Exterior complex scaling and the computation of continuum-continuum transition matrix elements involving converging or diverging operators

摘要

A general methodology and a numerical procedure are presented for the accurate computation of continuum-continuum transition matrix elements E prime , [Left Angle Bracket] l prime |T|E, l [Right Angle Bracket] , where T couples any two scattering states of a single-variable Hamiltonian. The procedure makes use of the exterior complex scaling of the r-coordinate [r->r0+(r-r0)exp(iθ)], widely known for its precious properties in resonance quantization. It is applicable whatever the potential interaction, provided its functional form is globally analytic asymptotically. As coupling T, any holomorphic function of the coordinate can be considered for which the matrix element is meaningful. This methodology enables one to handle systematically not only converging operators, but also diverging ones (such as the electric dipole moment in length form) for which standard real-coordinate integration fails. Graphical representation of the imaginary part of the θ-dependent complex matrix element against its real part shows that the exact value of the integral is clearly marked by a cusp, loop or inflection on a well defined θ-trajectory, analogously with resonance quantization.