The Siegert states have long been recognized as a potentially powerful tool in the formal scattering theory. Here we propose an efficient method to implement this power in practice. Our method yields bound states, complex-energy resonances, and scattering wave functions, i.e., a complete solution of the Schrödinger equation. We also obtain a representation of the Green function suitable for a variety of applications. The method is demonstrated by realistic examples of the eep and dtm three-body Coulomb systems. [S0031-9007(97)03930-6] PACS numbers: 34.10. + x, 03.65.Nk, 34.80.Bm, 36.10.Dr In 1939, in search of a formal derivation of the Breit-Wigner resonance formula, Siegert introduced [1] a class of solutions to the Schrödinger equation which now bear his name. The Siegert states satisfy outgoing wave boundary conditions, and, for the simplest generic scattering problem, they are defined by ∑ 2 1 2These equations can be satisfied simultaneously only for a discrete set of generally complex momenta k n ; thus one should consider Eqs. (1a) and (1b) as an eigenvalue problem defining k n and corresponding eigenfunctions f n ͑r͒. The eigenvalues k n coincide with the poles of the S matrix in the complex k plane. Pure imaginary k n with Im͑k n ͒ . 0 represent bound states of the system. Those lying close to the real k axis correspond to resonances. Others, though not observable directly, complement the set and, at least for finite range potentials, suffice to determine the S matrix in the entire k plane [2]. The significance of Siegert states within a general formalism of the scattering theory and their potential usefulness for computational implementations stem from the fact that they provide a possibility of unified description of bound states, resonances, and continuum spectrum in terms of a purely discrete set of states. For example, by virtue of the Mittag-Leffler expansion theorem (if applicable), one can construct the Green function simply in the form of a sum over the Siegert states, avoiding the annoying integral over the continuum [3]. Of course, there is a price to be paid for such a discretization. First, one must move away from the real energy axis to the complex energy plane. Second, for practical applications one must solve the problem of exponential divergence of f n ͑r͒ at r !`, say, for instance, by making the radial variable r also complex as in the complex rotation method [4]. However, the most important point is that in order to achieve the completeness required for representing the continuum, one has to generate not just one or a few but many solutions of Eqs. (1a) and (1b). This causes an essential practical difficulty, because the nonlinearity of Eqs. (1a) and (1b) with respect to the eigenvalue k prohibits a direct application of the variational methods, which renders the problem tractable only by means of an iterative procedure. In fact, this difficulty was restricting the power of Siegert states to studying only individual resonances [5].Here, we define Siegert pseudo-states by applying the...
