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gular eigenfunctions φ$_{n}$ and their corresponding eigen- values λ$_{n}$ . The three body wave function is then written as ψ = 1 ρ 5 / 2 ∑ $_{n}$f$_{n}$ ( ρ ) φ$_{n}$ ( x , y ), where n labels each of the adiabatic terms. The radial functions f$_{n}$ ( ρ ) are obtained after solving a coupled set of radial equations where the λ$_{n}$ angular eigenvalues enter as effective adiabatic poten- tials. The coupling between the different adiabatic terms appears through the functions P$_{nn}$ ′ ( ρ ) and Q$_{nn}$ ′ ( ρ ) de- fined for instance in [3]. 

Each adiabatic potential describes a specific relative structure of the three particles for a given root mean square radius, ρ , with wave function φ and eigenvalue λ . When only one adiabatic potential is considered, the coupled set of radial equations reduces to 

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where Q is the diagonal coupling term Q$_{nn}$ , which is in general not zero (contrary to what happens with P$_{nn}$ ′ , whose diagonal terms are zero). V 3 $_{b}$( ρ ) is the three-body potential usually used in three-body calculations to take into account all those effects that go beyond the two- body interactions. The total wave function, ψ , and Q are given by 

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The expectation value is over angular coordinates, Ω, ex- cluding only ρ . When only s -waves contribute for both x and y the angular wave function φ only depends on ρ and α . For short-range attractive two-body interactions the angular eigenvalue λ ( ρ ) would be monotonously in- creasing towards a constant asymptotic value. 

Narrow two-body resonance. When the α 1 − α 2 two- body s -wave interaction supports a bound state, the adi- abatic potential approaches the bound state energy at large distance. This reflects a two-body structure corre- sponding to particle n far away from the α 1 − α 2 bound state. For less α 1 − α 2 attraction this state moves into the continuum. With a confining Coulomb barrier from repelling charges on the particles, the state would appear as a resonance at low energy E 2 . The adiabatic poten- tial would then approach the positive value equal to E 2 corresponding to the large-distance two-body structure of n far away from the resonance α 1 − α 2 . We illustrate in Fig.1 by the specific examples, $^{9}$Be( α + α + n ) and $^{9}$B( α + α + p ). 

This description is only correct when the two-body res- onance width Γ 2 is very small and the coupling to the three-body continuum states can be ignored. In general the first adiabatic potential is crossed by numerous oth- ers while ρ increases. However, the couplings are neg- ligibly small for three-body energies E until distances ρ ≃ $_{9}$√$_{E}$ − E 2 / Γ 2 where the energies are in MeV and ρ in fm. Thus for small Γ 2 , say eV or keV, the couplings 

for moderate energies below 1 MeV can be neglected far outside the distance where the short-range n − ( α 1 − α 2 ) interaction has vanished. Then the system can effectively be described as a two-body system until the two-body resonance eventually decays. 

Let us now consider the three-body system with inter- actions leading to a three-body state of positive energy E$_{r}$ but below E 2 . Effectively this is a bound n − ( α 1 − α 2 ) two-body state, or rather a resonance decaying precisely with the width Γ$_{r}$ = Γ 2 of the α 1 − α 2 resonance. For less n − ( α 1 − α 2 ) s -wave attraction this two-body bound state moves into the continuum above E 2 . The expecta- tion is that the proper description is as a virtual state with no width in contrast to a resonance [9]. However, this is not necessarily true. 

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The $^{9}$Be( 1 / 2 $^{+}$) example. The low-lying states of $^{9}$Be are expected to be well described as cluster states consist- ing of one neutron and two α -particles [16]. The α − α in- teraction, including short range attraction and Coulomb repulsion, produces a low-lying s -wave two-body reso- nance at 0 . 0918 MeV with a width around 9 eV. Adding one neutron in s -waves leads to angular momentum and parity 1 / 2 + of the resulting three-body system. Such a state is listed in the tables of energies [17] at an excita- tion energy of 1 . 684 MeV, or 0 . 110 MeV above threshold, with a width of 0 . 217 MeV (see the upper part of Fig.1a). Thus the state is above the two-body resonance energy by 0 . 018 MeV and s -waves are most likely the dominating composition. 

The listed width is much larger than the distance to the two-body resonance threshold and even about two times larger than the three-body energy itself. It is a peculiar resonance structure which apparently extends into the bound state region below the threshold. These values are 

