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.Comment: pages 12, figures 7, accepted by J.Phys.

Properties of a fractional derivative Schroedinger type wave equation and a new interpretation of the charmonium spectrum

Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schroedinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite well potential and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced with a ordinary Schroeodinger equation with an additional potential, which is of the form V ~ x for $\alpha<1$, corresponding to the confinement potential, introduced phenomenologically to the standard models for non relativistic interpretation of quarkonium-spectra. The ordinary Schroedinger equation is triple factorized and yields a fractional wave equation with internal SU(3) symmetry. The twofold iterated version of this wave equation shows a direct analogy to the fractional Schroedinger equation derived. The angular momentum eigenvalues are calculated algebraically. The resulting mass formula is applied to the charmonium spectrum and reproduces the experimental masses with an accuracy better than 0.1%. Extending the standard charmonium spectrum, three additional particles are predicted and associated with $\Sigma_c^0(2455)$ and Y(4260) observed recently and one X(4965), not yet observed. The root mean square radius for $\Sigma_c^0(2455)$ is calculated to be ~0.3[fm]. The derived results indicate, that a fractional wave equation may be an appropriate tool for a description of quark-like particles.Comment: 14 pages, 5 figure

Fractional spin - a property of particles described with a fractional Schroedinger equation

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.Comment: 5 page

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

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 $200 \leq N \leq 300$. 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.Comment: 20 pages, 5 figure

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

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.Comment: 8 pages, 2 figure

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

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 by using the specific fractional approach an additional factor 2 in accuracy of the derived results.Comment: 5 pages, 3 figures, draft for proceedings IFAC FDA12 in Nanjing, Chin

q-deformed Lie algebras and fractional calculus

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.Comment: 8 pages, 3 figure

Uniqueness of the fractional derivative definition

For the Riesz fractional derivative besides the well known integral representation two new differential representations are presented, which emphasize the local aspects of a fractional derivative. The consequences for a valid solution of the fractional Schroedinger equation are discussed.Comment: 5 pages, section on manifest covariant representation on R^N adde

Generalization of the fractional Poisson distribution

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function $E_{\alpha, \beta}(\lambda)$ 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 $\beta$ is suggested.Comment: 10 pages, 3 figure

The fractional Schr\"odinger equation and the infinite potential well - numerical results using the Riesz derivative

Based on the Riesz definition of the fractional derivative the fractional Schr\"odinger equation with an infinite well potential is investigated. First it is shown analytically, that the solutions of the free fractional Schr\"odinger equation are not eigenfunctions, but good approximations for large k and in the vicinity of \alpha=2. The first lowest eigenfunctions are then calculated numerically and an approximate analytic formula for the level spectrum is derived.Comment: revised version: eqs. (2.15)ff corrected, figures actualized, arbitrary size of potential wel