5 Stable Structures, Hilbert Sectors, Phases
25
g(δ(s)) ≡
0
−G
ϕ 0
(χ(s))
,
(5.8)
G ϕ 0 (χ) ≡ U (ϕ 0 + χ) − U (ϕ 0 ) − U
(ϕ 0 )χ.
(5.9)
The subscript ϕ 0 and the explicit dependence on x through ϕ 0 will often be omitted in
the sequel, using for brevity the notation G(x, χ(x)) or simply G(χ). Furthermore,
for brevity ∇ z G(x, z)| z=χ will be denoted by G
(χ).
The crux of the argument is that for ϕ 0 bounded, briefly ∈ L
∞
(R
s
), for the class of
potentials under consideration, G(χ) satisfies
i) G
(χ) is globally Lipschitz continuous, i.e. for any ρ > 0, there exists a constant
C(ρ) such that for any χ 1 , χ 2 ∈ H
1
(R
s
), with χ i H 1 ≤ ρ, i = 1, 2,
G
(χ 2 ) − G
(χ 1 ) L 2 ≤ C(ρ)χ 2 − χ 1 H 1 ,
(5.10)
ii) G satisfies a lower bound condition, i.e. there exists a non-negative constant γ,
such that
G(x, z) ≥ −γ|z|
2
, ∀z ∈ R
n
, x ∈ R
s
.
(5.11)
(The proof of i) and ii) is given in Lemma 5.3 and 5.4, respectively).
Now, if i), ii) hold, since g(0) = 0, property i) implies that g(χ) ∈ H
1
(R
s
) ⊕ L
2
(R
s
)
and therefore, since W (t) maps H
1
(R
s
) ⊕ L
2
(R
s
) into itself continuously in t, (see
Appendix 10.1),
δ(t) ∈ C
0
(H
1
⊕ L
2
, R) iff L(t) ∈ C
0
(H
1
⊕ L
2
, R).
(5.12)
The latter condition is equivalent to conditions a) and b), ((5.2), (5.3)), (see Lemma
5.3 below).
The proof that the sector is not empty and actually is a Hilbert space sector amounts
to proving that (5.6) has one and only one solution δ(t) ∈ C
◦
(H
1
⊕ L
2
, R) for any
initial data δ 0 ∈ H
1
(R
s
) ⊕ L
2
(R
s
).
A simple important case is when u 0 is a static solution of (4.6),
Δϕ 0 − U
(ϕ 0 ) = 0, ψ 0 = 0.
(5.13)
In this case L(t) = 0 and (5.6) has the same form of (4.6), for which the Cauchy
problem in H
1
⊕ L
2 has been solved by Segal.
20
In the general case L(t) = 0, a generalization of Segal theorem (see Appendix 10.4)
gives existence and uniqueness in H
1
⊕ L
2 .
Lemma 5.3 For any ϕ 0 ∈ L
∞
(R
s
), the function G
(χ) defined through (5.9) is globally Lipschitz continuous, (5.10).
20 See Chap. 4, footnote 12.
25
g(δ(s)) ≡
0
−G
ϕ 0
(χ(s))
,
(5.8)
G ϕ 0 (χ) ≡ U (ϕ 0 + χ) − U (ϕ 0 ) − U
(ϕ 0 )χ.
(5.9)
The subscript ϕ 0 and the explicit dependence on x through ϕ 0 will often be omitted in
the sequel, using for brevity the notation G(x, χ(x)) or simply G(χ). Furthermore,
for brevity ∇ z G(x, z)| z=χ will be denoted by G
(χ).
The crux of the argument is that for ϕ 0 bounded, briefly ∈ L
∞
(R
s
), for the class of
potentials under consideration, G(χ) satisfies
i) G
(χ) is globally Lipschitz continuous, i.e. for any ρ > 0, there exists a constant
C(ρ) such that for any χ 1 , χ 2 ∈ H
1
(R
s
), with χ i H 1 ≤ ρ, i = 1, 2,
G
(χ 2 ) − G
(χ 1 ) L 2 ≤ C(ρ)χ 2 − χ 1 H 1 ,
(5.10)
ii) G satisfies a lower bound condition, i.e. there exists a non-negative constant γ,
such that
G(x, z) ≥ −γ|z|
2
, ∀z ∈ R
n
, x ∈ R
s
.
(5.11)
(The proof of i) and ii) is given in Lemma 5.3 and 5.4, respectively).
Now, if i), ii) hold, since g(0) = 0, property i) implies that g(χ) ∈ H
1
(R
s
) ⊕ L
2
(R
s
)
and therefore, since W (t) maps H
1
(R
s
) ⊕ L
2
(R
s
) into itself continuously in t, (see
Appendix 10.1),
δ(t) ∈ C
0
(H
1
⊕ L
2
, R) iff L(t) ∈ C
0
(H
1
⊕ L
2
, R).
(5.12)
The latter condition is equivalent to conditions a) and b), ((5.2), (5.3)), (see Lemma
5.3 below).
The proof that the sector is not empty and actually is a Hilbert space sector amounts
to proving that (5.6) has one and only one solution δ(t) ∈ C
◦
(H
1
⊕ L
2
, R) for any
initial data δ 0 ∈ H
1
(R
s
) ⊕ L
2
(R
s
).
A simple important case is when u 0 is a static solution of (4.6),
Δϕ 0 − U
(ϕ 0 ) = 0, ψ 0 = 0.
(5.13)
In this case L(t) = 0 and (5.6) has the same form of (4.6), for which the Cauchy
problem in H
1
⊕ L
2 has been solved by Segal.
20
In the general case L(t) = 0, a generalization of Segal theorem (see Appendix 10.4)
gives existence and uniqueness in H
1
⊕ L
2 .
Lemma 5.3 For any ϕ 0 ∈ L
∞
(R
s
), the function G
(χ) defined through (5.9) is globally Lipschitz continuous, (5.10).
20 See Chap. 4, footnote 12.