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2 Opuscula Mathematica

2 2006

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Efficient computation of the MCTDHF approximation to the time-dependent Schrodinger equation

Koch O.

Opuscula Mathematica

2006

Vol. 26, no. 3

483-497

We discuss analytical and numerical properties of the multi-configuration time-dependent Hartree-Fock method for the approximate solution of the time-dependent multi-particle (electronic) Schrödinger equation which are relevant for an efficient implementation of this model reduction technique. Particularly, we focus on a discretization and Iow rank approximation in the evaluation of the meanfield terms occurring in the MCTDHF equations of motion, which is crucial for the computational tractability of the problem. We give error bounds for this approximation and demonstrate the achieved gain in performance.

Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations

Auzinger W., Karner E., Koch O., Weinmuller E.

Opuscula Mathematica

2006

Vol. 26, no. 2

229-241

We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim.