A covariant non-local extention if the stationary Schrodinger equation is presented and its solution in terms of Heisenbergss matrix quantum mechanics is proposed. For the special case of the Riesz fractional derivative, the calculation of corresponding matrix elements for the non-local kinetic energy term is performed fully analytically in the harmonic oscillator basis and leads to a new interpretation of non local operators in terms of generalized Glauber states. As a first application, for the fractional harmonic oscillator the potential energy matrix elements are calculated and the and the corresponding Schrodinger equation is diagonalized. For the special case of invariance of the non-local wave equation under Fourier-transforms a new symmetry is deduced, which may be interpreted as an extension of the standard parity-symmetry.
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