Quantum Operator of Momentum in One-dimensional Lévy Processes

For the quantum generalization of Levy processes, expressions for the one-dimensional 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 alpha. Some aspects of fractional derivatives are discussed.