62
N. Crampé and A. M. Grundland
The generators S z and S ± of the su(2) Lie algebra satisfy the commutation relations
[S
z , S
±
] = ±S
± ,
[S
+ , S
−
] = 2S
z ,
(20)
and they are identified with the following (2s + 1) × (2s + 1) matrices [8]
(σ
z ) ij = (s − i)δ ij ,
(21)
(σ
+ ) ij =
(2s − j + 1)j δ i,j −1 ,
0 ≤ i, j ≤ 2s
(22)
(σ
− ) ij =
(2s − i + 1)iδ i−1,j .
(23)
Hence the matrices S z and S ± can be decomposed as a linear combination of the
matrices σ z and σ ± , namely
⎛
⎝
S z
S +
S −
⎞
⎠ =
1
1 + ξ + ξ −
⎛
⎝
ξ + ξ − − 1 −ξ − −ξ +
2ξ −
ξ 2
− −1
2ξ +
−1 ξ 2
+
⎞
⎠
⎛
⎝
σ z
σ +
σ −
⎞
⎠ ,
(24)
where (S + ) † = S − and (S − ) † = S + . The eigenvalue problem for the spin matrix
S z is given by
S
z f k = (k − s)f k ,
S
z (S
± f k ) = (k ± 1 − s)(S
± f k ),
for 0 ≤ k ≤ 2s.
Under these circumstances the following holds
Proposition 2 (Recurrence Relations Associated with the CP 2s Models) For the
Veronese sequence of analytic solutions f k of the CP 2s model (9), the algebraic
recurrence relations for the vectors S z f k and S ± f k are given by [6]
S
+ f k =
−(1 + ξ + ξ − )f k+1 for 0 ≤ k ≤ 2s − 1,
0 for k = 2s,
(25)
S
− f k =
1
1 + ξ + ξ −
k(k − 1 − 2s)f k−1 for 0 ≤ k ≤ 2s.
(26)
In terms of the projectors P k , the recurrence relations (7) take the algebraic form
P k+1 = Π + (P k ) :=
S + P k S −
tr(S + P k S − )
,
P k−1 = Π − (P k ) :=
S − P k S +
tr(S − P k S + )
,
(27)
where tr(S + P k S − ) = 0.