# The fractional Schroedinger equation and the infinite potential well – numerical results using the Riesz derivative

author:
R. Herrmann

abstract:
Based on the Riesz definition of the fractional derivative  the fractional Schroedinger equation with an infinite well potential is investigated. First it is shown analytically, that the solutions of the free fractional Schroedinger equation are not eigenfunctions, but good approximations for large k and for $\alpha \approx 2$. The first lowest eigenfunctions are then calculated numerically and an approximate analytic  formula for the level spectrum is derived.

reference: Gam. Ori. Chron. Phys. (2013) 1(1) 1-12

# Curvature interaction in collective space

author:
R. Herrmann

abstract:
For the Riemannian space, built from the collective coordinates used within nuclear models, an additional interaction with the metric is investigated, using the collective equivalent to Einstein’s curvature scalar. The coupling strength is determined using a fit with the AME2003 ground state masses. An extended finite-range droplet model including curvature is introduced, which generates significant improvements for light nuclei and nuclei in the trans-fermium region.

reference: International Journal of Modern Physics E (2012) 21 1250103

# Infrared spectroscopy of diatomic molecules – a fractional calculus approach

author:
R. Herrmann

abstract:
The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically  solving the fractional Schr\”odinger equation based on the Riemann- and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

reference: International Journal of Modern Physics B (2013) 27(6) 1350019

# Covariant fractional extension of the modified Laplace-operator used in 3D-shape recovery

author:
R. Herrmann

abstract:
Extending the Liouville-Caputo definition of a fractional derivative to a nonlocal covariant generalization of arbitrary bound operators acting on multidimensional Riemannian spaces an appropriate approach for the 3D-shape recovery of aperture afflicted 2D slide sequences is proposed. We demonstrate, that the step from a local to a nonlocal algorithm yields an order of magnitude in accuracy and the use of the specific fractional approach an additional factor 2 in accuracy of the derived results.

reference: Fract. Calc. Appl. Anal. (2012) 15(2) 332-343

# Common aspects of q-deformed Lie algebras and fractional calculus

author:
R. Herrmann

abstract:
Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. A new class of fractional q-deformed Lie algebras is proposed, which for the first time allows a smooth transition between different Lie algebras. For the fractional harmonic oscillator, the corresponding fractional q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool to describe e.g. the ground state spectra of even-even nuclei. In addition, the equivalence of rotational and vibrational spectra for fractional q-deformed Lie algebras is shown and the $B_\alpha(E2)$ values for the fractional q-deformed symmetric rotor are calculated. A first interpretation of half integer representations of the fractional rotation group is given in terms of a description of $K=1/2^-$ band spectra of odd-even nuclei.

reference: Physica A (2010) 389 4613-4622

# Fractional quantum numbers deduced from experimental ground state meson spectra

author: R. Herrmann

abstract:
Based on the Caputo definition of the fractional derivative the ground state spectra of mesons are classified as multiplets of the  fractional rotation group. The comparison with the experimental values leads to the conclusion, that quarks carry an up to now unrevealed fractional multiplicative quantum number, which we call fractional hyper charge.

# Fractional phase transition in medium size metal clusters and some remarks on magic numbers in gravitationally and weakly bound clusters

author:
R. Herrmann

abstract:
Based on the Riemann- and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group $SO(3)$ to construct a higher dimensional representation of a fractional rotation group with mixed derivative types. An analytic extended symmetric rotor model is derived, which correctly predicts the sequence of magic numbers in metal clusters. It is demonstrated, that experimental data may be described assuming a sudden change in the fractional derivative parameter $\alpha$ which is interpreted as a second order phase transition in the region of cluster size with $200 \le N \le 300$.

Furthermore it is demonstrated, that the four different realizations of higher dimensional fractional rotation groups may successfully be connected to the four fundamental interaction types realized in nature and may be therefore used for a prediction of magic numbers and binding energies of clusters with gravitational force and weak force respectively bound constituents.

The results presented lead to the conclusion, that mixed fractional derivative operators might play a key role for a successful unified theoretical description of all four fundamental forces realized in nature.

reference: Physica A 389 (2010) 3307-3315

# Higher order fractional derivatives

author:
R. Herrmann

abstract:
Based on the Liouville-Weyl definition of the fractional derivative, a new direct fractional generalization of higher order derivatives is presented. It is shown, that the Riesz and Feller derivatives are special cases of this approach.

# Higher-dimensional mixed fractional rotation groups as a basis for dynamic symmetries generating the spectrum of the deformed Nilsson oscillator

author:
R. Herrmann

abstract:
Based on the Riemann- and Caputo definition of the fractional derivative we use the fractional extensions of the standard rotation group SO(3) to construct a higher dimensional representation of a fractional rotation group with mixed derivative types. An extended symmetric rotor model is derived, which predicts the sequence of magic proton and neutron numbers accurately. The ground state properties of nuclei are correctly reproduced within the framework of this model.