6.2 Spin Systems with Polynomial Local Hamiltonians Q N
121
for any N ∈ , be the selfadjoint element of A introduced in 5.1.3 as a selfadjoint
operator on H and identified now with s G π u (X
N
j ). Let us denote
b := max{1 + +X (ξ j ) : j = 1, 2, . . . n := dim G}.
(6.2.2)
We shall use the Einstein summation rule for the summation over repeated vector
indices in g and g
∗ . Let c
m
jk be the structure constants of g in the given basis:
[ξ j , ξ k ] = c
m
jk ξ m .
(6.2.3)
Then we have from (5.1.1):
[X
K
j , X
K
k ] = i c
m
jk X
K
m , for all K ∈ .
(6.2.4)
Let Q be a polynomial specified in 6.1.1, hence satisfying the property 6.1.1(SA).
Let Q be written in the form of linear combination of p monomials of the maximal
degree q with the upper bound M ≥ 1 of the absolute values of the coefficients. Let
Q
K be given by (6.1.1) for all K ∈ . Let us introduce the notation:
c := max{|c
m
jk | : j, k, m = 1, 2, . . . n};
(6.2.5)
a N := max(nc; 2|N |b), N ∈ ;
(6.2.6)
b(x) := max(b; ;x), x ∈ A.
(6.2.7)
We shall use the standard notation for the multiple commutators:
[y, x]
(m+1)
:= [y, [y, x]
(m)
], [y, x]
(0)
:= x, [y, x] := yx − x y,
(6.2.8)
for any x, y ∈ A
∗∗ . [We shall use also |J | := the number of elements of the set J .]
6.2.4 Lemma. The following estimate is valid for any x ∈ B
N
0 and for all positive
integers N , K (≥ N ), m:
[Q
K
, x]
(m)
<
b(x)
q
(m − 1)! (Mpq
2 b
q−1 a N )
m
.
(6.2.9)
Proof. Each multiple commutator in (6.2.9) can be written in the form of a finite
linear combination of monomials P
(m) in the variables X j K and y r , where y r ∈ B
N
0
is of one of the forms of the multiple commutators occurring in the two following
formulas:
[X
N
j 1
, [X
N
j 2
, . . . [X
N
j r
, x] . . . ]]] ≤ (2bN )
r
x, x ∈ A
N
;
(6.2.10)
[X
K
j 1
, [X
K
j 2
, . . . [X
K
j r
, X kL ] . . . ]]] ≤ (nc)
r b, L ∈ .
(6.2.11)