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Perturbation theory and control in classical or quantum mechanics by an inversion formula

Michel Vittot

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We consider a perturbation of an 'integrable' Hamiltonian and give an expression for the canonical or unitary transformation which 'simplifies' this perturbed system. The problem is to invert a functional defined on the Lie-algebra of observables. We give a bound for the perturbation in order to solve this inversion, and apply this result to a particular case of the control theory, as a first example, and to the 'quantum adiabatic transformation', as another example.


PACS

03.65.Fd Algebraic methods

02.20.Sv Lie algebras of Lie groups

MSC

17B45 Lie algebras of linear algebraic groups (See also 14Lxx and 20Gxx)

81R50 Quantum groups and related algebraic methods (See also 16W35, 17B37)

Subjects

Mathematical physics

Quantum information and quantum mechanics

Dates

Issue 24 (18 June 2004)

Received 10 November 2003

Published 2 June 2004



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