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- Title
On the variational principle for the non-linear Schrödinger equation.
- Authors
Mihálka, Zsuzsanna É.; Margócsy, Ádám; Szabados, Ágnes; Surján, Péter R.
- Abstract
While variation of the energy functional yields the Schrödinger equation in the usual, linear case, no such statement can be formulated in the general nonlinear situation when the Hamiltonian depends on its eigenvector. In this latter case, as we illustrate by sample numerical calculations, the points of the energy expectation value hypersurface where the eigenvalue equation is satisfied separate from those where the energy is stationary. We show that the variation of the energy at the eigensolution is determined by a generalized Hellmann–Feynman theorem. Functionals, other than the energy, can, however be constructed, that result the nonlinear Schrödinger equation upon setting their variation zero. The second centralized moment of the Hamiltonian is one example.
- Subjects
VARIATIONAL principles; NONLINEAR Schrodinger equation; EIGENVALUE equations; NONLINEAR functional analysis; NUMERICAL calculations
- Publication
Journal of Mathematical Chemistry, 2020, Vol 58, Issue 1, p340
- ISSN
0259-9791
- Publication type
Article
- DOI
10.1007/s10910-019-01082-5