on the nonlocality of the fractional schrodinger equation a number of papers over the past eight years have claimed to solve the fractional schrodinger equation for systems ranging from the one dimensional infinite square well to the coulomb potential to one dimensional scattering with a rectangular barrier however some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal preventing the fractional schrodinger equation from being solved in the usual piecewise fashion we focus on the one dimensional infinite square well and show that the purported ground state which is based on a piecewise approach is definitely not a solution of the fractional schrodinger equation for the general fractional parameter on a more positive note we present a solution to the fractional schrodinger equation for the one dimensional harmonic oscillator with 1

on anomalous diffusion and the fractional generalized langevin equation for a harmonic oscillator the fractional generalized langevin equation fgle is proposed to discuss the anomalous diffusive behavior of a harmonic oscillator driven by a two parameter mittag leffler noise the solution of this fgle is discussed by means of the laplace transform methodology and the kernels are presented in terms of the three parameter mittag leffler functions recent results associated with a generalized langevin equation are recovered

time fractional wave diffusion equation in an inhomogeneous half space we consider the fundamental solution of the time fractional wave diffusion equation in a three dimensional half space medium which contains an inhomogeneity in form of a plane parallel layer the corresponding green s function which is derived by means of the fourier and laplace transforms can be accurately and efficiently evaluated without recourse to the mittag leffler or the fox h function moreover it is shown that in the one dimensional case the fundamental solution in an inhomogeneous half space is no longer a probability density function in addition we consider the advection equation for the fractional laplacian and the caputo time fractional derivative of orders on a bounded domain simple algorithms for accurate evaluation of the m wright function and the mittag leffler function are enclosed at the end of this article

periodic orbits near bifurcations of codimension two classical mechanics semiclassics and stokes transitions we investigate classical and semiclassical aspects of codimension two bifurcations of periodic orbits in hamiltonian systems a classification of these bifurcations in autonomous systems with two degrees of freedom or time periodic systems with one degree of freedom is presented we derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurcations in conjunction with stokes transitions that become important in the ensuing diffraction catastrophe integrals

similarity solution to fractional nonlinear space time diffusion wave equation in this article the so called fractional nonlinear space time wave diffusion equation is presented and discussed this equation is solved by the similarity method using fractional derivatives in the caputo riesz feller and riesz senses some particular cases are presented and the corresponding solutions are shown by means of 2 d and 3 d plots

a geometric construction of solutions to 11d supergravity necessary and sufficient conditions are provided for a class of warped product manifolds with non vanishing flux to be supersymmetric solutions of 11d supergravity many non compact but complete solutions can be obtained in this manner including the multi membrane solution initially found by duff and stelle in a different direction an explicit 5 parameter moduli space of solutions to 11d supergravity is also constructed which can be viewed as non supersymmetric deformations of the duff stelle solution

significance of ghost orbit bifurcations in semiclassical spectra gutzwiller s trace formula for the semiclassical density of states in a chaotic system diverges near bifurcations of periodic orbits where it must be replaced with uniform approximations it is well known that when applying these approximations complex predecessors of orbits created in the bifurcation ghost orbits can produce pronounced signatures in the semiclassical spectra in the vicinity of the bifurcation it is the purpose of this paper to demonstrate that these ghost orbits can also undergo bifurcations resulting in complex nongeneric bifurcation scenarios we do so by studying an example taken from the diamagnetic kepler problem namely the period quadrupling of the balloon orbit by application of normal form theory we construct an analytic description of the complete bifurcation scenario which is then used to calculate the pertinent uniform approximation the ghost orbit bifurcation turns out to produce signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits would

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

characterizations and simulations of a class of stochastic processes to model anomalous diffusion in this paper we study a parametric class of stochastic processes to model both fast and slow anomalous diffusions this class called generalized grey brownian motion ggbm is made up of self similar with stationary increments processes h sssi and depends on two real parameters 0 2 and 0 1 it includes fractional brownian motion when 0 2 and 1 and time fractional diffusion stochastic processes when 0 1 the latter have a marginal probability density function governed by timefractional diffusion equations of order the ggbm is defined through the explicit construction of the underlying probability space however in this paper we show that it is possible to define it in an unspecified probability space for this purpose we write down explicitly all the finite dimensional probability density functions moreover we provide different ggbm characterizations the role of them wright function which is related to the fundamental solution of the time fractional diffusion equation emerges as a natural generalization of the gaussian distribution furthermore we show that the ggbm can be represented in terms of the product of a random variable which is related to the m wright function and an independent fractional brownian motion this representation highlights the h sssi nature of the ggbm and provides a way to study and simulate the trajectories for this purpose we developed a random walk model based on a finite difference approximation of a partial integro differential equation of a fractional type

some solutions to the space fractional schrodinger equation using momentum representation method the space fractional schrodinger equation with linear potential delta function potential and coulomb potential is studied under momentum representation using fourier transformation by use of mellin transform and its inverse transform we obtain the energy levels and wave functions expressed in h function for a particle in linear potential field the wave function expressed also by the h function and the unique energy level of the bound state for the particle of even parity state in delta function potential well which is proved to have no action on the particle of odd parity state is also obtained the integral form of the wave functions for a particle in coulomb potential field is shown and the corresponding energy levels which have been discussed in laskin s paper phys rev e 66 056108 2002 are proved to satisfy an equality of infinite limit of the h function all of these results contain the ones of the standard quantum mechanics as their special cases

uniform approximation for bifurcations of periodic orbits with high repetition numbers the semiclassical contribution of a periodic orbit to the quantum density of states diverges when the orbit bifurcates in this case one has to apply approximations which are uniformly valid both in and a parameter which describes the distance to the bifurcation the form of the approximation depends on the repetition number m of the orbit that bifurcates in a two dimensional system the approximations are different for m 1 up to m 5 and for they have the same form as for m 5 in this article we consider the case which occurs first when an integrable system is perturbed a uniform approximation for the contribution to the spectral density is derived which in the limit of large reduces to a sum of semiclassical contributions of isolated periodic orbits

differentiation to fractional orders and the fractional telegraph equation using methods of differential and integral calculus we present and discuss the calculation of a fractional green function associated with the one dimensional case of the so called general fractional telegraph equation with one space variable this is a fractional partial differential equation with constant coefficients the equation is solved by means of juxtaposition of transforms i e we introduce the laplace transform in the time variable and the fourier transform in the space variable several particular cases are discussed in terms of the parameters some known results are recovered as a by product of our main result we obtain two new relations involving the two parameter mittag leffler function

asymptotically conical calabi yau manifolds i this is the first of a series of articles on complete calabi yau manifolds asymptotic to riemannian cones at infinity we begin by proving general existence and uniqueness results the uniqueness part relaxes the decay condition o r n e needed in earlier work to o r e relying on some new ideas about harmonic functions we then look at two classes of examples crepant resolutions of cones this includes a new class of ricci flat small resolutions associated with flag varieties and affine deformations of cones one focus here is the question of the precise rate of decay of the metric to its tangent cone we prove that the optimal rate for the stenzel metric on t sn is 2 n n 1

fractional schrodinger operator with delta potential localized on circle we consider a system governed by the fractional schodinger operator with a delta potential supported by a circle in r2 we find out the function counting the number of bound states in particular we give the necessary and sufficient conditions for the absence of bound state in our system furthermore we reproduce the form of eigenfunctions and analyze the asymptotic behavior of eigenvalues for the strong coupling constant case

response characteristics of a fractional oscillator abstract the integral equation of motion of a driven fractional oscillator is obtained by generalizing the corresponding equation of motion of a driven harmonic oscillator to include integrals of arbitrary order according to the methods of fractional calculus the green s function solution for the fractional oscillator is obtained in terms of mittag leffler functions using laplace transforms the response and resonance characteristics of the fractional oscillator are studied for several cases of forcing function

on some fractional green s functions in this paper we discuss some fractional green s functions associated with the fractional differential equations which appear in several fields of science more precisely the so called wave reaction diffusion equation and some of its particular cases the methodology presented is the juxtaposition of integral transforms in particular the laplace and the fourier integral transforms some recent results involving the reaction diffusion equation are pointed out

fractional wave diffusion equation with periodic conditions we study a time space fractional wave diffusion equation with periodic conditions using laplace transforms and fourier series and presenting its solution in terms of three parameter mittag leffler functions as a particular case we recover a recent result we also present some graphics associated with particular values of the parameters

uniform approximation for period quadrupling bifurcations we derive a uniform approximation for semiclassical contributions of periodic orbits to the spectral density which is valid for generic period quadrupling bifurcations in systems with a mixed phase space these bifurcations involve three periodic orbits which coalesce at the bifurcation in the vicinity of the bifurcation the three orbits give a collective contribution to the spectral density while the individual contributions of gutzwiller s type would diverge at the bifurcation the uniform approximation is obtained by mapping the action function onto the normal form corresponding to the bifurcation this article is a continuation of previous work in which uniform approximations for generic period m tupling bifurcations with were derived

time and space fractional partial differential equations the fundamental solution for time and space fractional partial differential operator dt a2 2 0 is given in terms of the fox s h function here the time fractional derivative in the sense of generalized functions distributions dt is defined by the convolution dt f t t f t where t t 1 and f t 0 as t 0 and the fractional n dimensional laplace operator 2 is defined by its fourier transform with respect to spatial variable f 2g x f g x the solutions for initial value problems for time and space fractional partial differential equation in the sense of caputo and riemann liouville time fractional derivatives respectively are obtained by the fundamental solution

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

the fractional schrodinger equation for delta potentials the fractional schrodinger equation is solved for the delta potential and the double delta potential for all energies the solutions are given in terms of fox s h function

homogeneous fractional embeddings fractional equations appear in the description of the dynamics of various physical systems for lagrangian systems the embedding theory developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 033504 2007 provides a univocal way to obtain such equations stemming from a least action principle however no matter how equations are obtained the dimension of the fractional derivative differs from the classical one and may induce problems of temporal homogeneity in fractional objects in this paper we show that it is necessary to introduce an extrinsic constant of time then we use it to construct two equivalent fractional embeddings which retains homogeneity the notion of fractional constant is also discussed through this formalism finally an illustration is given with natural lagrangian systems and the case of the harmonic oscillator is entirely treated

parallel spinors and holonomy groups in this paper we complete the classification of spin manifolds admitting parallel spinors in terms of the riemannian holonomy groups more precisely we show that on a given n dimensional riemannian manifold spin structures with parallel spinors are in one to one correspondence with lifts to spinn of the riemannian holonomy group with fixed points on the spin representation space in particular we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors

resonant periodic orbits and the semiclassical energy spectrum the semiclassical density of states depends according to the periodic orbit sum formula on the linear stability of the orbits this means however that contributions from the marginally stable or resonant orbits which necessarily accompany stable ones diverge unphysically the remedy for a system of two degrees of freedom is found to lie in the classical non linear normal forms for periodic orbits which describe how satellite periodic orbits coalesce with the central one as resonance is approached in to 0 through these forms the resonant contributions are expressed as diffraction integrals the first few being diffraction catastrophes uniformly valid in in and h cross and finite even for in to 0 provided h cross 0 an extension is proposed to incorporate jointly multiple resonances found in repetitions of orbits

solution of the fractional langevin equation and the mittag leffler functions we introduce the fractional generalized langevin equation in the absence of a deterministic field with two deterministic conditions for a particle with unitary mass i e an initial condition and an initial velocity are considered for a particular correlation function that characterizes the physical process and using the methodology of the laplace transform we obtain the solution in terms of the three parameter mittag leffler function as particular cases some recent results are also presented

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

some physical applications of fractional schrodinger equation the fractional schrodinger equation is solved for a free particle and for an infinite square potential well the fundamental solution of the cauchy problem for a free particle the energy levels and the normalized wave functions of a particle in a potential well are obtained in the barrier penetration problem the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined in the quantum scattering problem according to the fractional schrodinger equation the green s function of the lippmann schwinger integral equation is given

bifurcations of periodic orbits and uniform approximations we derive uniform approximations for contributions to gutzwiller s periodic orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space there orbits lie close together and give collective contributions while the individual contributions of gutzwiller s type would diverge at the bifurcation new results for the tangent the period doubling and the period tripling bifurcation are given they are obtained by going beyond the local approximation and including higher order terms in the normal form of the action the uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results

fundamental solution of the tempered fractional diffusion equation in this paper we consider the space time fractional diffusion equation dt u x t k dx u x t 0 x r t 0 with the tempered riemann liouville derivative of order 0 1 in space and the caputo derivative of order 0 1 in time the fundamental solution which turns out to be a spatial probability density function is given in computable series form as well as in integral representation the spatial moments of the probability density function are determined explicitly for an arbitrary order n 0 moreover green s function of the untempered neutral fractional diffusion equation is analyzed in view of absolute and relative extreme points at the end of this article we point out a remarkably and important integral representation for accurate evaluation of the m wright mainardi function m x of order 0 1 and arguments x r0