phase space lattices with threefold symmetry this paper presents a semiclassical analysis of the spectrum of the hamiltonian ir cos p e cos ci v 3 5 cos f p a 5 which is a model for bloch electrons in a magnetic field the energy levels are determined to a first approximation by bohr sommerfeld quantisation because of the translational symmetry in phase space this leads to a lattice of infinitely degenerate states this degeneracy is lifted by tunnelling effects which can be described by an effective hamiltonian of the same form as that above but with different values of 0 and h the calculation of the effective hamiltonian therefore defines a renormalisation group transformation and it predicts that the spectrum has a complex recursive structure which is confirmed by numerical experiments the results complement earlier work on lattices with fourfold symmetry in which a similar type of spectrum occurs the greater complexity of the problem in lattices with threefold symmetry necessitated developing simpler canonically invariant methods of analysis

an exact renormalisation group for bloch electrons in a magnetic field bloch electrons in a magnetic field can be modelled by a one dimensional hamiltonian h h x p which is periodic in x and p when beta the ratio of planck s constant h to the area of a unit cell is a rational number p q the spectrum is a set of q bloch bands if beta is perturbed away from this rational value the spectrum becomes a cantor set but it is still possible to define a set of generalised bloch waves which form a complete basis set for a given band projecting these states into the band and taking matrix elements of the hamiltonian leads to a new effective hamiltonian h1v x p and planck constant h v describing the spectrum of the vth band since degrees of freedom have been eliminated by projecting into a band this is a renormalisation group rg transformation the rg transformation can be iterated indefinitely if the measure of the spectrum is zero or if beta is a liouville number the measure of the spectrum vanishes if h x p has centres of threefold or fourfold symmetry in the phase plane and the rg transformation explains the hierarchical structure of the spectrum observed in these cases

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

fractional sturm liouville eigen problems we first consider a regular fractional sturm liouville problem of two kinds rfslp i and rfslp ii of order 0 2 the corresponding fractional differential operators in these problems are both of riemann liouville and caputo type of the same fractional order 2 0 1 we obtain the analytical eigensolutions to rfslp i ii as non polynomial functions which we define as jacobi poly fractonomials these eigenfunctions are orthogonal with respect to the weight function associated with rfslp i ii subsequently we extend the fractional operators to a new family of singular fractional sturm liouville problems of two kinds sfslp i and sfslp ii we show that the primary regular boundary value problems rfslp i ii are indeed asymptotic cases for the singular counterparts sfslp i ii furthermore we prove that the eigenvalues of the singular problems are real valued and the corresponding eigenfunctions are orthogonal in addition we obtain the eigen solutions to sfslp i ii analytically also as non polynomial functions hence completing the whole family of the jacobi poly fractonomials in numerical examples we employ the new poly fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results

an exact effective hamiltonian for a perturbed landau level considers the effect of a scalar potential v x y on a landau level in two dimensions an exact effective hamiltonian is derived which describes the effect of the potential on a single landau level expressed as a power series in v ec where ec is the cyclotron energy the effective hamiltonian can be represented as a function h x p in a one dimensional phase space the function h x p resembles the potential v x y when the area of a flux quantum is much smaller than the square of the characteristic length scale of v then h approximately v also h x p retains the translational and rotational symmetries of v x y exactly but reflection symmetries are not retained beyond the lowest order of the perturbation expansion

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

general quantization of canonical maps on a two torus canonical maps on a two torus in phase space are quantized under most general conditions recent results by keating et al 1999 nonlinearity 12 579 are thus fully extended in two directions a the translational component of a general canonical map is included in the quantization b all values of planck s constant consistent with the toral boundary conditions bcs are considered generically these values are rational numbers whose numerator must satisfy a number theoretical condition besides the condition on planck s constant the quantization is possible only for particular allowed bcs on the torus the general equation determining these bcs is derived allowed bcs may not exist in some cases representative examples are the irrational skew translations and kronecker maps exact versions of egorov s theorem are shown to hold under some conditions composition and representation properties of the quantization scheme are studied

optimal entanglement criterion for mixed quantum states we develop a strong and computationally simple entanglement criterion the criterion is based on an elementary positive map phi which operates on state spaces with even dimension n or 4 it is shown that phi detects many entangled states with a positive partial transposition ppt and that it leads to a class of optimal entanglement witnesses this implies that there are no other witnesses which can detect more entangled ppt states the map phi yields a systematic method for the explicit construction of high dimensional manifolds of bound entangled states

on the multifractal spectrum of the fibonacci chain we consider the multifractal properties of a quasiperiodic tight binding hamiltonian where the hopping elements are arranged according to the fibonacci chain by using the trace map approach and an assumption relating the cycles of the trace map to the integrated density of states it is shown that the maximal scaling of the spectrum does not always occur at the edges or the centre of the spectrum which is confirmed by numerical simulations a good description of the multifractal spectrum is obtained by applying the 3z model which was recently developed for the harper model

application of differential transformation to eigenvalue problems using differential transformation to solve eigenvalue problems is introduced in this study first differential transformation is introduced briefly second taking differential transformation of the strum liouville problem a set of difference equations is derived doing some simple mathematics operations on these operations we can get any ith eigenvalue and eigenfunction at a time finally two eigenvalue problems are solved by the present method the calculated results are compared closely with the results obtained by another analytical method

fractional sturm liouville problem in this paper we define some fractional sturm liouville operators fslos and introduce two classes of fractional sturm liouville problems fslps namely regular and singular fslp the operators defined here are different from those defined in the literature in the sense that the operators defined here contain left and right riemann liouville and left and right caputo fractional derivatives for both classes we investigate the eigenvalue and eigenfunction properties of the fslos in the class of regular fslps we discuss two types of fslps as an operator for the class of singular fslps we introduce a fractional legendre equation fle and discuss its solution it is shown that the legendre polynomials resulting from an fle are the same as those obtained from the integer order legendre equation however the eigenvalues of the two equations differ using the legendre integral transform we demonstrate some applications of our results by solving two fractional differential equations one ordinary and the other partial it is our hope that this paper will initiate new research in the area of fslps and many of its variations

fundamental solutions and eigenvalues of one boundary value problem with transmission conditions in this study we consider one boundary value problem type sturm liouville with eigenparameter dependent boundary conditions and with transmission conditions at one inner point of considered finite interval we give operator theoretical formulation construct fundamental solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions of the considered problem

discontinuous fractional sturm liouville problems with transmission conditions abstract the main purpose of this study is the investigation of discontinuous sturm liouville problem with fractional derivatives we give an operator theoretic framework of the problem under consideration namely we define an operator a in the hilbert space l 2 1 1 the eigenvalues and corresponding eigenfunctions of which coincide with the eigenvalues and corresponding eigenfunctions of the boundary value problem respectively then we establish the characteristic function and prove that the eigenvalues of the considered problem coincide with the roots of this characteristic function

a new method for solving singular initial value problems in the second order ordinary differential equations singular initial value problems linear and nonlinear homogeneous and nonhomogeneous are investigated by using adomian decomposition method the solutions are constructed in the form of a convergent series a new general formula is established the approach is illustrated with few examples

the local structure of the spectrum of the one dimensional schrodinger operator let h v frac d text 2 dt text 2 q t omega be an one dimensional random schrodinger operator in 2 v v with the classical boundary conditions the random potentialq t has a formq t f x t wherex t is a brownian motion on the compact riemannian manifoldk andf k r 1 is a smooth morse function mathop min limits k f 0 let n v delta sum limits ei v in delta 1 where 0 e i v are the eigenvalues ofh v the main result theorem 1 of this paper is the following ifv e 0 0 k z anda 0 a is a fixed constant then p left n v left e 0 frac a 2v e 0 frac a 2v right k right xrightarrow v to infty e an e 0 an e 0 k k wheren e 0 is a limit state density ofh v v this theorem mean that there is no repulsion between energy levels of the operatorh v v the second result theorem 2 describes the phenomen of the repulsion of the corresponding wave functions

generalized wannier function and renormalization of harper s equation harper s equation a model for bloch electrons in a magnetic field has a band spectrum when the dimensionless magnetic field beta is a rational number p q this paper considers the definition of generalized wannier functions which can be used to represent the bloch bands of the spectrum of the rational harper equation by means of a von neumann lattice one representation of these bloch bands can be extended to the irrational case and taking matrix elements in this basis leads to a renormalization group transformation acting on the hamiltonian the results in the present paper considerably extend a previous analysis of this renormalization group transformation in that the formalism is suitable for systematic calculations of the renormalized hamiltonian and that the transformation preserves the rotational symmetry of the harper hamiltonian in phase space

concurrence of mixed bipartite quantum states in arbitrary dimensions we derive a lower bound for the concurrence of mixed bipartite quantum states valid in arbitrary dimensions as a corollary a weaker purely algebraic estimate is found which detects mixed entangled states with a positive partial transpose

variational methods for the fractional sturm liouville problem abstract this article is devoted to the regular fractional sturm liouville eigenvalue problem by applying the methods of fractional variational analysis we prove the existence of a countable set of orthogonal solutions and corresponding eigenvalues moreover we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional

von neumann lattices of wannier functions for bloch electrons in a magnetic field the problem of bloch electrons in a magnetic field in two dimensions can be reduced to a one dimensional problem with a hamiltonian h that is a periodic function of x and p wannier functions can be defined for the sub bands of the spectrum of this effective hamiltonian when the chern class quantized hall conductance integer of the sub band is zero the weyl wigner formalism can be used to represent these wannier functions by a von neumann lattice it is shown how this von neumann lattice of wannier functions can be defined for irrational as well as rational magnetic fields an important benefit from using the weyl wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective hamiltonian in phase space it is shown how the wannier functions can be defined so that their wigner functions have the point symmetries of the effective hamiltonian an example of how these results can prove useful is given if we take matrix elements of the hamiltonian between the wannier states of a sub band we derive a new effective hamiltonian describing this sub band which is again a periodic function of coordinate and momentum operators since by projecting onto a sub band we have also reduced the number of degrees of freedom this operation is a renormalization group transformation it is shown that the symmetry of the new effective hamiltonian in phase space is the same as that of the original one this preservation of symmetry helps to explain some unusual properties of the spectrum when the hamiltonian has fourfold symmetry

equation of motion for multiqubit entanglement in multiple independent noisy channels we investigate the possibility and conditions to factorize the entanglement evolution of a multiqubit system passing through multi sided noisy channels by means of a lower bound of concurrence lbc as an entanglement measure we derive an explicit formula of the lbc evolution of the n qubit generalized greenberger horne zeilinger gghz state under some typical noisy channels based on which two kinds of factorizing conditions for the lbc evolution are presented in this case the time dependent lbc can be determined by a product of initial lbc of the system and the lbc evolution of a maximally entangled gghz state under the same multi sided noisy channels we analyze the realistic situations where these two kinds of factorizing conditions can be satisfied in addition we also discuss the dependence of entanglement robustness on the number of qubits and that of the noisy channels

a new modification of the adomian decomposition method for linear and nonlinear operators in this paper we present an efficient modification of the adomian decomposition method that will facilitate the calculations we then conduct a comparative study between the new modification and the modified decomposition method the study is conducted through illustrative examples the new modification introduces a promising tool for many linear and nonlinear models

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

hofstadter rules and generalized dimensions of the spectrum of harper s equation we consider the harper model which describes two dimensional bloch electrons in a magnetic field for irrational flux through the unit cell the corresponding energy spectrum is known to be a cantor set with multifractal properties in order to relate the maximal and minimal fractal dimension of the spectrum of harper s equation to the irrational number involved we combine a refined version of the hofstadter rules with results from semiclassical analysis and tunnelling in phase space for quadratic irrationals with continued fraction expansion the maximal fractal dimension exhibits oscillatory behaviour as a function of n which can be explained by the structure of the renormalization flow the asymptotic behaviour of the minimal fractal dimension is given by as the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket our results imply that the time evolution of high order moments is sensible to the parity of n

on the total bandwidth for the rational harper s equation in the last years several contributions have been done around the total bandwidth of the spectrum for the harper s operator in particular an interesting conjecture has been proposed by thouless which gives also strong convincing arguments for the proof in special cases on the other hand in the study of the cantor structure of the spectrum b helffer and j sjostrand have justified an heuristic semiclassical approach proposed by m wilkinson the aim of this article is to explain how one can use the first step of this approach to give a rigorous semi classical proof of the thouless formula in some of the simplest cases we shall also indicate how one can hope with more effort to prove rigorously recent results of last and wilkinson on the same conjecture

energy spectra for one dimensional quasiperiodic potentials bandwidth scaling mapping and relation with local isomorphism energy spectra for one dimensional tight binding models with two types of quasiperiodic potentials are studied for which the incommensurability is characterised by quadratic irrationals one is the step potential model for which the structure is a generalised fibonacci chain for special structures scaling properties of the spectrum are found numerically a critical index delta for the total bandwidth is determined after deriving recursion relations for the trace of transfer matrices it is shown that generalised fibonacci chains have the same energy spectrum if and only if they are locally isomorphic the other potential is sinusoidal for which the critical index delta is determined at the critical point

symmetry properties of exact energy solutions to the harper equation and related q normalizations proofs are given that exact solutions to the wavefunctions of the harper equation can be established in an explicit manner by resorting to three term recurrence relations implied by a q calculus approach proposed previously the q normalization of wavefunctions resulting in the appearance of peaks is discussed the exact q 6 energy solution has also been derived and analysed in some more detail there are reasons to say that fractal structures concerning band and gap distributions have to be accounted for if q 6

vortex structure and characterization of quasiperiodic functions quasiperiodic functions qpfs are characterized by their full vortex structure in one unit cell this characterization is much finer and more sensitive than the topological one given by the total vorticity per unit cell the chern index it is shown that qpfs with an arbitrarily prescribed vortex structure exist by constructing explicitly such a standard qpf two qpfs with the same vortex structure are equivalent in the sense that their ratio is a function which is strictly periodic nonvanishing and at least continuous a general qpf can then be approximately reconstructed from its vortex structure on the basis of the standard qpf and the equivalence concept as another application of this concept a simple method is proposed for calculating the quasiperiodic eigenvectors of periodic matrices possible applications to the quantum chaos problem on a phase space torus are briefly discussed

first order correction to the renormalization of the phase space lattice hamiltonian we develop a renormalization group analysis for a hamiltonian which is a periodic function of and this is a model for bloch electrons in a magnetic field and includes harper s model as a special case the eigenfunctions are bloch waves when the ratio of to the area of the unit cell is a rational number p q the renormalization procedure produces an effective hamiltonian for a subset of the spectrum which can be expanded in the zeroth order term has been obtained in earlier papers here we obtain the first order term of this expansion in terms of the q dimensional vectors which define the rational bloch states k and are bloch wavevectors the effective hamiltonian is not invariant under gauge transformations multiplying the vector we show that the infinitesimal gauge transformation is equivalent to evolving under a gauge hamiltonian obtained to lowest order in by quantizing

a sum rule for the dispersion relations of the rational harper equation the authors derive an exact sum rule satisfied by the dispersion relations of the commensurable case of harper s equation 1955 they use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to thouless 1990 concerning the total bandwidth when the commensurability is a high order rational

total bandwidth for harper equation correction to renormalization analysis this paper studies the correction to the effective hamiltonian of the harper equation in respect of the total bandwidth the effective hamiltonian describes the further splitting from the spectrum of a lower order fraction when the spectrum of a higher order fraction is considered the total bandwidth for a split sub band is given by the thouless scaling law modified by the curvature of the effective hamiltonian at its saddle points in this paper it is shown that the leading term of the effective hamiltonian is exactly solvable moreover the correction to the curvature of the effective hamiltonian is obtained numerically and the explicit analytical expression is also found this gives the second sum rule which is of importance in deriving the thouless scaling law for a generic fraction

a six fold rotation operator for the wannier functions of the phase space lattice hamiltonian we consider a set of localized wannier functions which can be used to describe the bloch bands of one dimensional hamiltonians whose associated weyl function is periodic under a hexagonal lattice of translations in phase space these hamiltonians can be used to describe electrons on two dimensional hexagonal lattices penetrated by a magnetic field the effect on the wannier functions of a rotation of the phase plane is considered the resulting transformation is found to be surprisingly complicated when the quantized hall conductance integer for the bloch band is non zero

level statistics in a quasiperiodic system the authors examine the level statistics of harper s equation a simple model of a one dimensional quasiperiodic system through the distribution of the spacings between adjacent eigenvalues each normalised by the average local density of states they present exact results for the distribution for both small and large amplitude alpha of the incommensurate potential showing it has a simple form in each of these limits they show numerically that these distributions are preserved as alpha approaches its critical value 2 from each side and also that at the critical point the distribution of the normalised spacings has a distinct but simple form