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8.2 Nearly hyperscalar potentials

The potential in Eq. (8.1) is a special case of a more general class of

nearly hyperscalar potentials

∑ V (r1,r2,r3) = V0(ξ) + v(rij) , i<j
(8.24)

or, if one gives up the two-body character,

V (r1,r2,r3) = V0(ξ) + δV (r1,r2,r3) ,
(8.25)

or equivalently, in the more abstract terms of the hyperspherical expansion of the potential

V (r ,r ,r ) = π3∕2 [V (ξ)Q (Ω) + V (ξ)Q (Ω) + V (ξ)Q (Ω) + ...] , 1 2 3 0 0 4 4 6 6
(8.26)

where Q0(Ω) = π−3∕2, Q 4(Ω), etc, are the lowest scalar and fully symmetric hyperspherical harmonics, given in Section 5.2, and where the L > 0 components V 4, V 6,... are small. For simplicity, we use the same notations α = [56, 0+], [70, 1−], [20, 1+]…as for the harmonic-oscillator states to stress their angular momentum and permutation properties.

If the potential V is nearly hyperscalar, one may approximate the wave function of each state by its lowest hyperspherical component, say

Ψ (ρ,λ ) = uL,α-P (Ω ) , L,α ξ5∕2 L,α
(8.27)

where uL,α and the binding energy are given by the single radial equation (m = ℏ = 1),

′′ (L-+-3-∕2)(L-+-5∕2) u L,α − ξ2 uL,α + [E − UL,α]uL,α = 0 ,
(8.28)

∫ UL,α = P ∗L,α(Ω )V (r1, r2,r3)PL,α(Ω )dΩ.
(8.29)

Inserting the expansion (8.26) results in the following expressions for the UL,α(ξ):

(L = 0)
+ [56, 0 ] V0(ξ) ,
(8.30)

(L = 1)
[70, 1− ] V0(ξ) ,
(8.31)

(L = 2)
√ -- [20,1+ ] V (ξ) + 5--3V (ξ) , 0 1√5-- 4 3 [70,2+ ] V0(ξ) + ---V4(ξ) , 1√5-- + --3- [56,2 ] V0(ξ) − 315 V4 (ξ) , √ -- [70,0+ ] V0(ξ) − 5--3V4 (ξ) , 15
(8.32)

(L = 3)
√ -- √ -- [56,3− ] V (ξ) + --3V (ξ) + --2V (ξ ) , 0 √7--4 √7-- 6 3 2 [20,3− ] V0(ξ) + ---V4(ξ) − ---V6 (ξ ) , 7√ -- 7 − 5--3- [70,3 ] V0(ξ) − 21 V4(ξ) , √ -- [70,2− ] V0(ξ) + --3V4(ξ) , √3-- √ -- − 3 2 [56,1 ] V0(ξ) − -3-V4(ξ) + -2-V6 (ξ ) , √ -- √ -- [20,1− ] V (ξ) − --3V (ξ) − --2V (ξ ) , 0 3 4 2 6 [70,1− ] V0(ξ) .
(8.33)

These expressions can be obtained either by direct angular integration of Eq. (8.29), or by considering the special case V 0(ξ) = ξ2, and using the formula of Section 8.1 for the potentials v(rij) = rijn, with n = 0, 4 or 6. This enables us to switch on one after the other the L = 0, L = 4 and L = 6 multipoles of the potential. This is one more illustration of the many links between the hyperspherical formalism and the harmonic oscillator [43].

If one compares the Eqs. (8.32) with the N = 2 sequence in the harmonic oscillator as appearing in Eqs. (8.6) and (8.14), one notices the absence of the [56, 0+]. This state is a hyper-radial excitation of the L = 0 ground state. Similarly, the [70, 1−] state of the N = 3 band is a hyper-radial excitation of the L = 1 level.

Now, if the radial equation (8.28) is first solved with V 0(ξ) alone and if the terms in V 4(ξ) and V 6(ξ) are treated at first order, one gets very simple relations between the energies. In the positive-parity sector, one gets the splitting pattern of Fig. 8.2. The negative-parity states are shown in Fig. 8.4. For illustration we have chosen a linear potential V = Σ rij, treated to first order around its hyperscalar approximation, but the ratios between the various splittings in each column are model-independent.

PICT


Figure 8.4: Splitting pattern of the negative-parity states for a linear potential treated at first order around its hyperscalar approximation. The bottom line (70, 1−)′ corresponds to the hyper-radial excitation of the L = 1 state. The other states belong to the L = 3 multiplet. The figure exhibits the splitting pattern one gets when switching on the corrections V 4 and V 6 to the hyperscalar potential V 0.

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