nonlinear fractional dynamics on a lattice with long range interactions abstract a unified approach has been developed to study nonlinear dynamics of a 1d lattice of particles with long range power law interaction a classical case is treated in the framework of the generalization of the well known frenkel kontorova chain model for the non nearest interactions quantum dynamics is considered following davydov s approach for molecular excitons in the continuum limit the problem is reduced to dynamical equations with fractional derivatives resulting from the fractional power of the long range interaction fractional generalizations of the sine gordon nonlinear schrodinger and hilbert schrodinger equations have been found there exists a critical value of the power s of the long range potential below the critical value s 3 s 1 2 we obtain equations with fractional derivatives while for s 3 we have the well known nonlinear dynamical equations with space derivatives of integer order long range interaction impact on the quantum lattice propagator has been studied we have shown that the quantum exciton propagator exhibits transition from the well known gaussian like behavior to a power law decay due to the long range interaction a link between 1d quantum lattice dynamics in the imaginary time domain and a random walk model has been discussed

map of discrete system into continuous continuous limits of discrete systems with long range interactions are considered the map of discrete models into continuous medium models is defined a wide class of long range interactions that give the fractional equations in the continuous limit is discussed the one dimensional systems of coupled oscillators for this type of long range interactions are considered the discrete equations of motion are mapped into the continuum equation with the riesz fractional derivative

fractional variations for dynamical systems hamilton and lagrange approaches fractional generalization of an exterior derivative for calculus of variations is defined the hamilton and lagrange approaches are considered fractional hamilton and euler lagrange equations are derived fractional equations are obtained by fractional variation of lagrangian and hamiltonian that have only integer derivatives

spectral statistics for the dirac operator on graphs we determine conditions for the quantization of graphs using the dirac operator for both two and four component spinors according to the bohigas giannoni schmit conjecture for such systems with time reversal symmetry the energy level statistics are expected in the semiclassical limit to correspond to those of random matrices from the gaussian symplectic ensemble this is confirmed by numerical investigation the scattering matrix used to formulate the quantization condition is found to be independent of the type of spinor we derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation

semiclassical form factor for chaotic systems with spin we study the properties of the two point spectral form factor for classically chaotic systems with spin in the semiclassical limit with a suitable semiclassical trace formula as our principal tool to this end we introduce a regularized form factor and discuss the limit in which the so called diagonal approximation can be recovered the incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics provided this skew product is mixing we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory

beyond the heisenberg time semiclassical treatment of spectral correlations in chaotic systems with spin 1 2 the two point correlation function of chaotic systems with spin 1 2 is evaluated using periodic orbits the spectral form factor for all times thus becomes accessible equivalence with the predictions of random matrix theory for the gaussian symplectic ensemble is demonstrated a duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

multifractality of one electron eigen states in 1d disordered long range models abstract one investigates the spatial structure of one electron eigenstates for the one dimensional anderson model with long range off diagonal disorder h n m 1 1 n m where a b means a uniform random distribution between a and b two cases are considered according to the choices for the on site energy e n namely e n 0 and e n 0 1 for 1 all states are critical and the multifractal spectra f is computed for the states near the center of the band the level spacing distribution function is also obtained and its behavior for both small and large separations are studied

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

simple strong glass forming models mean field solution with activation we introduce simple models inspired by previous models for froths and covalent glasses with trivial equilibrium properties but dynamical behaviour characteristic of strong glass forming systems these models are also a generalization of backgammon or urn models to a non constant number of particles where entropic barriers are replaced by energy barriers allowing for the existence of activated processes we formulate a mean field version of the models which keeps most of the features of the finite dimensional ones and solve analytically the out of equilibrium dynamics in the low temperature regime where activation plays an essential role

large deviation techniques applied to systems with long range interactions we discuss a method to solve models with long range interactions in the microcanonical and canonical ensemble the method closely follows the one introduced by r s ellis physica d 133 106 1999 which uses large deviation techniques we show how it can be adapted to obtain the solution of a large class of simple models which can show ensemble inequivalence the model hamiltonian can have both discrete ising potts and continuous hmf free electron laser state variables this latter extension gives access to the comparison with dynamics and to the study of non equilibrium effects we treat both infinite range and slowly decreasing interactions and in particular we present the solution of the ising model in one dimension with 0 1

fractional ginzburg landau equation for fractal media abstract we derive the fractional generalization of the ginzburg landau equation from the variational euler lagrange equation for fractal media to describe fractal media we use the fractional integrals considered as approximations of integrals on fractals some simple solutions of the ginzburg landau equation for fractal media are considered and different forms of the fractional ginzburg landau equation or nonlinear schrodinger equation with fractional derivatives are presented the agrawal variational principle and its generalization have been applied

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

the spin contribution to the form factor of quantum graphs following the quantization of a graph with the dirac operator spin 1 2 we explain how additional weights in the spectral form factor k due to spin propagation around orbits produce higher order terms in the small asymptotics in agreement with symplectic random matrix ensembles we determine conditions on the group of spin rotations sufficient to generate cse statistics

leading off diagonal approximation for the spectral form factor for uniformly hyperbolic systems we consider the semiclassical approximation to the spectral form factor k for two dimensional uniformly hyperbolic systems with time reversal symmetry and derive the first off diagonal correction for small the result agrees with the 2 term of the form factor for the goe random matrix ensemble

fractal burgers equations abstract the paper studies local and global in time solutions to a class of multidimensional generalized burgers type equations with a fractional power of the laplacian in the principal part and with general algebraic nonlinearity such equations naturally appear in continuum mechanics our results include existence uniqueness regularity and asymptotic behavior of solutions to the cauchy problem as well as a construction of self similar solutions the role of critical exponents is also explained

breathers on lattices with long range interaction we analyze the properties of breathers time periodic spatially localized solutions on chains in the presence of algebraically decaying interactions 1 r s we find that the spatial decay of a breather shows a crossover from exponential short distances to algebraic large distances decay we calculate the crossover distance as a function of s and the energy of the breather next we show that the results on energy thresholds obtained for short range interactions remain valid for s 3 and that for s 3 fractal dispersion at the band edge nonzero thresholds occur for cases where the short range interaction system would yield zero threshold values

semiclassical approach to parametric spectral correlation with spin 1 2 the spectral correlation of a chaotic system with spin 1 2 is universally described by the gse gaussian symplectic ensemble of random matrices in the semiclassical limit in semiclassical theory the spectral form factor is expressed in terms of the periodic orbits and the spin state is simulated by the uniform distribution on a sphere in this paper instead of the uniform distribution we introduce brownian motion on a sphere to yield the parametric motion of the energy levels as a result the small time expansion of the form factor is obtained and found to be in agreement with the prediction of parametric random matrices in the transition within the gse universality class moreover by starting the brownian motion from a point distribution on the sphere we gradually increase the effect of the spin and calculate the form factor describing the transition from the gaussian orthogonal ensemble class to the gse class

intermediate statistics for a system with symplectic symmetry the dirac rose graph we study the spectral statistics of the dirac operator on a rose shaped graph a graph with a single vertex and all bonds connected at both ends to the vertex we formulate a secular equation that generically determines the eigenvalues of the dirac rose graph which is seen to generalize the secular equation for a star graph with neumann boundary conditions we derive approximations to the spectral pair correlation function at large and small values of spectral spacings in the limit as the number of bonds approaches infinity and compare these predictions with results of numerical calculations our results represent the first example of intermediate statistics from the symplectic symmetry class

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

sublinear elliptic problems with a hardy potential abstract in this paper we study the positive solutions of sub linear elliptic equations with a hardy potential which is singular at the boundary by means of ode techniques a fairly complete picture of the class of radial solutions is given local solutions with a prescribed growth at the boundary are constructed by means of contraction operators some of those radial solutions are then used to construct ordered upper and lower solutions in general domains by standard iteration arguments the existence of positive solutions is proved an important tool is the hardy constant

the semiclassical origin of the logarithmic singularity in the symplectic form factor sieber and richter achieved a breakthrough towards a proof of the universality of spectral fluctuations of chaotic quantum systems conjectured by bohigas giannoni and schmidt by calculating semiclassically the first term beyond the diagonal approximation of the orthogonal form factor in this letter the semiclassical origin of the logarithmic singularity of the symplectic form factor is deduced perturbatively by combining this result with the contribution that arises due to the spin this approach stands in contrast to the duality approach introduced by bogomolny and keating which is essentially non perturbative and where the structure around the heisenberg time is related to the structure for very small time which can be deduced using the diagonal approximation

an exact trace formula and zeta functions for an infinite quantum graph with a non standard weyl asymptotics we study a quantum hamiltonian that is given by the negative laplacian and an infinite chain of like potentials with strength 0 on the half line and which is equivalent to a one parameter family of laplacians on an infinite metric graph this graph consists of an infinite chain of edges with the metric structure defined by assigning an interval in 0 ln to each edge with length we show that the one parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each 0 and prove that the dirichlet laplacian is the limit of the one parameter family in the strong resolvent sense the spectrum of the resulting dirichlet quantum graph is also purely discrete the eigenvalues are given by n n2 with multiplicities d n where d n denotes the divisor function we can thus relate the spectral problem of this infinite quantum graph to dirichlet s famous divisor problem and infer the non standard weyl asymptotics for the eigenvalue counting function based on an exact trace formula the voronoi summation formula we derive explicit formulae for the trace of the wave group the heat kernel the resolvent and for various spectral zeta functions these results enable us to establish a well defined renormalized secular equation and a selberg like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the riemann hypothesis is true

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries

on mittag leffler type functions in fractional evolution processes abstract we review a variety of fractional evolution processes so defined being governed by equations of fractional order whose solutions turn out to be related to mittag leffler type functions the chosen equations are the simplest of the fractional calculus and include the abel integral equations of the second kind which are relevant in typical inverse problems and the fractional differential equations which govern generalized relaxation and oscillation phenomena