Normal form of a quantum Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point

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.