Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
On second order perturbation theory for embedded
eigenvalues
Singular
Mourre
theory
Jérémy Faupin
References
Institut de Mathématiques de Bordeaux
Université de Bordeaux 1
Joint work with
J.S. Møller and E. Skibsted
1 / 25
Second
order perturbation
theory
Outline of the talk
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
1
Regular Mourre theory with a self-adjoint conjugate operator
2
The Nelson model
3
Singular Mourre theory with a non self-adjoint conjugate
operator
Singular
Mourre
theory
References
2 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Part I
Regular Mourre theory with
a self-adjoint conjugate operator
3 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Regularity w.r.t. a self-adjoint operator
• H complex Hilbert space
• H, A self-adjoint operators on H
Definition
Let n ∈ N. We say that H ∈ C n (A) if and only if ∀z ∈ C \ σ(H), ∀φ ∈ H,
s 7→ e isA (H − z)−1 e −isA φ ∈ C n (R)
References
Remarks
• H ∈ C 1 (A) if and only if ∀z ∈ C \ σ(H), (H − z)−1 D(A) ⊆ D(A), and
∀φ ∈ D(H) ∩ D(A),
|hAφ, Hφi − hHφ, Aφi| ≤ C (kHφk2 + kφk2 )
• If H ∈ C 1 (A), then D(H) ∩ D(A) is a core for H, and the quadratic form
[H, A] defined on (D(H) ∩ D(A)) × (D(H) ∩ D(A)) extend by continuity
to a bounded quadratic form on D(H) × D(H) denoted [H, A]0
4 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Regularity w.r.t. a self-adjoint operator
• H complex Hilbert space
• H, A self-adjoint operators on H
Definition
Let n ∈ N. We say that H ∈ C n (A) if and only if ∀z ∈ C \ σ(H), ∀φ ∈ H,
s 7→ e isA (H − z)−1 e −isA φ ∈ C n (R)
References
Remarks
• H ∈ C 1 (A) if and only if ∀z ∈ C \ σ(H), (H − z)−1 D(A) ⊆ D(A), and
∀φ ∈ D(H) ∩ D(A),
|hAφ, Hφi − hHφ, Aφi| ≤ C (kHφk2 + kφk2 )
• If H ∈ C 1 (A), then D(H) ∩ D(A) is a core for H, and the quadratic form
[H, A] defined on (D(H) ∩ D(A)) × (D(H) ∩ D(A)) extend by continuity
to a bounded quadratic form on D(H) × D(H) denoted [H, A]0
4 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Regularity w.r.t. a self-adjoint operator
• H complex Hilbert space
• H, A self-adjoint operators on H
Definition
Let n ∈ N. We say that H ∈ C n (A) if and only if ∀z ∈ C \ σ(H), ∀φ ∈ H,
s 7→ e isA (H − z)−1 e −isA φ ∈ C n (R)
References
Remarks
• H ∈ C 1 (A) if and only if ∀z ∈ C \ σ(H), (H − z)−1 D(A) ⊆ D(A), and
∀φ ∈ D(H) ∩ D(A),
|hAφ, Hφi − hHφ, Aφi| ≤ C (kHφk2 + kφk2 )
• If H ∈ C 1 (A), then D(H) ∩ D(A) is a core for H, and the quadratic form
[H, A] defined on (D(H) ∩ D(A)) × (D(H) ∩ D(A)) extend by continuity
to a bounded quadratic form on D(H) × D(H) denoted [H, A]0
4 / 25
Second
order perturbation
theory
Mourre estimate
Jérémy
Faupin
Regular
Mourre
theory
Definition
Singular
Mourre
theory
Let I be a bounded open interval, I ⊂ σ(H). We say that H satisfies a Mourre
estimate on I with A as conjugate operator if ∃ c0 > 0 and K0 compact such
that
1I (H)[H, iA]0 1I (H) ≥ c0 1I (H) − K0 ,
References
in the sense of quadratic forms on H × H
Nelson
model
Remarks
• An equivalent formulation is
[H, iA]0 ≥ c00 − c10 1R\I (H)hHi − K00 ,
in the sense of quadratic forms on D(H) × D(H), with c00 > 0, c10 ∈ R,
and K00 compact
• If K0 = 0, we say that H satisfies a strict Mourre estimate on I
5 / 25
Second
order perturbation
theory
Mourre estimate
Jérémy
Faupin
Regular
Mourre
theory
Definition
Singular
Mourre
theory
Let I be a bounded open interval, I ⊂ σ(H). We say that H satisfies a Mourre
estimate on I with A as conjugate operator if ∃ c0 > 0 and K0 compact such
that
1I (H)[H, iA]0 1I (H) ≥ c0 1I (H) − K0 ,
References
in the sense of quadratic forms on H × H
Nelson
model
Remarks
• An equivalent formulation is
[H, iA]0 ≥ c00 − c10 1R\I (H)hHi − K00 ,
in the sense of quadratic forms on D(H) × D(H), with c00 > 0, c10 ∈ R,
and K00 compact
• If K0 = 0, we say that H satisfies a strict Mourre estimate on I
5 / 25
Second
order perturbation
theory
The Virial Theorem
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Theorem ([Mo ’81], [ABG ’96], [GG ’99])
Singular
Mourre
theory
Let φ be an eigenstate of H. If H ∈ C 1 (A), then
hφ, [H, iA]0 φi = 0
References
Corollary
Assume that H ∈ C 1 (A) and that H satisfies a Mourre estimate on I. Then
the number of eigenvalues of H in I is finite, and each such eigenvalue has a
finite multiplicity
6 / 25
Second
order perturbation
theory
The Virial Theorem
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Theorem ([Mo ’81], [ABG ’96], [GG ’99])
Singular
Mourre
theory
Let φ be an eigenstate of H. If H ∈ C 1 (A), then
hφ, [H, iA]0 φi = 0
References
Corollary
Assume that H ∈ C 1 (A) and that H satisfies a Mourre estimate on I. Then
the number of eigenvalues of H in I is finite, and each such eigenvalue has a
finite multiplicity
6 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Limiting Absorption Principle
Theorem ([Mo ’81], [ABG ’96], [Ge ’08])
Assume that H ∈ C 2 (A) and that H satisfies a strict Mourre estimate on I.
Then for all closed interval J ⊂ I and s > 1/2,
sup khAi−s (H − z)−1 hAi−s k < ∞,
z∈J ±
where J ± = {z ∈ C, Re z ∈ J, ±Im z > 0} and hAi = (1 + A2 )1/2 . In
particular the spectrum of H in I is purely absolutely continuous.
Moreover for 1/2 < s ≤ 1, the maps
J ± 3 z 7→ khAi−s (H − z)−1 hAi−s k ∈ B(H)
are Hölder continuous of order s − 1/2. In particular, for λ ∈ J, the limits
hAi−s (H − λ ± i0)−1 hAi−s := limhAi−s (H − λ ± i)−1 hAi−s
↓0
exist in the norm topology of B(H), and the corresponding functions of λ are
Hölder continuous of order s − 1/2
7 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Fermi Golden Rule criterion
Theorem ([AHS ’89], [HuSi ’00])
Suppose
1) (Regularity of H) H ∈ C 2 (A) and the quadratic forms [H, iA] and
[[H, iA], iA] extend by continuity to H-bounded operators
2) (Mourre estimate) H satisfies a Mourre estimate on I
Let λ ∈ I be an eigenvalue of H. Let P = 1{λ} (H) be the associated
eigenprojection and P̄ = I − P. Let J ⊂ I be a closed interval such that
σpp (H) ∩ J = {λ}. Let W be a symmetric and H-bounded operator. Suppose
3) (Regularity of eigenstates) Ran(P) ⊆ D(A2 )
4) (Regularity of the perturbation) [W , iA] and [[W , iA], iA] extend by
continuity to H-bounded operators
If the Fermi Golden Rule criterion is satisfied, i.e.
PW Im((H − λ − i0)−1 P̄)WP ≥ cP
with c > 0, then ∃ σ0 > 0 such that ∀ 0 < |σ| ≤ σ0 ,
σpp (H + σW ) ∩ J = ∅
8 / 25
Second
order perturbation
theory
Regularity of bound states
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Theorem ([Ca ’05], [CGH ’06])
Let n ∈ N. Assume that H ∈ C n+2 (A) and that adkA (H) are H-bounded for all
1 ≤ k ≤ n + 2. Assume that H satisfies a Mourre estimate on I. Let λ ∈ I be
an eigenvalue of H and let P = 1{λ} (H) be the associated eigenprojection.
Then we have that
Ran(P) ⊆ D(An )
Remark
In fact H ∈ C n+1 (A) is sufficient for the conclusion of the previous theorem to
hold and this is optimal ([FMS’ 10], [MW’ 10]).
9 / 25
Second
order perturbation
theory
Regularity of bound states
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Theorem ([Ca ’05], [CGH ’06])
Let n ∈ N. Assume that H ∈ C n+2 (A) and that adkA (H) are H-bounded for all
1 ≤ k ≤ n + 2. Assume that H satisfies a Mourre estimate on I. Let λ ∈ I be
an eigenvalue of H and let P = 1{λ} (H) be the associated eigenprojection.
Then we have that
Ran(P) ⊆ D(An )
Remark
In fact H ∈ C n+1 (A) is sufficient for the conclusion of the previous theorem to
hold and this is optimal ([FMS’ 10], [MW’ 10]).
9 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Part II
The Nelson model
10 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Definition of the model
• Hilbert space: H = L2 (R3 ) ⊗ F ' L2 (R3 ; F) where F is the symmetric
Fock space over L2 (R3 ) defined by
F =C⊕
+∞
M
n
L2 (R3 )⊗s
n=1
• Hamiltonian:
Hg = Hel ⊗ 1 + 1 ⊗ Hf + g φ(h(x))
where
∗
Hel = −∆ + V (x) + U(x)
References
with V ∆ and U(x) ≥ c0 |x|α − c1 , c0 > 0, α > 4
∗
Hf = dΓ(|k|)
∗
φ(h(x)) = a∗ (h(x)) + a(h(x))
where ∀ x ∈ R3 , h(x) ∈ L2 (R3 , dk) is given by
h(x, k) =
χ(k)
1
|k| 2 −
e −ik·x ,
3
χ ∈ C∞
0 (R ),
>0
11 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Definition of the model
• Hilbert space: H = L2 (R3 ) ⊗ F ' L2 (R3 ; F) where F is the symmetric
Fock space over L2 (R3 ) defined by
F =C⊕
+∞
M
n
L2 (R3 )⊗s
n=1
• Hamiltonian:
Hg = Hel ⊗ 1 + 1 ⊗ Hf + g φ(h(x))
where
∗
Hel = −∆ + V (x) + U(x)
References
with V ∆ and U(x) ≥ c0 |x|α − c1 , c0 > 0, α > 4
∗
Hf = dΓ(|k|)
∗
φ(h(x)) = a∗ (h(x)) + a(h(x))
where ∀ x ∈ R3 , h(x) ∈ L2 (R3 , dk) is given by
h(x, k) =
χ(k)
1
|k| 2 −
e −ik·x ,
3
χ ∈ C∞
0 (R ),
>0
11 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Definition of the model
• Hilbert space: H = L2 (R3 ) ⊗ F ' L2 (R3 ; F) where F is the symmetric
Fock space over L2 (R3 ) defined by
F =C⊕
+∞
M
n
L2 (R3 )⊗s
n=1
• Hamiltonian:
Hg = Hel ⊗ 1 + 1 ⊗ Hf + g φ(h(x))
where
∗
Hel = −∆ + V (x) + U(x)
References
with V ∆ and U(x) ≥ c0 |x|α − c1 , c0 > 0, α > 4
∗
Hf = dΓ(|k|)
∗
φ(h(x)) = a∗ (h(x)) + a(h(x))
where ∀ x ∈ R3 , h(x) ∈ L2 (R3 , dk) is given by
h(x, k) =
χ(k)
1
|k| 2 −
e −ik·x ,
3
χ ∈ C∞
0 (R ),
>0
11 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Definition of the model
• Hilbert space: H = L2 (R3 ) ⊗ F ' L2 (R3 ; F) where F is the symmetric
Fock space over L2 (R3 ) defined by
F =C⊕
+∞
M
n
L2 (R3 )⊗s
n=1
• Hamiltonian:
Hg = Hel ⊗ 1 + 1 ⊗ Hf + g φ(h(x))
where
∗
Hel = −∆ + V (x) + U(x)
References
with V ∆ and U(x) ≥ c0 |x|α − c1 , c0 > 0, α > 4
∗
Hf = dΓ(|k|)
∗
φ(h(x)) = a∗ (h(x)) + a(h(x))
where ∀ x ∈ R3 , h(x) ∈ L2 (R3 , dk) is given by
h(x, k) =
χ(k)
1
|k| 2 −
e −ik·x ,
3
χ ∈ C∞
0 (R ),
>0
11 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Definition of the model
• Hilbert space: H = L2 (R3 ) ⊗ F ' L2 (R3 ; F) where F is the symmetric
Fock space over L2 (R3 ) defined by
F =C⊕
+∞
M
n
L2 (R3 )⊗s
n=1
• Hamiltonian:
Hg = Hel ⊗ 1 + 1 ⊗ Hf + g φ(h(x))
where
∗
Hel = −∆ + V (x) + U(x)
References
with V ∆ and U(x) ≥ c0 |x|α − c1 , c0 > 0, α > 4
∗
Hf = dΓ(|k|)
∗
φ(h(x)) = a∗ (h(x)) + a(h(x))
where ∀ x ∈ R3 , h(x) ∈ L2 (R3 , dk) is given by
h(x, k) =
χ(k)
1
|k| 2 −
e −ik·x ,
3
χ ∈ C∞
0 (R ),
>0
11 / 25
Second
order perturbation
theory
Fermi Golden Rule
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• Let H0 be the ’unperturbed‘ operator. Under different assumptions, it is
established that, for sufficiently small values of g , Fermi Golden Rule
holds for excited unperturbed eigenvalues ([BFS ’99], [BFSS ’99], [DJ
’01], [Go ’09]). In particular the spectrum of Hg is purely absolutely
continuous in a neighborhood of the excited unperturbed eigenvalues
• Problem: show that ’generically’ Hg does not have eigenvalue above the
ground state energy for an arbitrary value of g . More precisely, assuming
that λ is an eigenvalue of Hg for a given g ∈ R, we want to show that λ
is unstable under small perturbations according to Fermi Golden Rule
12 / 25
Second
order perturbation
theory
Fermi Golden Rule
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• Let H0 be the ’unperturbed‘ operator. Under different assumptions, it is
established that, for sufficiently small values of g , Fermi Golden Rule
holds for excited unperturbed eigenvalues ([BFS ’99], [BFSS ’99], [DJ
’01], [Go ’09]). In particular the spectrum of Hg is purely absolutely
continuous in a neighborhood of the excited unperturbed eigenvalues
• Problem: show that ’generically’ Hg does not have eigenvalue above the
ground state energy for an arbitrary value of g . More precisely, assuming
that λ is an eigenvalue of Hg for a given g ∈ R, we want to show that λ
is unstable under small perturbations according to Fermi Golden Rule
12 / 25
Second
order perturbation
theory
Choice of the conjugate operator
Jérémy
Faupin
• Generator of dilatations in Fock space
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
A1 = 1 ⊗ dΓ(a1 ) = 1 ⊗ dΓ(
i
(∇k · k + k · ∇k ))
2
Formal commutator with Hg :
[Hg , iA1 ] = dΓ(|k|) − g φ(ia1 h(x))
see [FGS ’08]. Difficulty when g is not supposed to be small
• Generator of radial translation in Fock space
A2 = 1 ⊗ dΓ(a2 ) = 1 ⊗ dΓ(
i
k
k
(∇k ·
+
· ∇k ))
2
|k|
|k|
Formal commutator with Hg :
[Hg , iA2 ] = dΓ(1) − g φ(ia2 h(x))
Mourre estimate established in [GGM ’04] for arbitrary g
13 / 25
Second
order perturbation
theory
Choice of the conjugate operator
Jérémy
Faupin
• Generator of dilatations in Fock space
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
A1 = 1 ⊗ dΓ(a1 ) = 1 ⊗ dΓ(
i
(∇k · k + k · ∇k ))
2
Formal commutator with Hg :
[Hg , iA1 ] = dΓ(|k|) − g φ(ia1 h(x))
see [FGS ’08]. Difficulty when g is not supposed to be small
• Generator of radial translation in Fock space
A2 = 1 ⊗ dΓ(a2 ) = 1 ⊗ dΓ(
i
k
k
(∇k ·
+
· ∇k ))
2
|k|
|k|
Formal commutator with Hg :
[Hg , iA2 ] = dΓ(1) − g φ(ia2 h(x))
Mourre estimate established in [GGM ’04] for arbitrary g
13 / 25
Second
order perturbation
theory
Difficulties
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• A2 is not self-adjoint, only maximal symmetric. Mourre theory with a non
self-adjoint conjugate operator initiated in [HüSp ’95] (the conjugate
operator is supposed to be the generator of a C0 -semigroup)
• [Hg , iA2 ] is not controlled by Hg (the quadratic form is not bounded on
D(Hg ) × D(Hg )). This situation is referred to as ’singlular’ Mourre
theory ([Sk ’98], [MS ’03], [GGM ’04])
• Each time we commute with iA2 , the singularity in the field operator is
increased by a power of |k|. As far as the infrared singularity is
concerned, it is crucial to minimize the number of commutators of Hg
with A2 we need to estimate
14 / 25
Second
order perturbation
theory
Difficulties
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• A2 is not self-adjoint, only maximal symmetric. Mourre theory with a non
self-adjoint conjugate operator initiated in [HüSp ’95] (the conjugate
operator is supposed to be the generator of a C0 -semigroup)
• [Hg , iA2 ] is not controlled by Hg (the quadratic form is not bounded on
D(Hg ) × D(Hg )). This situation is referred to as ’singlular’ Mourre
theory ([Sk ’98], [MS ’03], [GGM ’04])
• Each time we commute with iA2 , the singularity in the field operator is
increased by a power of |k|. As far as the infrared singularity is
concerned, it is crucial to minimize the number of commutators of Hg
with A2 we need to estimate
14 / 25
Second
order perturbation
theory
Difficulties
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• A2 is not self-adjoint, only maximal symmetric. Mourre theory with a non
self-adjoint conjugate operator initiated in [HüSp ’95] (the conjugate
operator is supposed to be the generator of a C0 -semigroup)
• [Hg , iA2 ] is not controlled by Hg (the quadratic form is not bounded on
D(Hg ) × D(Hg )). This situation is referred to as ’singlular’ Mourre
theory ([Sk ’98], [MS ’03], [GGM ’04])
• Each time we commute with iA2 , the singularity in the field operator is
increased by a power of |k|. As far as the infrared singularity is
concerned, it is crucial to minimize the number of commutators of Hg
with A2 we need to estimate
14 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Part III
Singular Mourre theory with
a non self-adjoint conjugate operator
15 / 25
Second
order perturbation
theory
Framework
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• H complex Hilbert space
1
1
• H, M self-adjoint operators, M ≥ 0, G = D(M 2 ) ∩ D(|H| 2 )
• R symmetric operator, D(R) ⊇ D(H)
• A closed operator, densely defined, maximal symmetric. Assuming that A
has deficiency indices (N, 0), this implies that A generates a
C0 -semigroup of isometries {Wt }t≥0
Definition
The map [0, ∞) 3 t 7→ Wt ∈ B(H) is called a C0 -semigroup if W0 = I ,
Wt Ws = Wt+s and w − limt→0 Wt = I . The generator of a C0 -semigroup is
defined by
˘
¯
1
D(A) = u ∈ H, Au := lim (Wt u − u)exists
t→0 it
16 / 25
Second
order perturbation
theory
Framework
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
• H complex Hilbert space
1
1
• H, M self-adjoint operators, M ≥ 0, G = D(M 2 ) ∩ D(|H| 2 )
• R symmetric operator, D(R) ⊇ D(H)
• A closed operator, densely defined, maximal symmetric. Assuming that A
has deficiency indices (N, 0), this implies that A generates a
C0 -semigroup of isometries {Wt }t≥0
Definition
The map [0, ∞) 3 t 7→ Wt ∈ B(H) is called a C0 -semigroup if W0 = I ,
Wt Ws = Wt+s and w − limt→0 Wt = I . The generator of a C0 -semigroup is
defined by
˘
¯
1
D(A) = u ∈ H, Au := lim (Wt u − u)exists
t→0 it
16 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Regularity with respect to C0 -semigroups
Definition
Let {W1,t } and {W2,t } be two C0 -semigroups in Hilbert spaces H1 and H2
with generators A1 and A2 respectively. A bounded operator B ∈ B(H1 ; H2 ) is
said to be in C 1 (A1 , A2 ) if
kW2,t B − BW1,t kB(H1 ;H2 ) ≤ Ct,
Singular
Mourre
theory
References
0≤t≤1
Remarks
• B ∈ C 1 (A1 ; A2 ) iff the quadratic form defined on D(A∗2 ) × D(A1 )
ihB ∗ φ, A1 ψiH1 − ihA∗2 φ, BψiH2
extends by continuity to a bounded quadratic form on H2 × H1
• The bounded operator in B(H1 ; H2 ) associated to the previous quadratic
form is denoted by [B, iA]0 , and we have that
[B, iA]0 = s − lim
t→0
1
(BW1,t − W2,t B)
t
• If B ∈ C 1 (A1 ; A2 ) and [B, iA]0 ∈ C 1 (A1 ; A2 ) we say that B ∈ C 2 (A1 ; A2 )
17 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Regularity with respect to C0 -semigroups
Definition
Let {W1,t } and {W2,t } be two C0 -semigroups in Hilbert spaces H1 and H2
with generators A1 and A2 respectively. A bounded operator B ∈ B(H1 ; H2 ) is
said to be in C 1 (A1 , A2 ) if
kW2,t B − BW1,t kB(H1 ;H2 ) ≤ Ct,
Singular
Mourre
theory
References
0≤t≤1
Remarks
• B ∈ C 1 (A1 ; A2 ) iff the quadratic form defined on D(A∗2 ) × D(A1 )
ihB ∗ φ, A1 ψiH1 − ihA∗2 φ, BψiH2
extends by continuity to a bounded quadratic form on H2 × H1
• The bounded operator in B(H1 ; H2 ) associated to the previous quadratic
form is denoted by [B, iA]0 , and we have that
[B, iA]0 = s − lim
t→0
1
(BW1,t − W2,t B)
t
• If B ∈ C 1 (A1 ; A2 ) and [B, iA]0 ∈ C 1 (A1 ; A2 ) we say that B ∈ C 2 (A1 ; A2 )
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Second
order perturbation
theory
Assumptions (I)
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
(Regularity of H with respect to A)
• Wt G ⊆ G, Wt∗ G ⊆ G, and ∀φ ∈ G,
sup kWt φk < ∞,
0<t<1
sup kWt∗ φk < ∞
0<t<1
This implies that
∗ Wt |G is a C0 -semigroup whose generator is denoted by AG
∗ Wt extends to a C0 -semigroup in G ∗ whose generator is denoted by AG ∗
• H ∈ C 2 (AG ; AG ∗ ) and for all φ ∈ D(H) ∩ D(M),
[H, iA]0 φ = (M + R)φ
(Regularity of H with respect to M)
H ∈ C 1 (M) and [H, iM]0 is H-bounded
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Second
order perturbation
theory
Assumptions (I)
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
(Regularity of H with respect to A)
• Wt G ⊆ G, Wt∗ G ⊆ G, and ∀φ ∈ G,
sup kWt φk < ∞,
0<t<1
sup kWt∗ φk < ∞
0<t<1
This implies that
∗ Wt |G is a C0 -semigroup whose generator is denoted by AG
∗ Wt extends to a C0 -semigroup in G ∗ whose generator is denoted by AG ∗
• H ∈ C 2 (AG ; AG ∗ ) and for all φ ∈ D(H) ∩ D(M),
[H, iA]0 φ = (M + R)φ
(Regularity of H with respect to M)
H ∈ C 1 (M) and [H, iM]0 is H-bounded
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Second
order perturbation
theory
The Virial Theorem
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Remark
Under the previous assumptions,
hφ1 , (M + R)φ2 i = ihHφ1 , Aφ2 i − ihA∗ φ1 , Hφ2 i
for all φ1 ∈ D(H) ∩ D(M) ∩ D(A∗ ) and φ2 ∈ D(H) ∩ D(M) ∩ D(A)
Theorem ([GGM ’04])
Assume that the previous hypotheses hold. If ψ is an eigenstate of H such
1
that ψ ∈ D(M 2 ), then
1
hψ, (M + R)ψi := kM 2 ψk2 + hψ, Rψi = 0
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Second
order perturbation
theory
The Virial Theorem
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Remark
Under the previous assumptions,
hφ1 , (M + R)φ2 i = ihHφ1 , Aφ2 i − ihA∗ φ1 , Hφ2 i
for all φ1 ∈ D(H) ∩ D(M) ∩ D(A∗ ) and φ2 ∈ D(H) ∩ D(M) ∩ D(A)
Theorem ([GGM ’04])
Assume that the previous hypotheses hold. If ψ is an eigenstate of H such
1
that ψ ∈ D(M 2 ), then
1
hψ, (M + R)ψi := kM 2 ψk2 + hψ, Rψi = 0
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Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Assumptions (II)
(Mourre estimate)
∃ an interval I ⊆ R such that ∀ η ∈ I, ∃ c0 > 0, C1 ∈ R, K0 compact, and a
function fη ∈ C0∞ (R; [0, 1]) such that fη = 1 in a neighborhood of η and
M + R ≥ c0 − C1 fη⊥ (H)2 hHi − K0 ,
in the sense of quadratic forms on D(H) ∩ D(M), where fη⊥ = 1 − fη
References
(Regularity of bound states and the perturbation) (∗)
For all compact interval J ⊆ I, ∃ γ > 0 and a set Bγ such that
˘
Bγ ⊆ V symmetric and H-bounded, V ∈ C 1 (AG ; AG ∗ )
¯
kV k1 := kV (H − i)−1 k + k[V , iA]0 (H − i)−1 k ≤ γ ,
{0} ⊂ Bγ , Bγ is star-shaped and symmetric w.r.t. 0, and the following holds:
∃ C > 0, ∀ V ∈ Bγ , ∀ λ ∈ J, ∀ ψ ∈ D(H), (H + V − λ)ψ = 0, we have that
ψ ∈ D(A) ∩ D(M)
and
kAψk ≤ C kψk
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Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Assumptions (II)
(Mourre estimate)
∃ an interval I ⊆ R such that ∀ η ∈ I, ∃ c0 > 0, C1 ∈ R, K0 compact, and a
function fη ∈ C0∞ (R; [0, 1]) such that fη = 1 in a neighborhood of η and
M + R ≥ c0 − C1 fη⊥ (H)2 hHi − K0 ,
in the sense of quadratic forms on D(H) ∩ D(M), where fη⊥ = 1 − fη
References
(Regularity of bound states and the perturbation) (∗)
For all compact interval J ⊆ I, ∃ γ > 0 and a set Bγ such that
˘
Bγ ⊆ V symmetric and H-bounded, V ∈ C 1 (AG ; AG ∗ )
¯
kV k1 := kV (H − i)−1 k + k[V , iA]0 (H − i)−1 k ≤ γ ,
{0} ⊂ Bγ , Bγ is star-shaped and symmetric w.r.t. 0, and the following holds:
∃ C > 0, ∀ V ∈ Bγ , ∀ λ ∈ J, ∀ ψ ∈ D(H), (H + V − λ)ψ = 0, we have that
ψ ∈ D(A) ∩ D(M)
and
kAψk ≤ C kψk
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Second
order perturbation
theory
Upper semicontinuity of point spectrum
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Theorem ([FMS’ 10])
Assume that the previous hypotheses hold. Let J ⊆ I be a compact interval
such that σpp (H) ∩ J = {λ}. There exists 0 < γ 0 ≤ γ such that if V ∈ Bγ
and kV k1 ≤ γ 0 , then the total multiplicity of the eigenvalues of H + V in J is
at most dim Ker(H − λ)
References
Remark
In the case where σpp (H) ∩ J = ∅, Hypothesis (∗) on the regularity of bound
states and the perturbation is not necessary to conclude that
σpp (H + V ) ∩ J = ∅. It is sufficient to assume that
• V ∈ C 2 (AG ; AG ∗ ) and V , [V , iA]0 are H-bounded
or
• V ∈ C 1 (AG ; AG ∗ ), V and [V , iA]0 are H-bounded, and the possibly
1
existing eigenstates of H + V belong to D(M 2 )
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Second
order perturbation
theory
Upper semicontinuity of point spectrum
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
Theorem ([FMS’ 10])
Assume that the previous hypotheses hold. Let J ⊆ I be a compact interval
such that σpp (H) ∩ J = {λ}. There exists 0 < γ 0 ≤ γ such that if V ∈ Bγ
and kV k1 ≤ γ 0 , then the total multiplicity of the eigenvalues of H + V in J is
at most dim Ker(H − λ)
References
Remark
In the case where σpp (H) ∩ J = ∅, Hypothesis (∗) on the regularity of bound
states and the perturbation is not necessary to conclude that
σpp (H + V ) ∩ J = ∅. It is sufficient to assume that
• V ∈ C 2 (AG ; AG ∗ ) and V , [V , iA]0 are H-bounded
or
• V ∈ C 1 (AG ; AG ∗ ), V and [V , iA]0 are H-bounded, and the possibly
1
existing eigenstates of H + V belong to D(M 2 )
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Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Fermi Golden Rule criterion
(Further technical hypothesis)
1
D(M 2 ) ∩ D(H) ∩ D(A∗ ) is a core for A∗
Theorem ([FMS ’10])
Assume that the previous hypotheses hold. Let J ⊆ I be a compact interval
such that σpp (H) ∩ J = {λ}. Let P = 1{λ} (H) and P̄ = I − P. Let V ∈ Bγ
be such that
`
´
PV Im (H − λ − i0)−1 P̄ VP ≥ cP, c > 0.
There exists σ0 > 0 such that for all 0 < |σ| ≤ σ0 ,
σpp (H + σV ) ∩ J = ∅
Remark
Hypothesis (∗) on the regularity of bound states and the perturbation can be
replaced by the following two assumptions:
• Ran(P) ⊆ D(A2 )
• V ∈ C 2 (AG ; AG ∗ ) and V , [V , iA]0 are H-bounded
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Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Fermi Golden Rule criterion
(Further technical hypothesis)
1
D(M 2 ) ∩ D(H) ∩ D(A∗ ) is a core for A∗
Theorem ([FMS ’10])
Assume that the previous hypotheses hold. Let J ⊆ I be a compact interval
such that σpp (H) ∩ J = {λ}. Let P = 1{λ} (H) and P̄ = I − P. Let V ∈ Bγ
be such that
`
´
PV Im (H − λ − i0)−1 P̄ VP ≥ cP, c > 0.
There exists σ0 > 0 such that for all 0 < |σ| ≤ σ0 ,
σpp (H + σV ) ∩ J = ∅
Remark
Hypothesis (∗) on the regularity of bound states and the perturbation can be
replaced by the following two assumptions:
• Ran(P) ⊆ D(A2 )
• V ∈ C 2 (AG ; AG ∗ ) and V , [V , iA]0 are H-bounded
22 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Fermi Golden Rule criterion
(Further technical hypothesis)
1
D(M 2 ) ∩ D(H) ∩ D(A∗ ) is a core for A∗
Theorem ([FMS ’10])
Assume that the previous hypotheses hold. Let J ⊆ I be a compact interval
such that σpp (H) ∩ J = {λ}. Let P = 1{λ} (H) and P̄ = I − P. Let V ∈ Bγ
be such that
`
´
PV Im (H − λ − i0)−1 P̄ VP ≥ cP, c > 0.
There exists σ0 > 0 such that for all 0 < |σ| ≤ σ0 ,
σpp (H + σV ) ∩ J = ∅
Remark
Hypothesis (∗) on the regularity of bound states and the perturbation can be
replaced by the following two assumptions:
• Ran(P) ⊆ D(A2 )
• V ∈ C 2 (AG ; AG ∗ ) and V , [V , iA]0 are H-bounded
22 / 25
Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Second order expansion of eigenvalues
(simple case)
Theorem ([FMS ’10])
Assume that the previous hypotheses hold. Let J ⊆ I be a compact interval
such that σpp (H) ∩ J = {λ}. Let P = 1{λ} (H) and P̄ = I − P. Let V ∈ Bγ .
Suppose that
P = |ψihψ|.
For all > 0, there exists σ0 > 0 such that if |σ| ≤ σ0 and λσ ∈ J is an
eigenvalue of H + σV , then
˛
˛
˛
˛
2
−1
2
˛λσ − λ − σhψ, V ψi + σ hV ψ, (H − λ − i0) P̄V ψi˛ ≤ σ ,
and there exists a normalized eigenstate ψσ , Hσ ψσ = λσ ψσ , such that
‚
‚
‚
‚
−1
≤ |σ|
‚ψσ − ψ + σ(H − λ − i0) P̄V ψ ‚
∗
D(A)
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Second
order perturbation
theory
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
Second order expansion of eigenvalues
(general case)
If Hypothesis (∗) on the regularity of bound states and the perturbation is
replaced by the following two assumptions:
• Ran(P) ⊆ D(A2 )
• V ∈ C 2 (AG ; AG ∗ ) and V , [V , iA]0 are H-bounded
then the following theorem holds:
Theorem ([FMS ’10])
Let J ⊆ I be a compact interval such that σpp (H) ∩ J = {λ}. Let
P = 1{λ} (H) and P̄ = I − P. There exist C ≥ 0 and σ0 > 0 such that if
|σ| ≤ σ0 and λσ ∈ J is an eigenvalue of Hσ = H + σV , then there exists
ψ ∈ Ran(P), kψk = 1, such that
˛
˛
˛λσ − λ − σhψ, V ψi + σ 2 hV ψ, (H − λ − i0)−1 P̄V ψi˛ ≤ C |σ| 25
24 / 25
Second
order perturbation
theory
References
Jérémy
Faupin
Regular
Mourre
theory
Nelson
model
Singular
Mourre
theory
References
[AHS ’89] S. Agmon, I. Herbst, E. Skibsted, Perturbation of embedded eigenvalues in the generalized N-body problem, Comm. Math.
Phys., 122, (1989), 411–438.
[ABG ’96] W. Amrein, A. Boutet de Monvel, V. Georgescu, C0 -groups, commutator methods and spectral theory of N-body
Hamiltonians, Basel–Boston–Berlin, Birkhäuser, 1996.
[BFS ’98] V. Bach, J. Fröhlich, I.M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. Math., 137, (1998),
299–395.
[BFSS ’99] V. Bach, J. Fröhlich, I.M. Sigal, A. Soffer, Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and
molecules, Comm. Math. Phys., 207, (1999), 557–587.
[Ca ’05] L. Cattaneo, Mourre’s inequality and embedded boundstates, Bull. Sci. Math., 129, (2005), 591–614.
[CGH ’06] L. Cattaneo, G.M. Graf, W. Hunziker, A general resonance theory based on Mourre’s inequality, Ann. Henri Poincaré 7,
(2006), 583–601.
[DJ ’01] J. Dereziński, V. Jakšić, Spectral theory of Pauli-Fierz operators, J. Funct. Anal., 180, (2001), 243–327.
[FMS ’10] J. Faupin, J.S. Møller, E. Skibsted, Regularity of embedded bound states, (2010), Preprint.
[FMS ’10] J. Faupin, J.S. Møller, E. Skibsted, Second order perturbation theory for embedded eigenvalues, (2010), Preprint.
[FGS ’08] J. Fröhlich, M. Griesemer, I.M. Sigal, Spectral Theory for the Standard Model of Non-Relativistic QED, Comm. Math. Phys.,
283, (2008), 613–646.
[GG ’99] V. Georgescu, C. Gérard, On the virial theorem in quantum mechanics, Comm. Math. Phys., 208, (1999), 275–281.
[GGM ’04] V. Georgescu, C. Gérard, J.S. Møller, Commutators, C0 –semigroups and resolvent estimates, J. Funct. Anal., 216, (2004),
303–361.
[GGM ’04] V. Georgescu, C. Gérard, J.S. Møller, Spectral theory of massless Pauli-Fierz models, Comm. Math. Phys., 249, (2004),
29–78.
[Ge] C. Gérard, A proof of the abstract limiting absorption principle by energy estimates, J. Funct. Anal., 254, (2008), 2707–2724.
[Go ’09] S. Golénia, Positive commutators, Fermi Golden Rule and the spectrum of 0 temperature Pauli-Fierz Hamiltonians, J. Funct.
Anal., 256, (2009), 2587–2620.
[HuSi ’00] W. Hunziker and I.M. Sigal, The quantum N-body problem, J. Math. Phys., 41, (2000), 3448–3510.
[HüSp ’95] M. Hübner, H. Spohn, Spectral properties of the spin-boson Hamiltonian, Ann. Inst. Henri Poincaré, 62, (1995), 289–323.
[Mo ’81] É. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Comm. Math. Phys., 78, (1981),
391–408.
[MS ’04] J.S. Møller, E. Skibsted, Spectral theory of time-periodic many-body systems, Advances in Math., 188, (2004), 137–221.
[Sk ’98] E. Skibsted, Spectral analysis of N-body systems coupled to a bosonic field, Rev. Math. Phys., 10, (1998), 989–1026.
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