The Complex-Scaling Coupled-Channel Methods for Atomic and Molecular Resonances in Intense External Fields

Abstract

Resonance states are characterized by complex energies corresponding to poles of the resolvent operator (E-Ĥ)⁻¹ in the complex-energy plane of a non-physical higher Riemann sheet. Numerous techniques have been developed for computing these poles. One of the most powerful techniques popularized in the last decade is the method known as the complex scaling (coordinate-rotation, complex-coordinate, or dilatation) transformation. ^1,2 As a result of the complex scaling transformation, r → re^1α, the eigenvalues corresponding to the bound states of Ĥ stay invariant, while the branch cuts associated with the continuous spectrum of Ĥ are rotated about their respective thresholds by an angle −2α (assuming 0<α<π/2), exposing the complex resonance states in appropriate strips of the complex energy plane. A crucial point from the computational point of view is that the eigen-functions associated with the complex-scaling resonance wave functions are localized, i.e. square integrable. The square integrability led to the extension of well-established bound-state techniques to the determination of resonance energies (E_R) and widths (Г) of metastable states.