On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

Abstract

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system ẋ=A+εQt, where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as Qt. Applications to the Schrödinger equation are also given.

Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems

“…From Theorem 2, we see that, for most small ε > 0, Equation ( 44) is changed into a constant coefficient system. Hence, similar to Xue ( [12]), by an analytic almost periodic transformation, Equation ( 44) is transformed into…”

“…, make the change of variable P(t) = SW(t)S −1 , and define R(t) = S −1 M(t)S. Equation (12) becomes…”

“…Furthermore, JP J = JP T J = 0. From the uniqueness of solution of (12) with P = 0, it follows that JP J = JP T J ; hence, P is Hamiltonian.…”

“…Later, many authors ( [3][4][5][6][7][8][9][10][11][12]) paid attention to the reducibility for the following quasiperiodic linear system:…”