From Schrödinger equation to tempered distribution space on Minkowsky chronotope and Schwartz Linear Algebra

In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the Schrodinger wave functions (wave functions classically belonging to the domain of the classic Schrodinger equation) to the space of tempered distributions defined upon the Minkowski space time. It’s ex tremely natural to consider the space E of complex valued smooth functions as the nat ural domain of the Schrodinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrodinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert‘s pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H2 is not good for quantum me chanics and the de Broglie solutions of the Schrodinger equation do not belong to L2. We show in this article that moving inside the very good space E - and finally inside the huge tempered distribution space S′- represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.