Discrete variable method for non-integrable quantum systems

The states of an atom in external electric fields become quasi-bound since the electron can ionize by tunneling through the potential barrier into the continuum. Due to the external electric field the ionization threshold of the atom is lowered from the field-free value. This process becomes importa nt for states close to the classical ionization energy or above. These reso nance states can be studied using the complex coordinate method. In this me thod the Hamiltonian of the system is continued into the complex plane by a complex dilatation, therefore the Hamiltonian is no longer Hermitian and c an support complex eigenenergies associated with decaying states. Resonance s are uncovered by the rotated continuum spectra with complex eigenvalue an d square-integrable (complex rotated) eigenfunctions. The basic idea is to combine this complex coordinate rotation method with the finite element met hod, and the discrete variable technique. These two methods have been succe ssfully used to compute atomic data for the hydrogen atom in external magne tic and electric fields. We obtain a complex symmetric Hamiltonian matrix, which we solve using the implicitly restarted Arnoldi method (ARPACK). Thes e methods have been extended to alkali atoms in external strong magnetic an d electric fields by including model potentials and have also been successf ully used in studying various effective one-particle problems. (C) 1999 Els evier Science B.V. All rights reserved.