Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 046, 9 pages      arXiv:0805.4024
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Hamiltonian Systems Inspired by the Schrödinger Equation

Vasyl Kovalchuk and Jan Jerzy Slawianowski
Institute of Fundamental Technological Research, Polish Academy of Sciences, 21, Swietokrzyska str., 00-049 Warsaw, Poland

Received October 30, 2007, in final form April 25, 2008; Published online May 27, 2008

Described is n-level quantum system realized in the n-dimensional ''Hilbert'' space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics of G are calculated. The usual, first- and second-order modified Schrödinger equations are obtained.

Key words: Schrödinger equation; Hamiltonian systems on manifolds of scalar products; n-level quantum systems; scalar product as a dynamical variable; essential non-perturbative nonlinearity; conservation laws; GL(n,C)-invariance.

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