Sep 19 2012

## 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

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Nov 20 2011

## 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

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Sep 20 2011

## 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

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Mar 20 2010

## 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.

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Sep 20 2009

## 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

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Jun 20 2009

## 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.

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Jan 20 2009

## 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.

reference: Physica A 389 (2010) 693

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Mar 20 2008

## Fractional spin – a property of particles described with a fractional Schroedinger equation

author:
R. Herrmann

abstract:
It is shown, that the requirement of invariance under spatial rotations reveales an intrinsic fractional extended translation-rotation-like property for particles described with the fractional Schroedinger equation, which we call fractional spin.

reference published as chapter 7 in doi:Physics Letters A 372 (2008) 5515-5522

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Nov 01 2007

## 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. For the fractional harmonic oscillator, the corresponding q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool e.g. to describe 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.
reference: Physica A 389 (2010) 4613

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Oct 20 2007

## The fractional symmetric rigid rotor

Based on the Riemann fractional derivative the Casimir operators and multipletts for the fractional extension of the rotation group SO(n) are calculated algebraically. The spectrum of the corresponding fractional symmetric rigid rotor is discussed. It is shown, that the rotational, vibrational and $\gamma$-unstable limits of the standard geometric collective models are particular limits of this spectrum. A comparison with the ground state band spectra of nuclei shows an agreement with experimental data better than 2%. The derived results indicate, that the fractional symmetric rigid rotor is an appropriate tool for a description of low energy nuclear excitations.