On the application of the variational principle to a type of nonlinear ’’Schrödinger equation’’

Abstract

’’Schrödinger equations’’ in which the potential of the ’’Hamiltonian’’ H (φ) =H0+V (φ) depends on the first order density σσ− have received some attention in recent literature. The direct variation of the functional <φ‖H (φ) ‖φ≳/<φ‖φ≳ does not lead to the eigenequation H (φ) φ=εφ. A variational functional J (φ) is proposed which has the form J (φ) =H (φ) −(q/q+1) V(q)(φ), q being the degree in φφ*. The functional j (φ) leads to solutions of the eigenequation H (φ) φ=εφ and, provided the potential Vq(φ) fulfil <φ‖V(q)(φ) ‖φ≳? (q+1) < (q+1) <φ‖V‖φ≳, its expectation value is bounded by the lowest exact eigenvalue E0, i.e., <σ‖J (φ) ‖φ≳?E0.