Symplectic integrators for the numerical solution of the Schrِdinger equation

The solution of the one-dimensional time-independent Schrِdinger equation is considered by symplectic integrators. The Schrِdinger equation is first transformed into a Hamiltonian canonical equation. The concept of asymptotic symplecticness is introduced and asymptotically symplectic methods of order up to 3 are developed. Numerical results are obtained for the one-dimensional harmonic oscillator, the hydrogen atom and the one dimensional double-well anharmonic oscillator.