Fractional corresponding operator in quantum mechanics and applications: A uniform fractional Schrödinger equation in form and fractional quantization methods

In this paper we use Dirac function to construct a fractional operator called fractional corresponding operator, which is the general form of momentum corresponding operator. Then we give a judging theorem for this operator and with this judging theorem we prove that R–L, G–L, Caputo, Riesz fractional derivative operator and fractional derivative operator based on generalized functions, which are the most popular ones, coincide with the fractional corresponding operator. As a typical application, we use the fractional corresponding operator to construct a new fractional quantization scheme and then derive a uniform fractional Schrödinger equation in form. Additionally, we find that the five forms of fractional Schrödinger equation belong to the particular cases. As another main result of this paper, we use fractional corresponding operator to generalize fractional quantization scheme by using Lévy path integral and use it to derive the corresponding general form of fractional Schrödinger equation, which consequently proves that these two quantization schemes are equivalent. Meanwhile, relations between the theory in fractional quantum mechanics and that in classic quantum mechanics are also discussed. As a physical example, we consider a particle in an infinite potential well. We give its wave functions and energy spectrums in two ways and find that both results are the same.     

Initialized fractional differential equations with Riemann-Liouville fractional-order derivative

The initial value problem of fractional differential equations and its solving method are studied in this paper. Firstly, for easy understanding, a different version of the initialized operator theory is presented for Riemann-Liouville’s fractional-order derivative, addressing the initial history in a straightforward form. Then, the initial value problem of a single-term fractional differential equation is converted to an equivalent integral equation, a form that is easy for both theoretical and numerical analysis, and two illustrative examples are given for checking the correctness of the integral equation. Finally, the counter-example proposed in a recent paper, which claims that the initialized operator theory results in wrong solution of a fractional differential equation, is checked again carefully. It is found that solving the equivalent integral equation gives the exact solution, and the reason behind the result of the counter-example is that the calculation therein is based on the conventional Laplace transform for fractional-order derivative, not on the initialized operator theory. The counter-example can be served as a physical model of creep phenomena for some viscoelastic materials, and it is found that it fits experimental curves well.     

Trace of a difference of singular Sturm-Liouville operators with a potential containing Dirac δ-functions

In L ₂[0,+∞) we consider the Sturm-Liouville operator generated by the expression involving delta functions and evaluate the trace of the difference between two operators of this kind which differ by a multiplication operator by a locally summable function on the semiaxis.     

Modelling of transmission through a chiral slab using fractional curl operator

Transmission of electromagnetic plane wave through a slab of reciprocal chiral medium has been modelled using fractional curl operator. It is noted that when order of the fractional curl operator becomes zero, the equivalent situation may correspond to absence of the chiral slab. Variation of the order of fractional curl operator may explore situations which may be regarded as intermediate step of a situation dealing with no chiral slab and a situation dealing with a chiral slab. It is noted that real order of the fractional curl operator may model the optical rotation while complex order of the fractional curl operator may model optical rotation as well as circular dichroism.     

An inverse spectral theory of Gel'fand-Levitan-type for higher order differential operators

An inverse spectral theory is presented for certain linear ordinary differential operators of arbitrary even order n which generalizes the Gel'fand-Levitan theory for Sturm-Liouville operators. It is proved that the coefficients in these operators are uniquely determined by n–1 distinct spectral matrices. Our method of proof makes use of a transformation due to M.K. Fage which generalizes the Povzner-Levitan transformations for Sturm-Liouville operators     

Spectral Properties of a <Emphasis Type="Italic">q</Emphasis>-Sturm–Liouville Operator

We study the spectral properties of a class of Sturm-Liouville type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, we construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. It is also shown that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively.     

Fractional Number Operator and Associated Fractional Diffusion Equations

In this paper, we study the fractional number operator as an analog of the finite-dimensional fractional Laplacian. An important relation with the Ornstein-Uhlenbeck process is given. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented. By means of the Mittag-Leffler function and the Laplace transform, we give the solution of the Caputo time fractional diffusion equation and Riemann-Liouville time fractional diffusion equation in infinite dimensions associated to the fractional number operator.     

New fractional derivative with sigmoid function as the kernel and its models

Based on the idea of the fractional derivative with respect to another function, a new fractional derivative operator with sigmoid function as the kernel in this article, is proposed for the first time. Then, we make use of this new fractional operator to model various nonlinear phenomena from different fields of applications in science, such as the population growth, the shallow water wave phenomena and reaction-diffusion processes, and so on. As a result, we hope that the new fractional operator can be used to discover more evolutionary mechanisms of these phenomena.     

Induced fractional spin and statistics in three-dimensional QED

It is shown that in background magnetic fields the fermionic Fock vacuum of (2+1)-dimensional quantum electrodynamics is a gauge invariant eigenstate of the full angular-momentum operator. The eigenvalue is computed and its relation to fractional statistics is discussed. It is argued that anomalous induced angular momentum persists to all orders in perturbation theory.     

Optical complex integration-transform for deriving complex fractional squeezing operator

We find a new complex integration-transform which can establish a new relationship between a two-mode operator's matrix element in the entangled state representation and its Wigner function. This integration keeps modulus invariant and therefore invertible. Based on this and the Weyl–Wigner correspondence theory, we find a two-mode operator which is responsible for complex fractional squeezing transformation. The entangled state representation and the Weyl ordering form of the two-mode Wigner operator are fully used in our derivation which brings convenience.     

Inverse problems for matrix Sturm-Liouville operators

Inverse spectral problems for nonselfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied. As a main spectral characteristic, we introduce the so-called Weyl matrix and prove that the specification of the Weyl matrix uniquely determines the matrix potential and the coefficients of the boundary conditions. Moreover, for a finite interval, we also study the inverse problems of recovering matrix Sturm-Liouville operators from discrete spectral data (eigenvalues and “weight” numbers) and from a system of spectra. The results thus obtained are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators. Dedicated to the memory of B. M. Levitan     

Analogs of Bohr-Sommerfeld-Maslov quantization conditions on Riemann surfaces and spectral series of nonself-adjoint operators

In the paper, the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators important for applications is studied (for the Sturm-Liouville operator with complex potential and the operator of induction). It turns out that the asymptotic behavior can be calculated using the quantization conditions, which can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the Bohr-Sommerfeld-Maslov formulas), to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different series.     

Analytic Study of Properties of Holographic Superconductors with Weyl Corrections

We analytically investigate the properties of holographic superconductors with Weyl corrections in AdS ₅-Schwarzschild background by two approaches, one based on the Sturm-Liouville eigenvalue problem and the other based on the matching of the solutions to the field equations near the horizon and near the asymptotic AdS region. The relation between the critical temperature and the charge density has been obtained and the dependence of the expectation value of the condensation operator on the temperature has been found. We find that the critical temperature of holographic superconductor with Weyl corrections increases as we amplify the Weyl coupling parameter γ, indicating the condensation will be harder when the parameter γ decreases. The critical exponent of the condensation also comes out to be 1/2 which is the universal value in the mean field theory.     

Path integral representation of fractional harmonic oscillator

Fractional oscillator process can be obtained as the solution to the fractional Langevin equation. There exist two types of fractional oscillator processes, based on the choice of fractional integro-differential operators (namely Weyl and Riemann-Liouville). An operator identity for the fractional differential operators associated with the fractional oscillators is derived; it allows the solution of fractional Langevin equations to be obtained by simple inversion. The relationship between these two fractional oscillator processes is studied. The operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes. Relevant quantities such as two-point and n-point functions can be calculated from the generating functions.     

A Physical experimental study of variable-order fractional integrator and differentiator

Recent research results have shown that many complex physical phenomena can be better described using variable-order fractional differential equations. To understand the physical meaning of variable-order fractional calculus, and better know the application potentials of variable-order fractional operators in physical processes, an experimental study of temperature-dependent variable-order fractional integrator and differentiator is presented in this paper. The detailed introduction of analogue realization of variable-order fractional operator, and the influence of temperature to the order of fractional operator are presented in particular. Furthermore, the potential applications of variable-order fractional operators in PI ^λ(t) D ^μ(t) controller and dynamic-order fractional systems are suggested.     

Surface wave modes in PEMC backed chiral slab

The characteristics of surface wave modes in a PEMC backed chiral slab are studied theoretically. First, the analytical solution of electromagnetic fields and dispersion relations are carried out. Then, the fractional field solutions are found using the fractional curl operator. The numerical results are given by assuming that wave numbers k and k_± are either real or imaginary. These results are also evaluated at real and imaginary values of fractional parameter describing the order of curl operator. The discussion contains fractional dispersion curves at various cut-off frequencies and the fractional surface waves in chiral-PEMC and achiral-PMC slabs respectively. For numerical analysis it is assumed that the fractional order of the curl operator is related to chiral admittance, thickness of the slab, and PEMC admittance. For the values of the fractional order equal to 0, 1, and 2 geometry corresponds to PMC backed ordinary dielectric slab, PEMC backed chiral slab, and PEC backed chiral slab respectively. Consequently TE, HE (even), and HE (odd) modes are produced in the respective geometries.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

From Complex Fractional Fourier Transform to Complex Fractional Radon Transform

We show that for n-dimensional complex fractional Fourier transform the corresponding complex fractional Radon transform can also be derived, however, it is different from the direct product of two n-dimensional real fractional Radon transforms. The complex fractional Radon transform of two-mode Wigner operator is calculated.     

On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

By a quantum mechanical analysis of the additive rule F_α[F_β[f]] = F_α+β[f], which the fractional Fourier transformation (FrFT) F_α[f] should satisfy, we reveal that the position-momentum mutualtransformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.     

Conservation laws for the Korteweg-de Vries equation and the theory of partitions

We show how the terms appearing in the expressions for the densities and the fluxes for the Korteweg-de Vries equation may be found by combinatorial methods. Our basic device consists in associating partitions and their Ferrers graphs to the first density and to the first flux, and then in proceeding inductively following very simple rules. Furthermore, we use unrestricted partitions and a recurrence relation to specify every term of every integral power of the Sturm-Liouville (or one-dimensional Schrödinger) operator.