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

The use of a complex absorbing potential (CAP) of the form −iηW 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)=r²], the eigenstates of H=−(1∕2)d²∕dr²−iηW(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 η→0⁺.