62
10 Appendix
and by the Riesz representation theorem this implies that there exists a h ∈ L
q such
that
h(g) =
h∇g d
s x.
Hence, h(g) = 0, ∀g ∈ C
∞
0 (R) implies 0 =
h∇g d
s x =−
∇hgd
s x, i.e. ∇h = 0,
i.e. h = const, i.e. h = 0 as a functional on H.
Finally, if f ∈ H, there exists a sequence { f j ∈ C
∞
0 (R
s
)} with f j → f in H; this
implies that ∇ f j converges in L
p
(R
s
) and, by Sobolev’s inequality
f j L q ≤ const ∇ f j L p ,
f j → ˜
f in L
q and ˜
f belongs to the same equivalence class of f , i.e. f = ˜
f + const.
ii) Since ˜
ϕ(r, ω) is continuous in r and U ∈ C
2
lim
r →∞
U
( ˜
ϕ(r, ω)) = U
( ˜
ϕ(∞, ω)) = U
(ϕ ∞ ) .
Equation (10.21) implies that also lim
r →∞
Δ ˜
ϕ(r, ω) exists and it is independent of ω.
Furthermore, ∀ f (r ) ∈ D(R
+
), with
∞
0 f (r )dr = 1
U
(ϕ ∞ ) = lim
r →∞
Δ ˜
ϕ(r, ω) = lim
a→∞
(Δ ˜
ϕ)(r + a, ω) =
= lim
a→∞
∞
0
dr f (r )(Δ ˜
ϕ)(r + a, ω) =
= lim
a→∞
∞
0
dr(Δf (r )) ˜
ϕ(r + a, ω) =
= ϕ(∞)
∞
0
drΔf (r ) = 0.
iii) If ϕ ∞ is an absolute minimum
˜
U (ϕ) ≡ U (ϕ) − U (ϕ ∞ ) ≥ 0
and the solutions of (10.21) are stationary points of the functional
H (ϕ) =
[|∇ϕ|
2
+ ˜
U (ϕ)]d
s x.
Now, by putting ϕ λ (x) ≡ ϕ(λx), λ ≥ 0, we get
H λ =
[|∇ϕ λ |
2
+ ˜
U (ϕ λ )]d
s x =
[λ
−1
|∇ϕ|
2
+ λ
−3 ˜
U (ϕ)]d
s x