118
17 Examples
≤ ||A|| (2
X 0
||Φ(X )||)
n e
nN(Δ) e
N (V 0 )
≤ C t
−n
0 ,
(17.11)
where we have used that x
n
/n! ≤ e
x and put
C ≡ ||A|| e
N (V 0 )
, t
−1
0 ≡ 2
X 0
||Φ(X )|| e
N (Δ)
.
The above estimate is enough to get the result. In fact,
||α
V 1
t (A) − α
V 2
t (A)|| ≤ ||
N
n=0
(A
V 1
n − A
V 2
n ) t
n
||
+
∞
n=N +1
||A
V 1
n t
n
|| +
∞
n=N +1
||A
V 2
n t
n
||.
Now, the first term on the r.h.s. of the inequality can be made as small as we like,
since, for fixed n, A
V
n becomes independent of V , for V large enough, by the finite
range; moreover, by the estimate (17.11), the second and third terms are smaller than
C
∞
n=N +1 |t/t 0 |
n , which can be made as small as we like, for t ≤ t 1 < t 0 .
The group law α t α t (A) = α t +t (A), which is easily proved for t, t
, t + t
∈
[−t 1 , t − 1], allows to extend α t for all t. From the estimate (17.11) and the convergence of the series (17.9), it follows that α t is strongly continuous on A L and
therefore also on A.
We shall now discuss the case of an interaction potential Φ 1 (X ), not necessarily
of finite range, satisfying the absolute summability condition
||Φ 1 || ≡
X 0
||Φ 1 (X )|| <
X 0
||Φ(X )||,
(17.12)
with Φ of finite range, and involving only a finite number ¯
N of k-body interactions
Φ 1 (X ) = 0, if N (X ) > ¯
N .
(17.13)
For the multiple commutator B n (A), A ∈ A(V 0 ) (see (17.10)),
B Φ,n (A) =
X ⊂V
[Φ(X ), B Φ,n−1 (A)], B Φ,1 (A) =
X ⊂V
[Φ(X ), A], B Φ,0 = A,
one easily proves the following algebraic identity
X ⊂V
[Φ 1 (X ), B Φ 1 ,n−1 (A)] −
X ⊂V
[Φ(X ), B Φ,n−1 (A)] =
17 Examples
≤ ||A|| (2
X 0
||Φ(X )||)
n e
nN(Δ) e
N (V 0 )
≤ C t
−n
0 ,
(17.11)
where we have used that x
n
/n! ≤ e
x and put
C ≡ ||A|| e
N (V 0 )
, t
−1
0 ≡ 2
X 0
||Φ(X )|| e
N (Δ)
.
The above estimate is enough to get the result. In fact,
||α
V 1
t (A) − α
V 2
t (A)|| ≤ ||
N
n=0
(A
V 1
n − A
V 2
n ) t
n
||
+
∞
n=N +1
||A
V 1
n t
n
|| +
∞
n=N +1
||A
V 2
n t
n
||.
Now, the first term on the r.h.s. of the inequality can be made as small as we like,
since, for fixed n, A
V
n becomes independent of V , for V large enough, by the finite
range; moreover, by the estimate (17.11), the second and third terms are smaller than
C
∞
n=N +1 |t/t 0 |
n , which can be made as small as we like, for t ≤ t 1 < t 0 .
The group law α t α t (A) = α t +t (A), which is easily proved for t, t
, t + t
∈
[−t 1 , t − 1], allows to extend α t for all t. From the estimate (17.11) and the convergence of the series (17.9), it follows that α t is strongly continuous on A L and
therefore also on A.
We shall now discuss the case of an interaction potential Φ 1 (X ), not necessarily
of finite range, satisfying the absolute summability condition
||Φ 1 || ≡
X 0
||Φ 1 (X )|| <
X 0
||Φ(X )||,
(17.12)
with Φ of finite range, and involving only a finite number ¯
N of k-body interactions
Φ 1 (X ) = 0, if N (X ) > ¯
N .
(17.13)
For the multiple commutator B n (A), A ∈ A(V 0 ) (see (17.10)),
B Φ,n (A) =
X ⊂V
[Φ(X ), B Φ,n−1 (A)], B Φ,1 (A) =
X ⊂V
[Φ(X ), A], B Φ,0 = A,
one easily proves the following algebraic identity
X ⊂V
[Φ 1 (X ), B Φ 1 ,n−1 (A)] −
X ⊂V
[Φ(X ), B Φ,n−1 (A)] =