Realizing fractional derivatives of elementary and composite functions through the generalized Euler's integral transform and integer derivative series: building the mathematical framework to model the fractional Schrödinger equation in fractional spacetime

Abstract

Since the engenderment of fractional derivatives in 1695 as a continuous transformation between integer order derivatives, the physical applicability of fractional derivatives has been questioned. While it is true that they share a set of distinguishing characteristics, no two fractional derivatives are alike. With each definition mathematically valid but results of one fractional derivative inconsistent with another, the theory of fractional calculus slowly evolved to create an interconnected web of ideas, limits, and insight. In time fractional derivatives came to be recognized as a powerful and ubiquitous tool. For example, fractional derivatives easily characterize the dynamics of anomalous diffusion in experimental settings where particles are allowed to jump farther than in a Gaussian-distributed random walk. With experimental evidence confirming the physical realization of fractional derivatives, the emphasis in research has been on developing both analytic and numerical tools to treat specific problems in fractional calculus. Similarly in this work we approach fractional derivatives from analytic and numerical perspectives. From large classical systems where it is easy to see the contribution of fractional derivatives we transition to fractional quantum mechanics, where the physical interpretation of fractional derivatives becomes more ambiguous. We concentrate on deriving the fractional Schroedinger equation via the Feynman path integral, under the assumption that space and time coordinates scale at different rates. This generalization is particularly useful for quantum systems where the underlying potential is characterized by a scaling relation. Scaling relations common to dynamics on self-similar geometries do not themselves justify the replacement of integer order derivatives by fractional derivatives. Instead, we seek to describe the evolution of a quantum particle in a particular class of nonlocal potentials, where to realize the kinetic energy of a particle we need to consider a large finite neighborhood. We study symmetry properties of the fractional Schroedinger equation and conclude that only a small subset of fractional derivatives ensures the Hamiltonian is parity- and time-reversal symmetric. To coalesce several fractional derivatives and further emphasize the similarities between them, we cast the finite difference fractional derivative into a sum of integer order derivatives. This expansion is particularly useful for approximating fractional derivatives of functions that would normally be represented by Taylor series with a finite radius of convergence. In the case when fractional derivatives are computed by first expanding the function into its Taylor series, we find that if the Taylor series diverges so does the fractional derivative. The integer derivative expansion allows for the fractional derivative to go beyond the function's finite radius of convergence. In an effort to come up with a universal way of ensuring the convergence of one fractional integral, we generalize the well-known Euler's integral transform. Euler's integral transform integrates a power law with a linear argument hypergeometric function, the result of which is a hypergeometric function with two additional parameters. We show that when the hypergeometric function has a polynomial argument, the result of the integral is a hypergeometric function with the number of added parameters equal to the order of the polynomial. With this ansatz we are able to calculate the fractional derivative of a function if it is indeed expressible as a polynomial argument hypergeometric function, which includes trigonometric, hyperbolic, and Gaussian functions. Next we examine the fractional derivative of a composite function which generalizes Leibniz's product rule. The product rule for a fractional derivative of a composite function is formed in terms of integer derivatives of one function and integrals of a fractional derivative of the other function. Finally in the Appendix we consider a preliminary numerical study that explores the Lax-Richtmyer stability of explicit and implicit Euler schemes to simulate a space-fractional Schroedinger equation. With the framework of fractional calculus enriched by new methods of calculating fractional derivatives, we look to refine our understanding of the fractional Schroedinger equation, and in particular, set the stage for how it may be realizable in multiscale systems.