Complex rotation method for the Dirac Hamiltonian

Abstract

We show that the method of the complex rotation can be applied to the Dirac Hamiltonian. We compute positions and widths of resonance levels of hydrogenlike ions ~with Z51 and Z510), described relativisti- cally, in the presence of the uniform dc electric field ~relativistic Stark effect !. We show that, for small Z, positions of the resonance levels, given by the relativistic approach, agree well with known nonrelativistic results. The relativistic treatment introduces corrections on the same order both for the resonance positions and widths, and may, therefore, alter widths considerably. The complex rotation method is a well-known tool, allow- ing to calculate positions and widths of resonances in atomic systems @1-7#. To mention just a few applications, the method has been used in nonrelativistic quantum-mechanical calculations of resonance parameters of doubly excited states @8,9# or Stark resonances @10-12# in such systems as He, H 2 ,o r Ps 2 . The above mentioned applications of the complex rotation procedure neglected relativistic effects. Recent developments of the experimental technique perfected experimental data to a degree where relativistic effects become observable. For example, in recent publications @13,14# of 2lnl8 resonances in helium the authors show that theoretical predictions based on the nonrelativistic approach differ considerably from the experimental data. Only by inclusion of the spin-orbit effects in the scheme of an R-matrix multichannel quantum-defect calculation @15# were the experimental results explained. Another manifestation of the relativistic effects has been reported in Ref. @16#, where a peak in the electron-He 1 (1s) scattering has been observed, which is forbidden in the pure LS-coupling scheme. The recent experimental studies @17# of the effect of the strong electric field on helium hint at the important role played by the relativistic effects. Also, forbid- den in the LS coupling, doubly excited states have been ob- served in a single-photon excitation from the ground state of helium @18#. These findings indicate that, in the spectra of the doubly excited states, or in the process of the interaction of atomic states with electric field, relativistic effects may play quite an important role. Incorporation of the relativistic effects into the scheme of calculation and, in particular, proper modifi- cations of the complex rotation procedure, allowing to take full account of the relativistic effects, seem to be necessary. In the present paper we propose a modification of the complex rotation method, taking full account of the relativ- istic effects. We develop the complex rotation procedure for the Dirac Hamiltonian.