Xabier Telleria-Allika

Arbitrary order calculus is a natural generalisation of usual calculus in which the order of differentiation and integration operators in not restricted to integer numbers. In engineering fractional order derivatives are used for describing the behaviour of materials with memory (i.e. viscoelastic materials) due to the fact that these materials lay somewhere in between Hookean springs and Newtonean fluids[1,2]; there are also many dynamical systems which can be better described when arbitrary order derivatives are included [3].

Lanskin [4] formulated the first Fractional Schrödinger Equation (FSE) along with the Fractional Continuation Equation in 2002; however, we are still far from fully understanding the effect pf the FSE on physical properties such as: Tunnelling [5], Diffraction [6] and Scattering [7]. Due to the properties of fractional derivatives, many jobs have been done in which relativistic properties and effects of extrinsic magnetic fields are obtained by incorporating an arbitrary order to the kinetic energy in the Hamiltonian [8,9].

Further studies of the FSE applied on astrophysically interesting systems such as \(H_2^+\) [10] and even hydrogen atom [11] seem to be promising. We shall take the FSE for a particle in a ring (1) as a first step into this world for which the eigenvalues are (2) and the eigenfunctions (3)

\(
\left[\frac{1}{2mr^2}\right]^{\alpha-1}\left[i\hslash \partial_\theta \right]^\alpha \Psi_\alpha(\theta;r)=\lambda_\alpha\Psi_\alpha(\theta;r) \tag{1}
\)

\(
\lambda_\alpha = \left[\frac{1}{2mr^2}\right]^{1-\alpha} N^\alpha \hslash^\alpha \exp \left[ i \pi \alpha \left(n+1\right)\right]\; | \; N, \; n \in \mathbb{N}+\{0\} \tag{2}
\)

\(
\Psi_\alpha(\theta;r)=C\exp \left(-i N^\alpha\left[ \frac{\hslash}{2mr^2}\right]^{\alpha-1}\exp \left(i \pi \alpha \left(n+1\right)\right)\theta \right). \tag{3}
\)

References
[1] M. Stiassnie, 1979, Appl. Math. Modelling, 3, 300.

[2] M. Du et al., 2013, Scientific Reports, 3, 3431.

[3] V. E. Tarasov, 2013, Int. J. Mod. Phys. B, 2013, 9, 1330005.

[4]N. Laskin, Physics Review E, 2000, 66, 056108.

[5] E. Capelas et al., 2011, J. Phys. A 44, 185303.

[6] Y. Zhang et al., 2015, Scientific Reports 6, 23645.

[7] A. Liemert, 2016, Mathematics, 4, 31.

[8] J. Lorinczi and J. Malecki, 2012, J. Diff. Eq., 253, 2846.

[9] J. Blackledge and B. Babajanov, 2013, Math. Aeterna, 3, 601.

[10] A. Turbiner et al., 1999, JETP Letters, 11, 69.

[11] A. I. Arbab, 2012, J. Modern Physics, 3, 1737.