Schwartz-Green's families for quantum mechanics

This paper is devoted to the concept of Green families associated with a linear continuous endomorphism defined on the space of tempered distributions S'n. Our concept of Green family is an operative and rigorous version (counterpart) of some formal type of Green function used in Theoretical Physics, Quantum Mechanics, Quantum Chemistry and Engineering and sometime also in Mathematical Physics to solve some kind of Linear Differential Equations. The relationships between our method and the approach by fundamental solutions is immediately showed, to solve inhomogeneous linear equations.