Why complex absorbing potentials work: A discrete-variable-representation perspective

Abstract

The use of a complex absorbing potential (CAP) of the form $\ensuremath{-}i\ensuremath{\eta}W$ to calculate the Siegert energy of a resonance state rests on a solid mathematical foundation [U. V. Riss and H.-D. Meyer, J. Phys. B 26, 4503 (1993)]. In this paper, in order to facilitate a better understanding of the basic principles underlying the CAP method, a radial one-particle Hamiltonian with a model potential supporting resonances is analyzed. Using a purely quadratic CAP $[W(r)={r}^{2}]$, the eigenstates of $H=\ensuremath{-}(1∕2){d}^{2}∕d{r}^{2}\ensuremath{-}i\ensuremath{\eta}W(r)$ are employed to construct a discrete variable representation. The introduction of this grid method makes it transparent how using a CAP is related to the method of complex scaling, and why, in the limit of an infinite basis set, the exact Siegert energy may emerge in the spectrum as $\ensuremath{\eta}\ensuremath{\rightarrow}{0}^{+}$.