Lie systems and Schr"odinger equations (original) (raw)

2016, arXiv: Mathematical Physics

Abstract

We prove that ttt-dependent Schr\"odinger equations on finite-dimensional Hilbert spaces determined by ttt-dependent Hermitian Hamiltonian operators can be described through Lie systems admitting a Vessiot--Guldberg Lie algebra of K\"ahler vector fields. This result is extended to other related Schr\"odinger equations, e.g. projective ones, and their properties are studied through Poisson, presymplectic and K\"ahler structures. This leads to derive nonlinear superposition rules for them depending in a lower (or equal) number of solutions than standard linear ones. Special attention is paid to applications in nnn-qubit systems.

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