7.4 Particle Detection—A “Nonideal” Measurement
183
ϕ|e
−i(t−iε)H
|ψ =
∞
a
dλe
−i(t−iε)λ
ϕ|E H (λ)|ψ ≡ ≡ϕ|e
−iz H
|ψ, Im z ≤ 0,
(7.4.14)
which is analytic in the open lower complex half-plain of z and continuous in the
closed lower half-plain, hence also on the real line z = t − iε → t − i0+. Assume
that ϕ|e
−it H
|ψ ≡ 0, ∀t ∈ I ⊂ R, where I is an interval of positive length. Then,
according to the Schwarz reflection principle, the analytic function z → →ϕ|e
−iz H
|ψ
is complex-analytic also on this interval I , hence it is identically zero also in lower
complex half-plain. Due to its continuity on R, the function t → →ϕ|e
−it H
|ψ ≡
0 (∀t ∈ R).
In the other cases, there is no interval of nonzero length I ⊂ R on which the
function t → →ϕ|e
−it H
|ψ identically vanishes. Since it is continuous, it is = 0 on
open intervals composing an open dense subset of R. But union of all these intervals
is a set of total Lebesgue measure on R, as is shown in [151]. Hence the function
t → →ϕ|e
−it H
|ψ = 0 a.e. with respect to the Lebesgue measure.
Linearity of the set of the ψ’s satisfying (a) is clear. That this subspace is closed
in H v follows from the norm-continuity of the matrix elements ψ → →ϕ|e
−it H
|ψ;
the last assertion follows from the other proved assertions of this Lemma.
7.4.5 Lemma. The condition ϕ ∈ D(R
3
) for the choice of the vector ϕ occurring
in the definition of the interaction Hamiltonian in (7.4.4), as well as the condition
ψ ∈ H B ∩ L
1
(R
3
) for the choice of the particle’s initial vector ψ, both imposed in
the Theorem 7.4.8, guarantee the following properties of the functions t → F
0
ϕ (ψ)(t)
(7.4.6a) of the time variable t ∈ R:
F
0
ϕ (ψ) ∈ L
2
(R) ∩ L
1
(R), g ≡ F
0
ϕ (ϕ) ∈ L
2
(R) ∩ L
1
(R).
(7.4.15)
The set L
1
(R
3
) ∩ L
2
(R
3
) ⊃ D(R
3
) is dense in H B together with D(R
3
).
Proof. According to the Theorem IX.30 of [262], there is for ψ ∈ L
1
(R
3
) ∩ L
2
(R
3
):
ess sup
x∈R 3
|e
−it H B ψ(x)| ≡ ≡e
−it H B ψ ∞ ≤ |t
−
3
2 | |ψ 1 .
(7.4.16)
The function ϕ has finite support, say ϕ(x) = 0 ⇒ |x| < R < ∞. Let us denote by
B R ⊂ R
3 the ball of radius R containing the support of ϕ. Due to the implication
χ ∈ L
2
(R
3
) ⇒ |χ| ∈ L
2
(R
3
), we have
||ϕ|e
−it H B ψ ≤
B R
d
3 x |ϕ(x)| · |e
−it H B ψ(x)| ≤
B R
d
3 x |ϕ(x)| ·
ψ 1
|t
3
2 |
=
ϕ 1 ψ 1
|t
3
2 |
.
(7.4.17)
183
ϕ|e
−i(t−iε)H
|ψ =
∞
a
dλe
−i(t−iε)λ
ϕ|E H (λ)|ψ ≡ ≡ϕ|e
−iz H
|ψ, Im z ≤ 0,
(7.4.14)
which is analytic in the open lower complex half-plain of z and continuous in the
closed lower half-plain, hence also on the real line z = t − iε → t − i0+. Assume
that ϕ|e
−it H
|ψ ≡ 0, ∀t ∈ I ⊂ R, where I is an interval of positive length. Then,
according to the Schwarz reflection principle, the analytic function z → →ϕ|e
−iz H
|ψ
is complex-analytic also on this interval I , hence it is identically zero also in lower
complex half-plain. Due to its continuity on R, the function t → →ϕ|e
−it H
|ψ ≡
0 (∀t ∈ R).
In the other cases, there is no interval of nonzero length I ⊂ R on which the
function t → →ϕ|e
−it H
|ψ identically vanishes. Since it is continuous, it is = 0 on
open intervals composing an open dense subset of R. But union of all these intervals
is a set of total Lebesgue measure on R, as is shown in [151]. Hence the function
t → →ϕ|e
−it H
|ψ = 0 a.e. with respect to the Lebesgue measure.
Linearity of the set of the ψ’s satisfying (a) is clear. That this subspace is closed
in H v follows from the norm-continuity of the matrix elements ψ → →ϕ|e
−it H
|ψ;
the last assertion follows from the other proved assertions of this Lemma.
7.4.5 Lemma. The condition ϕ ∈ D(R
3
) for the choice of the vector ϕ occurring
in the definition of the interaction Hamiltonian in (7.4.4), as well as the condition
ψ ∈ H B ∩ L
1
(R
3
) for the choice of the particle’s initial vector ψ, both imposed in
the Theorem 7.4.8, guarantee the following properties of the functions t → F
0
ϕ (ψ)(t)
(7.4.6a) of the time variable t ∈ R:
F
0
ϕ (ψ) ∈ L
2
(R) ∩ L
1
(R), g ≡ F
0
ϕ (ϕ) ∈ L
2
(R) ∩ L
1
(R).
(7.4.15)
The set L
1
(R
3
) ∩ L
2
(R
3
) ⊃ D(R
3
) is dense in H B together with D(R
3
).
Proof. According to the Theorem IX.30 of [262], there is for ψ ∈ L
1
(R
3
) ∩ L
2
(R
3
):
ess sup
x∈R 3
|e
−it H B ψ(x)| ≡ ≡e
−it H B ψ ∞ ≤ |t
−
3
2 | |ψ 1 .
(7.4.16)
The function ϕ has finite support, say ϕ(x) = 0 ⇒ |x| < R < ∞. Let us denote by
B R ⊂ R
3 the ball of radius R containing the support of ϕ. Due to the implication
χ ∈ L
2
(R
3
) ⇒ |χ| ∈ L
2
(R
3
), we have
||ϕ|e
−it H B ψ ≤
B R
d
3 x |ϕ(x)| · |e
−it H B ψ(x)| ≤
B R
d
3 x |ϕ(x)| ·
ψ 1
|t
3
2 |
=
ϕ 1 ψ 1
|t
3
2 |
.
(7.4.17)