Discrete variable method for non-integrable quantum systems

Abstract

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 important for states close to the classical ionization energy or above. These resonance states can be studied using the complex coordinate method. In this method the Hamiltonian of the system is continued into the complex plane by a complex dilatation, therefore the Hamiltonian is no longer Hermitian and can support complex eigenenergies associated with decaying states. Resonances are uncovered by the rotated continuum spectra with complex eigenvalue and square-integrable (complex rotated) eigenfunctions. The basic idea is to combine this complex coordinate rotation method with the finite element method, and the discrete variable technique. These two methods have been successfully used to compute atomic data for the hydrogen atom in external magnetic and electric fields. We obtain a complex symmetric Hamiltonian matrix, which we solve using the implicitly restarted Arnoldi method (ARPACK). These methods have been extended to alkali atoms in external strong magnetic and electric fields by including model potentials and have also been successfully used in studying various effective one-particle problems.