Title in English: Smooth and F-smooth systems with applications to Covariant Quantum Mechanics We review the geometric theory of smooth systems of smooth maps, of smooth systems of smooth sections of a smooth double fibred manifold and of smooth systems of smooth connections of a smooth fibred manifold. Moreover, after reviewing the concept of F-smooth space due to A. Froelicher, we discuss the F-smooth systems of smooth maps, the F-smooth systems of fibrewisely smooth sections of a smooth double fibred manifold, the F-smooth systems of fibrewisely smooth connections of a smooth fibred manifold and of F-smooth connections of an F-smooth system of fibrewisely smooth sections. Further, we discuss three applications of the general geometric theory, which are taken in the framework of Covariant Quantum Mechanics. Namely, we discuss the smooth upper quantum connection, the F-smooth sectional quantum bundle and the Schroedinger operator regarded as an F-smooth connection of the F-smoth sectional quantum bundle.

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