J. Phys. A: Math. Gen. 30 (1997) 5543–5549. Printed in the UK PII: S0305-4470(97)83018-1 Double poles of the S -matrix in a two-channel model W Vanroose†, P Van Leuven‡, F Arickx† and J Broeckhove† † Departement Wiskunde-Informatica, Universiteit Antwerpen, B-2020 Antwerpen, Belgium ‡ Departement Natuurkunde, Vrije Universiteit Brussel, B-1050 Brussel, Belgium Received 3 April 1997 Abstract. The poles of the S-matrix for a two-channel model with square well potentials are calculated. It is found that the trajectories of these poles in the complex k-plane for varying coupling strength show avoided crossings. At the critical parameters the poles coincide to form a double pole. Its presence is revealed by a symmetry property of the cross section. 1. Introduction In this paper we address the problem of degeneracy of two resonant states or, more generally, to the occurrence of double complex poles of the S -matrix. In recent literature a number of cases of interfering resonances leading to degeneracy have been described. Hernandez and Mondragon [1] considered a doublet of unbound states in 8 Be as an example of accidental degeneracy of resonances. Kylstra and Joachain [2] discussed double poles in the case of laser-assisted electron–atom scattering. Latinne et al [3] studied degeneracies involving autoionizing states in complex atoms. Double poles have been investigated as examples of non-exponential decay laws. Lassila and Ruuskanen [4] examined atomic resonance fluorescence using a parametrized form of the T -matrix. Bell and Goebels [5] proposed a one-channel double well potential model and a Lee-type model of unstable particles as cases with double poles. Here we describe another scenario for the origin of double poles using a simple two- channel model with square well potentials. Although, in the past, such a model has been used [6] in the context of resonance scattering theory, it has, to our knowledge, not been reported in connection with double poles. We focus our investigation on the role of the coupling strength in the generation of double poles. We present a study of the trajectories of the poles of the S -matrix and show that critical values exist of the model parameters which give rise to a double pole. The situation of coupled channels, one of which is closed, is encountered a.o. in the study of continuum effects of monopole and quadrupole degrees of freedom in 4 He [7]. It reveals the appearence of Feshbach resonances above the monopole but below the quadrupole threshold. However, the relation with the quadrupole eigenstates (in the uncoupled problem) is not clear: some give rise to a resonance, some do not. In order to gain insight into this situation we turn to an exactly soluble model, in particular one that allows us to vary the coupling strength between the channels because we feel this strength will be a determining factor. Indeed, the trajectory as a function of the coupling strength of a pole of the S -matrix reveals the relation of resonances in the coupled problem to states (bound or continuum) in the uncoupled problem. Changing the potential parameters (other than the coupling 0305-4470/97/155543+07$19.50 c  1997 IOP Publishing Ltd 5543