The One-dimensional Fractional Supersymmetric Quantum Mechanical Operator of Momentum

In the case of the quantum generalization of stable Lévy processes, expressions for the Hermitian operator of momentum and its eigenfunctions are proposed. The normalization constant has been determined and its relation to the translation operator is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality α. The simplest nonlocal superalgebra is introduced.