122
6 Dynamics of Quantum Mechanical Macroscopic Systems
These estimates of y r are easy consequences of the definitions as well as of the
relations (6.2.4). Let r ∈ Z + be called the degree of any of the variables denoted by
y r . Then the sum
j r j of degrees of all the variables y r j occurring in any of the
monomials P
(m) is less or equal to m. The maximal degree of any of the monomials
P
(m) is m(q − 1) + 1, hence we have the estimate:
P
(m)
≤ b(x)(a N b
q−1
)
m
,
(6.2.12)
where we have used the fact that a variable y r of the form given in (6.2.10) occurs in
any of the monomials P
(m) at most in the first power (what implies the first power of
b(x) in (6.2.12)), as well as the inequalities a N > b > 1 were used in the derivation
of (6.2.12).
The maximal value of coefficients at the monomials P
(m) is < M
m . The maximal
number of monomials P
(m) occurring in the expression of [Q
K
, x]
(m) can be calculated recursively, using the derivation property of the commutators. One has the
identity
[x j 1 . . . x j q , y k 1 . . . y k s ] =
q
i=1
s
j=1
x j 1 . . . x j i−1 y k 1 . . . y k j−1
[x j i , y k j ]y k j+1 . . . y k s x j i+1 . . . x j q ,
(6.2.13)
in which the commutator of two monomials of degrees q and s is expressed as a
sum of qs monomials of degree q + s − 1 (some of the monomials could be equal
to zero). If [Q
K
, x]
(m) is a sum of n m monomials const.P
(m) of the maximal degree
s m := m(q − 1) + 1, then [Q
K
, x]
(m+1) is a sum of n m+1 monomials, where
n m+1 ≤ n m pqs m ≤ n m mpq
2
.
(6.2.14)
Since n 1 ≤ pq, we obtain the estimate:
n m ≤
(m − 1)!
q
( pq
2
)
m
.
(6.2.15)
After the multiplication of the right hand side of (6.2.12) by the right hand side
of (6.2.15) and by the upper bound M
m of the coefficients at P
(m) , we obtain the
estimate (6.2.9).
6.2.5 Lemma. Let us define
κ N := (Mpq
2 b
q a N )
−1
, for all N ∈ .
(6.2.16)
Let |t| ≤ κ N , x ∈ B
N
0 for a given N ∈ . Then:
6 Dynamics of Quantum Mechanical Macroscopic Systems
These estimates of y r are easy consequences of the definitions as well as of the
relations (6.2.4). Let r ∈ Z + be called the degree of any of the variables denoted by
y r . Then the sum
j r j of degrees of all the variables y r j occurring in any of the
monomials P
(m) is less or equal to m. The maximal degree of any of the monomials
P
(m) is m(q − 1) + 1, hence we have the estimate:
P
(m)
≤ b(x)(a N b
q−1
)
m
,
(6.2.12)
where we have used the fact that a variable y r of the form given in (6.2.10) occurs in
any of the monomials P
(m) at most in the first power (what implies the first power of
b(x) in (6.2.12)), as well as the inequalities a N > b > 1 were used in the derivation
of (6.2.12).
The maximal value of coefficients at the monomials P
(m) is < M
m . The maximal
number of monomials P
(m) occurring in the expression of [Q
K
, x]
(m) can be calculated recursively, using the derivation property of the commutators. One has the
identity
[x j 1 . . . x j q , y k 1 . . . y k s ] =
q
i=1
s
j=1
x j 1 . . . x j i−1 y k 1 . . . y k j−1
[x j i , y k j ]y k j+1 . . . y k s x j i+1 . . . x j q ,
(6.2.13)
in which the commutator of two monomials of degrees q and s is expressed as a
sum of qs monomials of degree q + s − 1 (some of the monomials could be equal
to zero). If [Q
K
, x]
(m) is a sum of n m monomials const.P
(m) of the maximal degree
s m := m(q − 1) + 1, then [Q
K
, x]
(m+1) is a sum of n m+1 monomials, where
n m+1 ≤ n m pqs m ≤ n m mpq
2
.
(6.2.14)
Since n 1 ≤ pq, we obtain the estimate:
n m ≤
(m − 1)!
q
( pq
2
)
m
.
(6.2.15)
After the multiplication of the right hand side of (6.2.12) by the right hand side
of (6.2.15) and by the upper bound M
m of the coefficients at P
(m) , we obtain the
estimate (6.2.9).
6.2.5 Lemma. Let us define
κ N := (Mpq
2 b
q a N )
−1
, for all N ∈ .
(6.2.16)
Let |t| ≤ κ N , x ∈ B
N
0 for a given N ∈ . Then: