On singular Lagrangian underlying the Schrödinger equation

We analyze the properties that manifest Hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution of the Schrödinger equation with time independent potential can be presented in the form Ψ = (- frac(ℏ², 2 m) Δ + V) φ{symbol} + i ℏ ∂_t φ{symbol}, where the real field φ{symbol} (t, xⁱ) is some solution of nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field φ{symbol}. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.