ABSTRACT
There may or may not be a solution of (7.1), and if there is one it may not
be unique [188]. Since (7.1) is intractable, reduced or equivalent models that are
tractable are then introduced. We see that Hartree-Fock equations and Kohn-Sham
equations are the most widely used models in electronic structure calculations. Note
that electrons are fermions. One reduced model for a type of bosons is the so-called
Gross-Pitaevskii equation, which is used to model a Bose-Einstein condensation
(BEC) of ultracold dilute gas with N identical bosons confined in an external trap. Such kind of models are nonlinear eigenvalue problems in R3, which usually
require a so-called self-consistent field (SCF) iteration to linearize in computation.
Consequently, the central computation in the application is the solution of the fol-
lowing linear Schrödinger equation
