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
author: R. Herrmann
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.
The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charmonium spectrum may be classified by the derived angular momentum eigenvalues for stepsize=2/3.
authors: R. Herrmann, G. Plunien, M. Greiner, W. Scheid
The free Schrödinger equation in Riemannian collective spaces with an arbitrary number of dimensions is linearized. The coupling of external fields is discussed. The operators to classify collective spin states are introduced and the collective Pauli equation is derived.
reference:International Journal of Modern Physics A, Volume 4, Issue 18, pp. 4961-4975 (1989).