Quantum operators and their action on tempered distributions

The aim of this paper is to realize that in a quantum system the observables belonging to a certain class of operators (that we call quan- tum operators) act on the entire space of tempered distributions in an extremely natural way for the applications to physics. To this end we must dene some new concepts and we must establish some their properties. The new concepts are: 1) semiquantum operators; 2) weak adjoint operators, that we will prove to be a generalization of the formal adjoints of dierential operators; 3) quantum operators; 4) value of a quantum operator on a tempered distribution. All the concepts are applied to quantum mechanics and in particular they give for the rst time a precise mathematical meaning to the so called \improper vectors" of a \physical Hilbert system", i.e. the states of a quantum system that aren't normalizable, as it's possible to nd in [Di] , [Pa], [Sha] and other.