scaling and eigenstates for a class of one dimensional quasiperiodic lattices numerical and analytical calculations of the electronic properties of a one dimensional quasiperiodic lattice are presented the lattice consists of two types of bonds a and b whose distribution follows the recursion formula sl 1 sml 1snl where m n 1 2 3 and sl is a building block sequence for l or 2 an exact evaluation of the wavefunctions yields the scaling for several pairs of values of m and n numerical calculations of the wavefunction and the resistance reveal a rich self similar structure as well as localisation for one of the quasiperiodic sequences studied

existence of quasiperiodic solutions and littlewood s boundedness problem of super linear impact oscillators abstract so far most application of kolmogorov arnold moser kam theory has been restricted to smooth dynamical systems in this paper it is shown by a series of transformations that how kam theory can be used to analyze the dynamical behavior of duffing type equations with impact the analysis is carried out for the example 0 1 x x 2 n 1 p t for x t 0 x t 0 x t 0 x t 0 if x t 0 0 with p c 5 being periodic we prove that all solutions are bounded and that there are infinitely many periodic and quasiperiodic solutions in this case

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

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

on the characteristic polynomials of fibonacci chains special diatomic linear chains with elastic nearest neighbour interaction and the two masses distributed according to the binary fibonacci sequence are studied

boundedness of solutions for a class of reversible systems in this paper we are concerned with the boundedness of all solutions for the following differential systemx email protected p y e t y email protected q x f x f p y f x e t where f r s s r 2s r 1 f x is odd e t c 4 r z is even max 1 p 2 p 1

on littlewood s boundedness problem for relativistic oscillators with anharmonic potentials abstract in this paper we are concerned with the littlewood s boundedness problem and the existence of quasi periodic solutions for the nonlinear differential equation of the motions of relativistic oscillators with anharmonic potentials d d t x 1 x 2 x 1 x p t where the 1 periodic function p t is a smooth function and is a positive constant

on littlewood s boundedness problem for relativistic oscillators with singular potentials abstract in this paper we study the following nonlinear differential equations of motions of relativistic oscillators with singular potentials d d t x 1 x 2 v x p t where v is a singular potential and p is a 1 periodic function we will prove the boundedness of all solutions and the existence of infinitely many quasi periodic solutions via moser s twist theorem

boundedness of solutions for sublinear reversible systems in this paper the boundedness of all the solutions for the second order differential equationx f x x email protected p x e t is studied where f x is in an odd c 3 function f p u u p 2u 1

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

purely singular continuous spectrum for cmv operators generated by subshifts we prove uniform absence of point spectrum for cmv operators corresponding to the period doubling subshift we also prove almost sure absence of point spectrum for cmv operators corresponding to a class of sturmian subshifts lastly we prove almost sure absence of point spectrum for cmv operators corresponding to some subshifts generated by a coding of a rotation

boundedness of solutions for nonlinear reversible system at resonance abstract in this paper we are concerned with the problem of boundedness of solutions for the following nonlinear p laplacian p x p 1 n p p x f x p x g x e t where p u u p 2u p 2 for some n n f g c r are odd functions e t c r is odd and 2 p periodic

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

long time stability estimates for the non periodic littlewood boundedness problem we consider the nonlinear oscillator equation for 3 and non periodic forcing p for any solution x which is unbounded in the phase plane it is shown that for e 0 small enough there is a solution x e which is bounded in the phase plane and such that the actions of x e and x remain close on a time interval of length e 2 further precise information is available on the location of the zeros of x e in addition it is possible to take the bounded solution x e from a fixed countable family of such solutions

symbolic dynamics for the renormalization map of a quasiperiodic schrodinger equation a rigorous analysis is given of the dynamics of the renormalization map associated to a discrete schrodinger operatorh onl 2 defined byh n n 1 n 1 vf n n wherev is a real parameter f is a certain discontinuous period 1 function and sigma left 1 sqrt 5 right mathord left vphantom left 1 sqrt 5 right 2 right kern nulldelimiterspace 2 is the golden mean the renormalization map forh is a diffeomorphism t of 3 preserving a cubic surfaces v forv 8 we prove that the non wandering set of the restriction oft tos v is a hyperbolic set on whicht is conjugate to a subshift on six symbols it follows from results in dynamical systems theory that the optimally approximating periodic operators toh have spectra which obey a global scaling law we also define a set which we call the pseudospectrum of the operatorh we prove it to be a cantor set of measure zero and obtain bounds on its hausdorff dimension it is an open question whether the pseudospectrum coincides with the spectrum ofh

fractal interpolation surfaces derived from fractal interpolation functions based on the construction of fractal interpolation functions a new construction of fractal interpolation surfaces on arbitrary data is presented and some interesting properties of them are proved finally a lower bound of their box counting dimension is provided

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

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

boundedness of solutions for duffing s equations with semilinear potentials abstract in this paper we will prove the boundedness of all the solutions and the existence of quasiperiodic solutions for the second order scalar differential equation x arctan x ee t where e is a small parameter and the 1 periodic function e t is a smooth function

boundedness of solutions for reversible systems in this paper we study the boundedness of all the solutions for the following second order nonlinear differential equation p x f x p x g x e t ddtwhere p u u p 2u p 1 f g c r 0 and odd e c r z is odd g is subquasilinear sgn x g x g x p x 0 as x

invariant tori for semilinear reversible systems abstract in this paper we consider the problem of the boundedness of all solutions for the oscillator x f x x 2 x g x p t where 0 and n g x is bounded and p t p t 2 since the system in general is non hamiltonian we have to introduce reversibility assumptions to apply the twist theorem for reversible maps under some suitable conditions we then obtain the existence of invariant tori and thus the boundedness of all solutions

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

critical points for finite fibonacci chains of point delta interactions and orthogonal polynomials for a one dimensional schrodinger operator with a finite number n of point delta interactions with a common intensity the parameters are the intensity the n 1 intercenter distances and the mass critical points are points in the parameters space of the hamiltonian where one bound state appears or disappears the study of critical points for hamiltonians with point delta interactions arranged along a fibonacci chain is shown to be closely related to the study of the so called fibonacci operator a discrete one dimensional schrodinger type operator which occurs in the context of tight binding hamiltonians these critical points are the zeros of orthogonal polynomials previously studied in the context of special diatomic linear chains with elastic nearest neighbor interaction properties of the zeros location asymptotic behavior gaps are investigated the perturbation series from the solvable periodic case is determined the measure which yields orthogonality is investigated numerically from the zeros it is shown that the transmission coefficient at zero energy can be expressed in terms of the orthogonal polynomials and their associated polynomials in particular it is shown that when the number of point delta interactions is equal to a fibonacci number minus 1 i e when the intervals between point delta interactions form a palindrome all the fibonacci chains at critical points are completely transparent at zero energy

error analysis for bivariate fractal interpolation functions generated by 3 d perturbed iterated function systems based on a determined 3 d iterated function system ifs we introduce a perturbed ifs in r 3 the attractor of the perturbed ifs is the graph of a bivariate fractal interpolation function fif that interpolates arbitrarily given data on rectangular grids of r 2 we consider the error problem between the fif generated by the perturbed ifs and the fif generated by the original ifs an explicit relation of the difference between the two bivariate fifs is presented furthermore we investigate the error of moment integrals of the two fifs an upper bound estimate for the error of moments is obtained

boundedness for sublinear duffing equations with time dependent potentials in this paper we prove a sufficient and necessary condition for the boundedness of all solutions for the sublinear equation x e t x 1x p t where 0 1 p t and e t are smooth 1 periodic functions

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

ultradiffusion the relaxation of hierarchical systems the authors show that the dynamics of relaxation in hierarchical structures leads to an anomalous decay process which they term ultradiffusion using renormalisation group techniques they explicitly calculate the behaviour of the autocorrelation function in a one dimensional hierarchical system and show its universal power law decay as a function of temperature the authors also demonstrate analytically the emergence of a hierarchy of time scales this phenomenon is relevant to systems ranging from macromolecules to computing structures

on the spectrum of operator families on discrete groups over minimal dynamical systems it is well known that given an equivariant and continuous in a suitable sense family of selfadjoint operators in a hilbert space over a minimal dynamical system the spectrum of all operators from that family coincides as shown recently similar results also hold for suitable families of non selfadjoint operators in ell p mathbb z here we generalize this to a large class of bounded linear operator families on banach space valued ell p spaces over countable discrete groups we also provide equality of the pseudospectra for operators in such a family a main tool for our analysis are techniques from limit operator theory

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

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

the measure of the orthogonal polynomials related to fibonacci chains the periodic case the spectral measure for the two families of orthogonal polynomial systems related to periodic chains with n particle elementary unit and nearest neighbour harmonic interaction is computed using two different methods the interest is in the orthogonal polynomials related to fibonacci chains in the periodic approximation the relation of the measure to appropriately defined green s functions is established

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

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

missing moment and perturbative methods for polynomial iterated function systems abstract an iterated function system ifs over a compact metric space x is defined by a set of contractive maps wi x x i 1 n with associated nonzero probabilities pi 0 p i 1 the parallel action of the maps defines a unique compact invariant attractor set a x which supports an invariant measure and which is balanced with respect to the pi for linear w i on x r the invariance of yields a relation between the moments gn nd which permits their recursive computation from the initial value g0 1 for nonlinearwi however the moment relations are incomplete and do not permit a recursive computation this paper describes two methods of obtaining accurate estimates of the moments when the ifs maps wi are polynomials i application of the necessary hausdorff conditions on the gi to obtain convergent upper and lower bounds and ii a perturbation expansion approach the methods are applied to some model problems

the fractal dimensions of the spectrum of sturm hamiltonian abstract let 0 1 be irrational and 0 a 1 a 2 be the continued fraction expansion of let h v be the sturm hamiltonian with frequency and coupling v v be the spectrum of h v the fractal dimensions of the spectrum have been determined by fan liu and wen 2011 8 when a n n 1 is bounded the present paper will treat the most difficult case i e a n n 1 is unbounded we prove that for v 24 dim h v s v and dim b v s v where s v and s v are lower and upper pre dimensions respectively by this result we determine the fractal dimensions of the spectrums for all sturm hamiltonians we also show the following results s v and s v are lipschitz continuous on any bounded interval of 24 the limits s v ln v and s v ln v exist as v tends to infinity and the limits are constants only depending on s v 1 if and only if lim sup n a 1 a n 1 n which can be compared with the fact s v 1 if and only if lim inf n a 1 a n 1 n liu and wen 2004 13

cantor spectrum and singular continuity for a hierarchical hamiltonian we study the spectrum of the hamiltonianh onl2 given by h n n 1 n 1 v n n with the hierarchical ultrametric potentialv 2m 2l 1 1 rm 1 r corresponding to 1 2 and 3 dimensional coulomb potentials for 0 1 respectively in a suitably chosen valuation metric we prove that the spectrum is a cantor set and gaps open at the eigenvaluesen 1 en 2 en 2n 1 of the dirichlet problemh e 0 2n 0 n 1 in the gap opening aten k the integrated density of states takes on the valuek 2n the spectrum is purely singular continuous forr 1 when the potential is unbounded and the lyapunov exponent vanishes in the spectrum the spectrum is purely continuous forr 1 in h 2 2 and 0 here but one cannot exclude the presence of eigenvalues near the border of the spectrum we also propose an explicit formula for the green s function

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

a variational method for the existence of bounded solutions of a sublinear forced oscillator we prove that for every bounded and measurable forcing p t the differential equation ddot u u 1 3 p t has bounded solutions with arbitrarily large amplitude in general it is not possible to say that all solutions are bounded as shown by an example due to littlewood the proof is based on a variational method which can be seen as a dual version of nehari s method for boundary value problems on compact intervals

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

boundedness of solutions of nonlinear p laplacian in this paper we prove the boundedness of all the solutions for the p laplacian f p x g x e t where f p u u p 2u p 2 g satisfies some subquasilinear conditions and e t c 6 r z

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

boundedness problem for solutions of semilinear asymmetric equations in this paper we consider the boundedness problem for the solutions of the following semilinear differential equation f p x p 1 email protected p x email protected f p x g x e t where f p u u p 2u p 2 a b are positive constants satisfying a 1 p email protected 1 p email protected 1 g x c r is semilinear and e t c r z

inverse problems in fractal construction moment method solution abstract the inverse problem for several classes of fractals in one and two dimensions can be profitably solved via the application of the classical theory of moments this new perspective leads to important results on the reconstruction process when only a finite number of moments of the fractal measures are known while discussing the rigorous aspects of the theory we also implement it via efficient numerical algorithms

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

existence of quasi periodic solutions and littlewood s boundedness problem of sub linear impact oscillators abstract in this paper it is shown by a series of transformations that how moser s invariant curve theorem can be used to analyze the dynamical behavior of sub linear duffing type equations with impact we prove that all solutions are bounded and that there are infinitely many periodic and quasi periodic solutions in this impact case

dynamics in coupled oscillators with recurrent random forcing a pde approach the current paper is devoted to the study of coupled oscillators with recurrent random forcing special attention is given to the solutions having the same recurrence randomness as that of the forcing recurrent random solutions for short by embedding coupled oscillators into coupled parabolic equations it establishes a general theorem on the existence of recurrent random solutions it also finds conditions under which such solutions are unique when the recurrent forcing is actually quasi periodic or almost periodic recurrent solutions are refereed to as quasi periodic or almost periodic solutions in a weak sense and they are quasi periodic or almost periodic in the classical sense under the uniqueness conditions in addition applications of the general theory to coupled duffing type oscillators and josephson junctions are considered and the results obtained extend several existing ones for quasi periodic duffing oscillators

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

fractal interpolation surfaces and a related 2 d multiresolution analysis abstract we consider the theory of fractal interpolation surfaces algorithms are given allowing the construction of these surfaces over polygonal regions with arbitrary interpolation points a class of invariant measures supported on these surfaces is introduced and discussed as is the fractal dimension of some simple surfaces using these surfaces we construct a sequence of nested subspaces forming a generalized multiresolution analysis

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

existence of quasiperiodic solutions and littlewood s boundedness problem of duffing equations with subquadratic potentials where p t is periodic has been extensively investigated due to its relevance in applications much work has been carried out concerning the existence of periodic solutions see 3 8 also for further references in this paper we pay attention to more complicated solutions and the lagrangian stability problem proposed by littlewood 5 as remarked in 5 it has been conjectured that all solutions of eq 1 1 are bounded if either

spectral continuity for aperiodic quantum systems i general theory abstract how does the spectrum of a schrodinger operator vary if the corresponding geometry and dynamics change is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures in this work a positive answer is provided using the rather general setting of groupoid c algebras a characterization of the convergence of the spectra by the convergence of the underlying structures is proved in order to do so the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles with this at hand magnetic schrodinger operators on dynamical systems or delone systems fall into this unified setting various approximations used in computational physics like the periodic or the finite cluster approximations are expressed through the tautological groupoid which provides a universal model for fields of groupoids the use of the hausdorff topology turns out to be fundamental in understanding why and how these approximations work

a natural class of generalized fibonacci chains in this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typid properties with the quasiperiodic fibonacci chain for a subclass we explicitly con vct the atomic surface moreover scaling properties of the energy specmm are discussed in relation to the dynamics of trace maps

dimensions of multitype moran sets with lower limit of the contractions being zero abstract multitype moran sets were studied by liu and wen 2005 9 under the assumption of primitivity they proved that for any multitype moran set e with a positive lower bounded condition on contracting ratios dim h e s s dim p e dim b e where s and s are the lower and upper pre dimension according to the natural coverings in this paper we permit that the lower limit of contraction ratios is zero and study the two classes of them for one of the two by assuming some stronger condition than primitivity we show that the above formula still holds for the other under no assumption of primitivity we prove that dim h e s resp dim b e s by assuming a mild condition and show that the above formula also holds provided primitivity and these two mild conditions are satisfied

boundedness for sublinear reversible systems with a nonlinear damping and periodic forcing term abstract in this paper we are concerned with the sublinear reversible systems with a nonlinear damping and periodic forcing term x f x g x x 1 x p t where f x p t are odd functions p t is smooth 1 periodic function and 0 is a constant a sufficient and necessary condition for the boundedness of all solutions of the above equation is established moreover we show the existence of aubry mather sets as well

the density of states for almost periodic schrodinger operators and the frequency module a counter example we exhibit an example of a one dimensional discrete schrodinger operator with almost periodic potential for which the steps of the density of states do not belong to the frequency module this example is suggested by thek theory 3

escher like quasiperiodic heterostructures quasiperiodic heterostructures present unique structural electronic and vibrational properties connected to the existence of incommensurate periods we go beyond previous schemes such as fibonacci or thue morse based on substitutional sequences by introducing construction rules generated by tessellations of the unit disc by regular polygons we explore some of the properties exhibited by these systems

invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term abstract in this paper the boundedness of all solutions of the oscillator x f x x 2 x x p t is discussed where is a positive constant f x y x and p t are smooth functions and p t p t 2

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

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

almost sure frequency independence of the dimension of the spectrum of sturmian hamiltonians we consider the spectrum of discrete schrodinger operators with sturmian potentials and show that for sufficiently large coupling its hausdorff dimension and its upper box counting dimension are the same for lebesgue almost every value of the frequency

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

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

almost periodic one dimensional systems the schrodinger equation with potential v x 2 1cos x 2 2cos x is considered its solution reduces to the problem of finding the eigenvectors of a matrix the eigenvalues of this matrix show a band structure which is very sensitive to the value of the parameter the solutions of the schrodinger equation are presented and their physical meaning is discussed the potential v x has a complex multiwell structure and quantum tunneling occurs the accuracy of all approximations is carefully studied

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

boundedness of solutions for semilinear duffing equations in this paper we are concerned with the boundedness of all the solutions and the existence of quasi periodic solutions for some semilinear duffing equationx g x e t wheree t is a smooth 2 periodic function and0 g x x k

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

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