Extended Sobolev and Hilbert spaces and approximate stationary solutions for electronic systems within the non-linear Schrodinger equation

The definition of Sobolev spaces, which has already been shown to be a conv enient way to set up the Schrodinger equation for approximate stationary so lutions within extended Hilbert spaces, is readily generalized in order to express, in a similar way, the so-called nonlinear Schrodinger equation (NL SE). The unavoidable theory, related to extended Hilbert and Sobolev spaces , is previously described in order to design the formalism inherent to the approximate NLSE. Afterwards the nature of the NLSE stationary solutions is discussed. The procedure uses as a basic tool an implied N-electron quantu m self-similarity measure, provided with the structure of an overlap-like m easure form, involving the integral of the fourth power of the N-electron w avefunction. Computation of this theoretical element is sketched and a two- electron case is developed as an illustrative example within the LCAO MO fr amework. The N-electron Slater determinant situation is also presented unde r the additional help of the nested sums formalism. It is shown afterwards that addition of second order gradient terms on the extended wavefunction p rovides variation of mass with velocity corrections into the energy express ion. Finally, use into the Hamilton operator of exponential terms depending on the density functions in the extended Hilbert spaces formalism provides the theory with a general structure.