# Definition of the Riesz derivative and its application to space fractional quantum mechanics

@article{Bayin2016DefinitionOT, title={Definition of the Riesz derivative and its application to space fractional quantum mechanics}, author={Selccuk cS. Bayin}, journal={Journal of Mathematical Physics}, year={2016}, volume={57}, pages={123501} }

We investigate and compare different representations of the Riesz derivative, which plays an important role in anomalous diffusion and space fractional quantum mechanics. In particular, we show that a certain representation of the Riesz derivative, R x α , that is generally given as also valid for α = 1, behaves no differently than the other definition given in terms of its Fourier transform. In the light of this, we discuss the α → 1 limit of the space fractional quantum mechanics and its…

## 25 Citations

### Factorization of the Riesz-Feller Fractional Quantum Harmonic Oscillators

- MathematicsJournal of Physics: Conference Series
- 2020

Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz,…

### Energy-dependent noncommutative quantum mechanics

- Physics, MathematicsThe European Physical Journal C
- 2019

We propose a model of dynamical noncommutative quantum mechanics in which the noncommutative strengths, describing the properties of the commutation relations of the coordinate and momenta,…

### On solvability of differential equations with the Riesz fractional derivative

- MathematicsMathematical Methods in the Applied Sciences
- 2021

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent…

### On Riesz derivative

- MathematicsFractional Calculus and Applied Analysis
- 2019

Abstract This paper focuses on studying Riesz derivative. An interesting investigation on properties of Riesz derivative in one dimension indicates that it is distinct from other fractional…

### On the solution of a generalized Higgs boson equation in the de Sitter space-time through an efficient and Hamiltonian scheme

- MathematicsJ. Comput. Phys.
- 2020

### 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

- Mathematics
- 2018

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.…

### Remarks on the Generalized Fractional Laplacian Operator

- MathematicsMathematics
- 2019

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to…

### On the Generalized Riesz Derivative

- MathematicsMathematics
- 2020

The goal of this paper is to construct an integral representation for the generalized Riesz derivative R Z D x 2 s u ( x ) for k < s < k + 1 with k = 0 , 1 , ⋯ , which is proved to be a one-to-one…

### Cauchy Processes, Dissipative Benjamin–Ono Dynamics and Fat-Tail Decaying Solitons

- PhysicsFractal and Fractional
- 2021

In this paper, a dissipative version of the Benjamin–Ono dynamics is shown to faithfully model the collective evolution of swarms of scalar Cauchy stochastic agents obeying a…

## References

SHOWING 1-10 OF 21 REFERENCES

### Principles of Fractional Quantum Mechanics

- Physics
- 2010

A review of fundamentals and physical applications of fractional quantum mechanics has been presented. Fundamentals cover fractional Schr\'odinger equation, quantum Riesz fractional derivative, path…

### On the consistency of the solutions of the space fractional Schrödinger equation

- Mathematics
- 2012

Recently, it was pointed out that the solutions found in the literature for the space fractional Schrodinger equation in a piecewise manner are wrong, except the case with the delta potential. We…

### Comment on “On the consistency of the solutions of the space fractional Schrödinger equation” [J. Math. Phys.53, 042105 (2012)]

- Physics
- 2012

Recently we have reanalyzed the consistency of the solutions of the space fractional Schrodinger equation found in a piecewise manner, and showed that an exact and a proper treatment of the relevant…

### Fractals and quantum mechanics.

- PhysicsChaos
- 2000

The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics.

### On the consistency of the solutions of the space fractional Schroumldinger equation (vol 53, 042105, 2012)

- Mathematics
- 2012

Recently we have reanalyzed the consistency of the solutions of the space fractional Schr\"odinger equation found in a piecewise manner, and showed that an exact and a proper treatment of the…

### Nonlocally induced (fractional) bound states: Shape analysis in the infinite Cauchy well

- Mathematics
- 2015

Fractional (Levy-type) operators are known to be spatially nonlocal. This becomes an issue if confronted with a priori imposed exterior Dirichlet boundary data. We address spectral properties of the…

### Relativistic Quantum Mechanics

- Physics
- 1965

In this text the authors develop a propagator theory of Dirac particles, photons, and Klein-Gordon mesons and per- form a series of calculations designed to illustrate various useful techniques and…

### Consistency problem of the solutions of the space fractional Schrödinger equation

- Mathematics
- 2013

Recently, consistency of the infinite square well solution of the space fractional Schrodinger equation has been the subject of some controversy. Hawkins and Schwarz [J. Math. Phys.54, 014101 (Year:…

### The fundamental solution of the space-time fractional diffusion equation

- Mathematics
- 2007

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a…