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This article is cited in 4 scientific papers (total in 4 papers)
Nonholonomic mechanics
Normal Form of a Quantum Hamiltonian with One and a Half Degrees of Freedom Near a Hyperbolic Fixed Point
A. Yu. Anikin M.V. Lomonosov Moscow State University,
Leninskie Gory, Moscow, 119991 Russia
Abstract:
According to classical result of Moser [1] a real-analytic Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point can be reduced to the normal form by a real-analytic symplectic change of variables. In this paper the result is extended to the case of the non-commutative algebra of quantum observables.We use an algebraic approach in quantum mechanics presented in [2] and develop it to the non-autonomous case. We introduce the notion of quantum non-autonomous canonical transformations and prove that they form a group and preserve the structure of the Heisenberg equation. We give the concept of a non-commutative normal form and prove that a time-periodic quantum observable with one degree of freedom near a hyperbolic fixed point can be reduced to a normal form by a canonical transformation. Unlike traditional results, where only formal theory of normal forms is constructed, we prove a convergence of the normalizing procedure.
Keywords:
algebra of quantum observables, quantum normal forms, non-autonomous quantum dynamics.
Received: 21.05.2008 Accepted: 05.08.2008
Citation:
A. Yu. Anikin, “Normal Form of a Quantum Hamiltonian with One and a Half Degrees of Freedom Near a Hyperbolic Fixed Point”, Regul. Chaotic Dyn., 13:5 (2008), 377–402
Linking options:
https://www.mathnet.ru/eng/rcd585 https://www.mathnet.ru/eng/rcd/v13/i5/p377
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