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

The use of a complex absorbing potential (CAP) of the form $-i\eta W$ 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 $[W\left(r\right)=r^2]$, the eigenstates of $H=-(1∕2)d^2∕d{r}^2-i\eta W\left(r\right)$ 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 $\eta \to 0^{+}$.

Figure 1

The spectrum of $\mathit{H}\left(\eta \right)$ [Eq. (21)].

Figure 2

Eigenvalues in the vicinity of the resonance at $3.4-i0.013$. (a) Full $\mathit{H}\left(\eta \right)$, Eq. (21). (b) After deleting $-i\eta W$ from Eq. (21).