Sep
19
2012

**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.

**download:** arxiv: arXiv:1209.1630 [physics.gen-ph]

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

Nov
20
2011

**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.

**download:** arxiv: 1111.1311v1 [cs.CV]

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

Sep
20
2011

**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.

**download:** arxiv: arXiv:1007.1084v1 [physics.gen-ph]

**reference:** Physica A (2010) **389** 4613-4622

Mar
20
2010

**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.

**download:** arxiv: arXiv:1003.5246v1 [physics.gen-ph]

Sep
20
2009

**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.

**download:** arxiv: 0907.1953v1 [math-ph]

**reference**: Physica A **389** (2010) 3307-3315

Jun
20
2009

**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.

**download:** arxiv: 0906.2185v2 [math.GM]

Jan
20
2009

**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.

**download:** arxiv: 0806.2300v2 [physics.gen-ph]

**reference:** Physica A **389** (2010) 693

Mar
20
2008

**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.

**download: **arxiv: 0805.3434v1 [physics.gen-ph]

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

Nov
01
2007

**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.

**download: **arxiv: 0711.3701v1 [physics.gen-ph]

**reference:** Physica A **389** (2010) 4613

Oct
20
2007

**author:**

R. Herrmann

**abstract:**

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.

**download:arXiv:nucl-th/0610091v3**

**reference**: J. Phys. G.: Nucl.Part.Phys. 34 (2007) 607-625