A. Anikin
Publications:
Anikin A.
Normal Form of a Quantum Hamiltonian with One and a Half Degrees of Freedom Near a Hyperbolic Fixed Point
2008, vol. 13, no. 5, pp. 377402
Abstract
According to classical result of Moser [1] a realanalytic Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point can be reduced to the normal form by a realanalytic symplectic change of variables. In this paper the result is extended to the case of the noncommutative algebra of quantum observables.We use an algebraic approach in quantum mechanics presented in [2] and develop it to the nonautonomous case. We introduce the notion of quantum nonautonomous canonical transformations and prove that they form a group and preserve the structure of the Heisenberg equation. We give the concept of a noncommutative normal form and prove that a timeperiodic 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.
