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

Abstract:

The definition of Sobolev spaces, which has already been shown to be a convenient way to set up the Schrödinger equation for approximate stationary solutions within extended Hilbert spaces, is readily generalized in order to express, in a similar way, the so-called nonlinear Schrödinger equation (NLSE). 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 quantum self-similarity measure, provided with the structure of an overlap-like measure form, involving the integral of the fourth power of the N-electron wavefunction. Computation of this theoretical element is sketched and a two-electron case is developed as an illustrative example within the LCAO MO framework. The N-electron Slater determinant situation is also presented under the additional help of the nested sums formalism. It is shown afterwards that addition of second order gradient terms on the extended wavefunction provides variation of mass with velocity corrections into the energy expression. 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.

Año de publicación:

2001

Keywords:

• Sobolev spaces
• N-electron wavefunctions
• Kinetic energy distribution
• Quantum self-similarity measures
• Non-linear Schrödinger equation
• Extended Hilbert spaces
• Nested sums formalism

Fuente:

scopus