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Hamiltonian Systems Inspired by the Schrödinger Equation
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: 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; $\mathrm{GL}(n,\mathbb C)$-invariance. 
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     author = {Vasyl Kovalchuk and Jan Jerzy Slawianowski},
     title = {Hamiltonian {Systems} {Inspired} by the {Schr\"odinger} {Equation}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a45/}
}
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Vasyl Kovalchuk; Jan Jerzy Slawianowski. Hamiltonian Systems Inspired by the Schrödinger Equation. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a45/

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