Numerical Solution of the two-dimensional time independent Schrödinger Equation by symplectic schemes†

The solution of the two-dimensional time-independent Schrödinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and solved numerically by asymptotically symplectic methods. The problem is then transformed into an algebraic eigenvalue problem involving real, symmetric matrices. The eigenvalues of the two-dimensional harmonic oscillator and the twodimensional Henon-Heils potential are computed by the application of the methods developed. The results are compared with the results produced by full discretization. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)