On a dynamic system arising from a finite-dimensional approximation of the Schrödinger equation
The system of ordinary differential equations (1) \(J\dot z= \nabla_z H(z, \sigma)\), \(\dot\sigma J= [\sigma J, JH''_0]\) is studied. The author shows that system (1) is Hamiltonian with respect to a Poisson bracket that is the direct sum of the usual Poisson bracket on \(\mathbb{R}^{2n}\) and the linear Poisson bracket on the Lie algebra \(\text{sp}(2n, \mathbb{R})\), to which the matrix \(\sigma J\) belongs. The system (1) is studied in the case corresponding to a one-dimensional anharmonic oscillator. The normal form method is used to investigate the behavior of solutions; in particular, the family of periodic solutions is described.
scientific article; zbMATH DE number 830729