequences generated by circle maps whose rotation numbers are obtained from the precious means the nature of the electron spectra for the corresponding diatomic chains is studied in the nearest neighbour on site tight binding approximation using the transfer matrix technique by studying properties of the trace maps we find that the spectrum is purely singular continuous in most of the studied cases

linear response theory for magnetic schrodinger operators in disordered media abstract we justify the linear response theory for an ergodic schrodinger operator with magnetic field within the noninteracting particle approximation and derive a kubo formula for the electric conductivity tensor to achieve that we construct suitable normed spaces of measurable covariant operators where the liouville equation can be solved uniquely if the fermi level falls into a region of localization we recover the well known kubo str eda formula for the quantum hall conductivity at zero temperature