Category Archives: Publications

Solutions of the fractional Schrödinger equation via diagonalization – A plea for the harmonic oscillator basis part 1: the one dimensional case

author: R. Herrmann

abstract:

A covariant non-local extention if the stationary Schr\”odinger equation is presented and it’s solution in terms of Heisenbergs’s matrix quantum mechanics is proposed. For the special case of the Riesz fractional derivative, the calculation of corresponding matrix elements for the non-local kinetic energy term is performed fully analytically in the harmonic oscillator basis and leads to a new interpretation of non local operators in terms of generalized Glauber states.
As a first application, for the fractional harmonic oscillator the potential energy matrix elements are calculated and the and the corresponding Schr\”odinger equation is diagonalized. For the special case of invariance of the non-local wave equation under Fourier-transforms a new symmetry is deduced, which may be interpreted as an extension of the standard parity-symmetry.

reference:
arXiv:1805.03019

Fractional Cassini Coordinates

author: R. Herrmann

abstract:

Introducing a set {α_i}R of fractional exponential powers of focal distances an extension of symmetric Cassini-coordinates on the plane to the asymmetric case is proposed which leads to a new set of fractional generalized Cassini-coordinate systems. Orthogonality and classical limiting cases are derived. An extension to cylindrically symmetric systems in is investigated. The resulting asymmetric coordinate systems are well suited to solve corresponding two- and three center problems in physics.

reference: arXiv:1802.08142

Generalization of the fractional Poisson distribution

author: R. Herrmann

abstract:
A generalization of the Poisson distribution based on the generalized Mittag-Leffler function Eα,β(λ) is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter β is suggested.

reference: Fract. Calc. Appl. Anal. (2016) 19(4) 832-842

Reflection symmetric Erdelyi-Kober type operators – a quasi-particle interpretation

author: R. Herrmann

abstract:
Reflection symmetric Erdelyi-Kober type fractional integral operators are used to construct fractional quasi-particle generators. The eigenfunctions and eigenvalues of these operators are given analytically.
A set of fractional creation- and annihilation-operators is defined and the properties of the corresponding free Hamiltonian are investigated. Analogue to the classical approach for interacting multi-particle systems the results are interpreted as a fractional quantum model for a description of residual interactions of pairing type.

reference: Fract. Calc. Appl. Anal. (2014) 17(4) 1215-1228

A fractal approach to the dark silicon problem: a comparison of 3D computer architectures — standard slices versus fractal Menger sponge geometry

author:
R. Herrmann
abstract:
The dark silicon problem, which limits the power-growth of future computer generations, is interpreted as a heat energy transport problem when increasing the energy emitting surface area within a given volume. A comparison of two 3D-configuration models, namely a standard slicing and a fractal surface generation within the Menger sponge geometry is presented. It is shown, that for iteration orders n>3 the fractal model shows increasingly better thermal behavior. As a consequence cooling problems may be minimized by using a fractal architecture.
Therefore the Menger sponge geometry is a good example for fractal architectures applicable not only in computer science, but also e.g. in chemistry when building chemical reactors, optimizing catalytic processes or in sensor construction technology building highly effective sensors for toxic gases or water analysis.

download: arXiv: arXiv:1404.1891[cs.ET]
reference: Chaos Solitons Fractals (2015) 70,  38-41

Towards a geometric interpretation of generalized fractional integrals – Erdelyi-Kober type integrals on R^N as an example

author: R. Herrmann

abstract:

A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on R^N . Based on this geometric view, several extensions are discussed.

download: arXiv: arXiv:1401.6051
reference: Fract. Calc. Appl. Anal. (2014) 17(2) 361-370

On the origin of space

author: R. Herrmann

abstract:

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension $d$ of space evolves smoothly with time in the range 0 <= d(t) <=3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.

download: arXiv: arXiv:1308.4587
reference: Cent. Eur. J. Phys. (2013) 11(10) 1212-1220

Folded potentials in cluster physics – a comparison of Yukawa and Coulomb potentials with Riesz fractional integrals

author:
R. Herrmann

abstract:

In cluster physics a single particle potential to determine the microscopic part of the total energy of a collective configuration is necessary to calculate the shell- and pairing effects. In this paper we investigate the properties of the Riesz fractional integrals and compare their properties with the standard Coulomb and Yukawa potentials commonly used. It is demonstrated, that Riesz potentials may serve as a promising extension of standard potentials and may be reckoned as a smooth transition from Coulomb to Yukawa like potentials, depending of the fractional parameter $\alpha$. For the macroscopic part of the total energy the Riesz potentials treat the Coulomb-, symmetry- and pairing-contributions from a generalized point of view, since they turn out to be similar realizations of the same fractional integral at distinct $\alpha$ values.

download: arXiv: arXiv:1305.0890
reference: J. Phys. A.: Math. Theor. (2013) 46(40) 405203

Numerical solution of the fractional quantum mechanical harmonic oscillator based on the Riemann and Caputo derivative

author:
R. Herrmann

abstract:
Based on the Riemann- and Caputo definition of the fractional derivative we tabulate the lowest n=31 energy levels and generated graphs of the occupation probability of the fractional quantum mechanical harmonic oscillator with a precision of 32 digits for 0.50 < \alpha < 2.00, which corresponds to the transition from U(1) to SO(3).

reference: Gam. Ori. Chron. Phys. (2013) 1(1) 13-176