  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 <r>~0.3[fm]. The
derived results indicate, that a fractional wave equation may be an appropriate
tool for a description of quark-like particles.
