7.6 Radiating Finite Spin Chain
205
{ 0 :=
S
0 ⊗
F
0 , β n := |n ⊗
F
0 ,
β N (ψ) := |N ⊗ b
∗
(ψ))
F
0 ; n = 0, 1, . . . N − 1, ψ ∈ L
2
(R
3
, d
3 x)}.
Then it is valid:
7.6.3 Lemma. The space H min defined above is H -invariant: H H min ⊂ H min .
A proof of this Lemma is presented in [39]. Hence the description of our process
can be restricted to time evolution in the subspace H min ⊂ H. We shall choose the
parameters of the model, namely the operator h acting on L
2
(R
3
), and the quantities
ε 0 > 0, v ∈ R, σ ∈ L
2
(R
3
, d
3 x), so that with our Hamiltonian given by (7.6.2a) the
relation
lim
t→∞
β n |e
it H a
∗
N a N e
−it H
|β n = 1, n = 0, 1, . . . N − 1,
(7.6.3a)
or more specifically:
β n |e
it H a
∗
N a N e
−it H
|β n = 1 − o(t
−m
), n = 0, 1, . . . N − 1, for t → +∞, ∀m ∈ N,
(7.6.3b)
will be satisfied. The meaning of (7.6.3) is that the probability of emission of the
Fermi particle and simultaneous transition of the spin chain to the stationary state
β N (i.e. all the spins in the chain “are pointing up” and the Fermi field is again in
its vacuum state) approaches certainty ‘almost exponentially quickly’ if the time is
growing to infinity.
The dynamics is investigated by a repeated use of Fourier transform F, e.g. in
[39, Lemma 2].
7.6.4 Lemma. Let e
−it H be any (unitary) time evolution group. Then the Fourier
transform of its (truncated) matrix elements for given φ, ψ ∈ H is
F[θ(t)φ, e
it H
ψ](ξ) =
i
√
2π
φ, R H (ξ)ψ,
(7.6.4)
for ξ ∈ C : Im ξ < 0.
The function θ is here the Heaviside function, and R H (ξ) ≡ (H − ξ I )
−1
(ξ ∈ C, ξ /
∈
sp(H ) ≡ spectrum of H ) is the resolvent of the operator H .
Another useful result is that we obtain the resolvent R H (λ) as a solution of an
operator equation, [39, Lemma 3].
7.6.5 Lemma. Suppose H = H 0 + V ∈ L(H) and ξ /
∈ sp(H ) ∪ sp(H 0 ). Then the
resolvent R H (ξ) is the solution of the operator equation
R H (ξ) = R H 0 (ξ)(I − V R H (ξ)).
(7.6.5)
Hence, the Fourier transform of the (truncated) matrix elements of the time evolution
operator for Im ξ < 0 is given by:
205
{ 0 :=
S
0 ⊗
F
0 , β n := |n ⊗
F
0 ,
β N (ψ) := |N ⊗ b
∗
(ψ))
F
0 ; n = 0, 1, . . . N − 1, ψ ∈ L
2
(R
3
, d
3 x)}.
Then it is valid:
7.6.3 Lemma. The space H min defined above is H -invariant: H H min ⊂ H min .
A proof of this Lemma is presented in [39]. Hence the description of our process
can be restricted to time evolution in the subspace H min ⊂ H. We shall choose the
parameters of the model, namely the operator h acting on L
2
(R
3
), and the quantities
ε 0 > 0, v ∈ R, σ ∈ L
2
(R
3
, d
3 x), so that with our Hamiltonian given by (7.6.2a) the
relation
lim
t→∞
β n |e
it H a
∗
N a N e
−it H
|β n = 1, n = 0, 1, . . . N − 1,
(7.6.3a)
or more specifically:
β n |e
it H a
∗
N a N e
−it H
|β n = 1 − o(t
−m
), n = 0, 1, . . . N − 1, for t → +∞, ∀m ∈ N,
(7.6.3b)
will be satisfied. The meaning of (7.6.3) is that the probability of emission of the
Fermi particle and simultaneous transition of the spin chain to the stationary state
β N (i.e. all the spins in the chain “are pointing up” and the Fermi field is again in
its vacuum state) approaches certainty ‘almost exponentially quickly’ if the time is
growing to infinity.
The dynamics is investigated by a repeated use of Fourier transform F, e.g. in
[39, Lemma 2].
7.6.4 Lemma. Let e
−it H be any (unitary) time evolution group. Then the Fourier
transform of its (truncated) matrix elements for given φ, ψ ∈ H is
F[θ(t)φ, e
it H
ψ](ξ) =
i
√
2π
φ, R H (ξ)ψ,
(7.6.4)
for ξ ∈ C : Im ξ < 0.
The function θ is here the Heaviside function, and R H (ξ) ≡ (H − ξ I )
−1
(ξ ∈ C, ξ /
∈
sp(H ) ≡ spectrum of H ) is the resolvent of the operator H .
Another useful result is that we obtain the resolvent R H (λ) as a solution of an
operator equation, [39, Lemma 3].
7.6.5 Lemma. Suppose H = H 0 + V ∈ L(H) and ξ /
∈ sp(H ) ∪ sp(H 0 ). Then the
resolvent R H (ξ) is the solution of the operator equation
R H (ξ) = R H 0 (ξ)(I − V R H (ξ)).
(7.6.5)
Hence, the Fourier transform of the (truncated) matrix elements of the time evolution
operator for Im ξ < 0 is given by: