α continuity properties of one dimensional quasicrystals we apply the jitomirskaya last extension of the gilbert pearson theory to discrete one dimensional schrodinger operators with potentials arising from generalized fibonacci sequences we prove for certain rotation numbers that for every value of the coupling constant there exists an 0 such that the corresponding operator has purely continuous spectrum this result follows from uniform upper and lower bounds for the l norm of the solutions corresponding to energies from the spectrum of the operator

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

spectra of schreier graphs of grigorchuk s group and schroedinger operators with aperiodic order we study spectral properties of the laplacians on schreier graphs arising from grigorchuk s group acting on the boundary of the infinite binary tree we establish a connection between the action of g on its space of schreier graphs and a subshift associated to a non primitive substitution and relate the laplacians on the schreier graphs to discrete schroedinger operators with aperiodic order we use this relation to prove that the spectrum of the anisotropic laplacians is a cantor set of lebesgue measure zero we also use it to show absence of eigenvalues both almost surely and for certain specific graphs the methods developed here apply to a large class of examples

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

substitution hamiltonians with bounded trace map orbits we investigate discrete one dimensional schrodinger operators with aperiodic potentials generated by primitive invertible substitutions on a two letter alphabet we prove that the spectrum coincides with the set of energies having a bounded trace map orbit and show that it is a cantor set of zero lebesgue measure this result confirms a suggestion arising from a study of roberts and complements results obtained by bovier and ghez as an application we present a class of models exhibiting a purely singular continuous spectrum with probability one

substitution dynamical systems characterization of linear repetitivity and applications we consider dynamical systems arising from substitutions over a finite alphabet we prove that such a system is linearly repetitive if and only if it is minimal based on this characterization we extend various results from primitive substitutions to minimal substitutions this includes applications to random schrodinger operators and to number theory

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

some remarks on discrete aperiodic schrodinger operators we consider schrodinger operators onl2 with deterministic aperiodic potential and schrodinger operators on the l2 space of the set of vertices of penrose tilings and other aperiodic self similar tilings the operators onl2 fit into the formalism of ergodic random schrodinger operators hence their lyapunov exponent integrated density of states and spectrum are almost surely constant we show that they are actually constant the lyapunov exponent for one dimensional schrodinger operators with potential defined by a primitive substitution the integrated density of states and the spectrum in arbitrary dimension if the system is strictly ergodic we give examples of strictly ergodic schrodinger operators that include several kinds of almost periodic operators that have been studied in the literature for schrodinger operators on penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular penrose tiling under consideration

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

singularly continuous spectrum of a self similar laplacian on the half line we investigate the spectrum of the self similar laplacian which generates the so called pq random walk on the integer half line using the method of spectral decimation we prove that the spectral type of the laplacian is singularly continuous whenever p 12 this serves as a toy model for generating singularly continuous spectrum which can be generalized to more complicated settings we hope it will provide more insight into fibonacci type and other weakly self similar models

model updating via weighted reference basis with connectivity constraints abstract the paper considers the problem of updating an analytical model from experimental data using the reference basis approach in the general framework of the reference basis method certain quantities e g natural frequencies or modeshapes are considered to be completely accurate and the others are updated by solving a constrained optimization problem however the underlying structure known as connectivity existing in the model is not preserved and the method is not suited for parametric updating in this paper a method for introducing connectivity constraints into reference basis while maintaining its advantages is presented it brings the reference basis method closer to a broad class of updating methods that use parametric updating the notions of connectivity cost and parameterization cost are defined and used to obtain the best model for a given parameterization and to compare the outcomes of different parameterizations

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

quantum dynamics via complex analysis methods general upper bounds without time averaging and tight lower bounds for the strongly coupled fibonacci hamiltonian abstract we develop further the approach to upper and lower bounds in quantum dynamics via complex analysis methods which was introduced by us in a sequence of earlier papers here we derive upper bounds for non time averaged outside probabilities and moments of the position operator from lower bounds for transfer matrices at complex energies moreover for the time averaged transport exponents we present improved lower bounds in the special case of the fibonacci hamiltonian these bounds lead to an optimal description of the time averaged spreading rate of the fast part of the wavepacket in the large coupling limit this provides the first example which demonstrates that the time averaged spreading rates may exceed the upper box counting dimension of the spectrum

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

hierarchical structures in sturmian dynamical systems the paper is concerned with hierarchical stuctures in subshifis over a finite alphabet in particular we present a hierarchy based approach to sturmian systems this approach is then used to characterize the linearly repetitive sturmian systems among the sturmian systems by uniform positivity of certain weights more generally we discuss various bounds on weights and their relationship finally we combine hierarchies and bounds on a weights to study a thermodynamic formalism for lattice gas models associated to arbitrary linearly repetitive subshifts

continuity properties of the electronic spectrum of 1 d quasicrystals in this paper we consider operatorsh x defined onl 2 by h alpha x psi n sum limits m in mathbb z t m circ phi n alpha x psi n m where x x t m is in the algebra of bounded periodic functions on 2 generated by the characteristic functions of the sets phi n left alpha x in mathbb r 2 left left 1 alpha leqq x alpha bmod 1 right right right this class of hamiltonian includes the kohmoto model numerically computed by ostlund and kim where the potential is given by upsilon alpha x n lambda chi 1 alpha 1 x n alpha n in mathbb z x lambda alpha in mathbb r see b i s t we prove that the spectrum as a set ofh x varies continuously with respect to near each irrational for anyx we also show that the various strong limits obtained as converges to a rational numberp q describe either a periodic medium or a periodic medium with a localized impurity the corresponding spectrum has eigenvalues in the gaps and the right and left limits as p q do not coincide for the kohmoto model the results are obtained throughc algebra techniques

uniform spectral properties of one dimensional quasicrystals i absence of eigenvalues we consider discrete one dimensional schrodinger operators with sturmian potentials for a full measure set of rotation numbers including the fibonacci case we prove absence of eigenvalues for all elements in the hull

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

transport exponents of sturmian hamiltonians 2015 elsevier inc we consider discrete schrodinger operators with sturmian potentials and study the transport exponents associated with them under suitable assumptions on the frequency we establish upper and lower bounds for the upper transport exponents as an application of these bounds we identify the large coupling asymptotics of the upper transport exponents for frequencies of constant type we also bound the large coupling asymptotics uniformly from above for lebesgue typical frequency a particular consequence of these results is that for most frequencies of constant type transport is faster than for lebesgue almost every frequency we also show quasi ballistic transport for all coupling constants generic frequencies and suitable phases

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

almost periodic schrodinger operators along interval exchange transformations it is shown that schrodinger operators with potentials along the shift embedding of irreducible interval exchange transformations in a dense set have pure singular continuous spectrum for lebesgue almost all points of the interval such potentials are natural generalizations of the sturmian case

absence of localization in a class of schrodinger operators with quasiperiodic potential we prove that a class of discrete schrodinger operators with a quasiperiodic potential taking only a finite number of values exhibits purely continuous spectrum in particular they cannot have localized eigenvectors

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

construction of sturmian sequences it is well known that a sturmian sequence can be regarded as a rotation sequence or a balanced sequence in this paper by a rotation sequence we first construct a series of sequences with complexity kn 1 then from these sequences we reconstruct other sturmian sequences and discuss their relationships

fast convergent resummation algorithm and critical exponents of φ4 theory in three dimensions we develop an efficient algorithm for evaluating divergent perturbation expansions of field theories in the bare coupling constant gb for which we possess a finite number l of expansion coefficients plus two more information the knowledge of the large order behavior proportional to kk k gbk with a known growth parameter and the knowledge of the approach to scaling of the type c c gb with constants c c and a critical exponent of approach the latter information leads to an increase in the speed of convergence and a high accuracy of the results the algorithm is applied to the six and seven loop expansions for the critical exponents of o n symmetric 4 theories and the result for the critical exponent is compared with a recent satellite experiment

spectral and quantum dynamical properties of the weakly coupled fibonacci hamiltonian we consider the spectrum of the fibonacci hamiltonian for small values of the coupling constant it is known that this set is a cantor set of zero lebesgue measure here we study the limit as the value of the coupling constant approaches zero of its thickness and its hausdorff dimension we prove that the thickness tends to infinity and consequently the hausdorff dimension of the spectrum tends to one we also show that at small coupling all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly moreover for a sufficiently small coupling the sum of the spectrum with itself is an interval this last result provides a rigorous explanation of a phenomenon for the fibonacci square lattice discovered numerically by even dar mandel and lifshitz finally we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents

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

palindromic richness for languages invariant under more symmetries for a given finite group g consisting of morphisms and antimorphisms of a free monoid a we study infinite words with language closed under the group g we focus on the notion of g richness which describes words rich in generalized palindromic factors i e in factors w satisfying q w w for some antimorphism email protected g we give several equivalent descriptions which are generalizations of known characterizations of rich words in the terms of classical palindromes and show two examples of g rich words

absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian author s damanik d gorodetski a solomyak b abstract 2015 we prove for the square fibonacci hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants this is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact dimensional measures

singular continuous spectrum for the period doubling hamiltonian on a set of full measure we consider the discrete one dimensional schrodinger operator with potential generated by the period doubling substitution we show that for almost every element in the hull with respect to the unique ergodic measure on there are no eigenvalues combining this with a result proven by kotani we establish purely singular continuous spectrum on a set of full measure

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

dynamical upper bounds for one dimensional quasicrystals abstract following the killip kiselev last method we prove quantum dynamical upper bounds for discrete one dimensional schrodinger operators with sturmian potentials these bounds hold for sufficiently large coupling almost every rotation number and every phase

orthogonal polynomials on the unit circle with fibonacci verblunsky coefficients ii applications we consider cmv matrices with verblunsky coefficients determined in an appropriate way by the fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics in our first application we estimate the spreading rates of quantum walks on the line with time independent coins following the fibonacci sequence the estimates we obtain are explicit in terms of the parameters of the system in our second application we establish a connection between the classical nearest neighbor ising model on the one dimensional lattice in the complex magnetic field regime and cmv operators in particular given a sequence of nearest neighbor interaction couplings we construct a sequence of verblunsky coefficients such that the support of the lee yang zeros of the partition function for the ising model in the thermodynamic limit coincides with the essential spectrum of the cmv matrix with the constructed verblunsky coefficients under certain technical conditions we also show that the zeros distribution measure coincides with the density of states measure for the cmv matrix