Space-time fractional Schrödinger equation with time-independent potentials

We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.     

The effect of fractional calculus on the formation of quantum-mechanical operators

In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in 1 + 1 dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.     

The Hausdorff dimension of fractal sets and fractional quantum Hall effect

We consider Farey series of rational numbers in terms of fractal sets labeled by the Hausdorff dimension with values defined in the interval 1<h<2 and associated with fractal curves. Our results come from the observation that the fractional quantum Hall effect-FQHE occurs in pairs of dual topological quantum numbers, the filling factors. These quantum numbers obey some properties of the Farey series and so we obtain that the universality classes of the quantum Hall transitions are classified in terms of h. The connection between Number Theory and Physics appears naturally in this context.     

Numerical solutions of nonlinear fractional Schrödinger equations using nonstandard discretizations

In this article, numerical study for both nonlinear space‐fractional Schrödinger equation and the coupled nonlinear space‐fractional Schrödinger system is presented. We offer here the weighted average nonstandard finite difference method (WANSFDM) as a novel numerical technique to study such kinds of partial differential equations. The space fractional derivative is described in the sense of the quantum Riesz‐Feller definition. Stability analysis of the proposed method is studied. To show that this method is reliable and computationally efficient different numerical examples are provided. We expect that the proposed schemes can be applicable to different systems of fractional partial differential equations. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1399–1419, 2017     

Fractional monodromy of resonant classical and quantum oscillatorsMonodromie fractionnelle des oscillateurs résonnants classiques et quantiques

We introduce fractional monodromy for a class of integrable fibrations which naturally arise for classical nonlinear oscillator systems with resonance. We show that the same fractional monodromy characterizes the lattice of quantum states in the joint spectrum of the corresponding quantum systems. Results are presented on the example of a two-dimensional oscillator with resonance 1:(?1) and 1:(?2). To cite this article: N.N. Nekhoroshev et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 985–988.     

A new method for exact solutions of variant types of time‐fractional Korteweg‐de Vries equations in shallow water waves

The current article devoted on the new method for finding the exact solutions of some time‐fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time‐fractional equations viz. time‐fractional KdV‐Burgers and KdV‐mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Fractional nonholonomic Ricci flows

We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman’s functionals and derived the corresponding fractional evolution (Hamilton’s) equations. We apply in fractional calculus the nonlinear connection formalism originally elaborated in Finsler geometry and generalizations and recently applied to classical and quantum gravity theories. There are also analyzed the fractional operators for the entropy and fundamental thermodynamic values.     

Statistical theory of rarified gases in the coulomb model of substance: Adiabatic approximation and initial atoms

In the framework of the adiabatic approximation for a subsystem of nuclei with the average distance between them significantly exceeding the dimensions of the initial atom, we consider a nonrelativistic Coulomb system consisting of electrons and nuclei of one type for the temperature range where we can restrict ourself to using the ground state to describe the electron subsystem. We show that the equilibrium properties of such a system are equivalent to the thermodynamic properties of the one-component system of initial atoms interacting between themselves via a short-range potential that is the effective potential of the nucleus-nucleus interaction. In the framework of the applicability of Boltzmann statistics, we present quantum group expansions for the thermodynamic properties of a chemically reacting rarified gas that correspond to the method of initial atoms.     

Reanalysis of an open problem associated with the fractional Schrödinger equation

It was recently shown that there are some difficulties in the solution method proposed by Laskin for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schrödinger equation with an infinite potential well encountered in quantum mechanics. In fact, this problem is still open. We propose a new fractional approach that allows overcoming the limitations of some previously introduced strategies. In deriving the solution, we use a method based on the eigenfunction of the Weyl fractional derivative. We obtain a solution suitable for computations in a closed form in terms of Mittag–Leffler functions and fractional trigonometric functions. It is a simple extension of the results previously obtained by Laskin et al.     

The effect of time fractality on the transition coefficients: Historical Stern–Gerlach experiment revisited

In this article, the influence of time discreteness on the transition coefficients is investigated within the framework of time fractional development of quantum systems which has been developed recently by the present authors [22]. In this formalism, fractional mathematics which is a powerful tool to study the non-Markovian and non-Gaussian properties of physical processes is used in order to obtain time fractional evolution operator and transition probability. They are given in terms of Mittag–Leffler function which plays an important role in the mathematical structure as well as the physical interpretation of the phenomena under investigation. In order to place the presented formalism on a concrete basis, historical Stern–Gerlach experiment has been revisited with the purpose of studying transition coefficients which have a non-Markovian feature. The effect of the time fractionalization has been clearly illustrated in the figures via fractional derivative order α.     

Fractional Hamiltonian Monodromy

We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy–momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:( − 2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems. Communicated by Eduard Zehnder Submitted: February 25, 2005; Accepted: November 17, 2005     

Duality of graph invariants

We study a new set of duality relations between weighted,combinatoric invariants of a graph G.The dualities arise from a non-linear transform B,acting on the weight function p.We define B on a space of real-valued functions O and investigate its properties.We show that three invariants(the weighted independence number,the weighted Lovasz number,and the weighted fractional packing number) are fixed points of B~2,but the weighted Shannon capacity is not.We interpret these invariants in the study of quantum non-locality.     

Stability analysis and a numerical scheme for fractional Klein‐Gordon equations

Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order nonlinear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.     

Fractional Nottale’s Scale Relativity and emergence of complexified gravity

Fractional calculus of variations has recently gained significance in studying weak dissipative and nonconservative dynamical systems ranging from classical mechanics to quantum field theories. In this paper, fractional Nottale’s Scale Relativity (NSR) for an arbitrary fractal dimension is introduced within the framework of fractional action-like variational approach recently introduced by the author. The formalism is based on fractional differential operators that generalize the differential operators of conventional NSR but that reduces to the standard formalism in the integer limit. Our main aim is to build the fractional setting for the NSR dynamical equations. Many interesting consequences arise, in particular the emergence of complexified gravity and complex time.     

Control of non‐integer‐order dynamical systems using sliding mode scheme

This article deals with the problem of control of canonical non‐integer‐order dynamical systems. We design a simple dynamical fractional‐order integral sliding manifold with desired stability and convergence properties. The main feature of the proposed dynamical sliding surface is transferring the sign function in the control input to the first derivative of the control signal. Therefore, the resulted control input is smooth and without any discontinuity. So, the harmful chattering, which is an inherent characteristic of the traditional sliding modes, is avoided. We use the fractional Lyapunov stability theory to derive a sliding control law to force the system trajectories to reach the sliding manifold and remain on it forever. A nonsmooth positive definite function is applied to prove the existence of the sliding motion in a given finite time. Some computer simulations are presented to show the efficient performance of the proposed chattering‐free fractional‐order sliding mode controller. © 2015 Wiley Periodicals, Inc. Complexity 21: 224–233, 2016     

Laplacian pretty good fractional revival

We develop the theory of pretty good fractional revival in quantum walks on graphs using their Laplacian matrices as the Hamiltonian. We classify the paths and the double stars that have Laplacian pretty good fractional revival.     

The <Emphasis Type="Italic">p</Emphasis>-adic quantum differential

We propose the Fredholm module structure on the space L₂(ℙ¹(ℚ_p)). We study the properties of the corresponding quantum Connes differential.     

Ab-initio and modeling study of some amino acid-tRNA: NMR shielding and thermodynamic of solvent effect (Poster Presentation)

The tRNA molecule takes suitable amino acid to the ribosome for the formation of peptide bonds. In starting the peptide chain, the first amino acid which is taken to the ribosome is methionine. We have simulated methionine-amino acid bonding and amino acid-tRNA bonding, using a mixed study of quantum mechanics ab-initio and molecular mechanics. NMR shielding tensors and thermodynamic parameters and total energies are also calculated. It is important to understand the physical properties and the environmental conditions in which the dipeptides cause the tRNA to attach to a wrong amino acid. One of the properties studied is the dielectric environment of in which the bonding occurs. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fractional Brownian motion approximation based on fractional integration of a white noise

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We consider correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similarity properties of the approximation to fractional Brownian motion, namely, `τ^H laws' for the structure function and the range. We conclude that the models proposed serve as a convenient tool for modelling of natural processes and testing and improvement of methods aimed at analysis and interpretation of experimental data.