The Schr\"odinger equation and its generalized analogs derived by the quantum hydrodynamic representation

The derivation of the Schr\"odinger-like equations from the system of equations of the quantum hydrodynamic analogy (QHA) is analyzed in presence of fluctuations. If in absence of fluctuation each QHA solution can be tracked back to the equivalent one of the Schr\"odinger problem, in presence of fluctuations the work shows that the ensemble of solution of the QHA equations is wider than the Schr\"odinger one and the tracking-back procedure from the QHA problem to the original Schr\"odinger equation is not generally possible. The break-down of the mathematical correspondence is inspected and correlated to the characteristics of the states. Finally, the paper shows that if the form of Schr\"odinger equation is generalized, the QHA stochastic case as well as its classical limit can be translated into a Schr\"odinger-like representation.