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A covariant non-local extention if the stationary Schr\"odinger equation ispresented and it's solution in terms of Heisenbergs's matrix quantum mechanicsis proposed. For the special case of the Riesz fractional derivative, thecalculation of corresponding matrix elements for the non-local kinetic energyterm is performed fully analytically in the harmonic oscillator basis and leadsto a new interpretation of non local operators in terms of generalized Glauberstates.  As a first application, for the fractional harmonic oscillator the potentialenergy matrix elements are calculated and the and the correspondingSchr\"odinger equation is diagonalized. For the special case of invariance ofthe non-local wave equation under Fourier-transforms a new symmetry is deduced,which may be interpreted as an extension of the standard parity-symmetry.

Solutions of the fractional Schr\"odinger equation via diagonalization - A plea for the harmonic oscillator basis part 1: the one dimensional case