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Comptes Rendus
Mathématique
Anne Boutet de Monvel and Lech Zielinski
Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model
Volume 361 (2023), p. 1761-1766
Published online: 21 December 2023
https://doi.org/10.5802/crmath.515
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1761-1766
https://doi.org/10.5802/crmath.515
Mathematical physics, Spectral theory / Physique mathématique, Théorie spectrale
Asymptotic formula for large eigenvalues of
the two-photon quantum Rabi model
Formule asymptotique pour les grandes valeurs propres
du modèle quantique de Rabi à deux photons
Anne Boutet deMonvel a and Lech Zielinski ∗,b
a Institut de Mathématiques de Jusieu, Université Paris Cité, 8 place Aurélie Nemours, 75013 Paris, France
b Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville UR 2597, Université du Littoral Côte
d’Opale, 62228 Calais, France
E-mails: anne.boutet-de-monvel@imj-prg.fr, lech.zielinski@lmpa.univ-littoral.fr
Abstract. We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigen-
value sequences (E+m )∞m=0, (E
−
m )
∞
m=0 satisfying a three-term asymptotic formula with the remainder estimate
O(m−1 lnm) when m tends to infinity. By analogy to the one-photon quantum Rabi model, the leading three
terms of this asymptotic formula, describe a generalized rotating-wave approximation for large eigenvalues
of the two-photon quantum Rabi model.
Résumé. Nous démontrons que le spectre de l’hamiltonien du modèle quantique de Rabi à deux photons
est constitué de deux suites de valeurs propres (E+m )∞m=0, (E
−
m )
∞
m=0 vérifiant une formule asymptotique à
trois termes avec l’estimation de l’erreur O(m−1 lnm) quand m tend vers l’infini. Par analogie avec le modèle
quantique de Rabi à un photon, les trois termes dominants dans cette formule asymptotique décrivent
l’approximation de l’onde tournante pour les grandes valeurs propres du modèle quantique de Rabi à deux
photons.
Manuscript received 11 May 2023, accepted 16 May 2023.
1. Introduction
The quantum Rabi model describes the interactions between a two-level system and a single-
mode quantum field. It is considered as a fundamental model in various domains of theoretical
physics, e.g. cavity optics, a theory of nanostructured semiconductors, superconducting circuits,
trapped ions and quantum information. We refer to [15] for an exhaustive overview of theoretical
and experimental works in the domain.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1762 Anne Boutet de Monvel and Lech Zielinski
The papers Boutet de Monvel, Naboko, Silva [7, 8] and Yanovich [14] (see also [13]) investigate
the behaviour of large eigenvalues of operators related to the quantum Rabi model. The paper [9]
gives the three-term asymptotic formula for large eigenvalues of the one-photon Rabi model
(see Definition 1(1)). The two-term asymptotic formula for the two-photon Rabi model (see
Definition 1(2)) is given in [10] and this note gives the three-term asymptotic formula for this
model. Concerning the two-photon Rabi model, we refer to [2, 5, 15].
In Section 2 we give the definition of the basic quantum Rabi model in the one-photon and
two-photon case. Our main result is Theorem 2, stated in Section 3. We describe main ingredients
of the proof in Sections 4, 5 and 6.
We remark that our result is closely related to the method of the generalized rotating-wave
approximation (GRWA), used by a great number of physicists working with the quantum Rabi
model. The method takes its name from the famous paper of Irish [6], but the same idea was
also used in the paper Feranchuk, Komarov, Ulyanenkov [4] under the name of the zeroth order
approximation of the operator method (see also [3]). It appears that in the case of the one-photon
quantum Rabi model, the GRWA given by the formula (25) in [4], coincides with the three-term
asymptotic formula proved in [9]. In Section 6 we describe the relation between the assertion of
Theorem 2 and the GRWA for large eigenvalues of the two-photon quantum Rabi model.
2. Definition of the quantum Rabi model
In what follows, Z denotes the set of integers, N= {n ∈Z : n ≥ 0} and ℓ2(N) is the Hilbert space of
square summable sequences x :N→C. For s ≥ 0 we denote
ℓ2,s (N) =
{
x ∈ ℓ2(N) : ∑
k∈N
(1+k2)s |x(k)|2 <∞
}
.
We define the photon annihilation and creation operators â and â†, as linear maps ℓ2,1/2(N) →
ℓ2(N) satisfying
â†em =
p
m +1em+1 for m = 0, 1, 2, . . .
âem =
p
m −1em−1 for m = 1, 2, 3, . . . and âe0 = 0,
where {em}m∈N is the canonical basis of ℓ2(N).
Definition 1. We fix two real parameters: the energy spacing of the two-level system ∆ and the
coupling constant g .
(1) The Hamiltonian of the one-photon quantum Rabi model is given by the linear map
C2 ⊗ℓ2,1(N) →C2 ⊗ℓ2(N) of the form
H1 =
(∆
2 0
0 −∆2
)
⊗ Iℓ2(N) + IC2 ⊗ â†â +
(
0 g
g 0
)
⊗
(
â† + â
)
. (1)
(2) The Hamiltonian of the two-photon quantum Rabi model is given by the linear map
C2 ⊗ℓ2,1(N) →C2 ⊗ℓ2(N) of the form
H2 =
(∆
2 0
0 −∆2
)
⊗ Iℓ2(N) + IC2 ⊗ â†â +
(
0 g
g 0
)
⊗
(
(â†)2 + â2
)
. (2)
3. Main result
In what follows, we assume 0 < g < 1/2 and introduce
β :=
√
1−4g 2. (3)
Anne Boutet de Monvel and Lech Zielinski 1763
Let H 02 denote the operator given by (2) with ∆ = 0. If 0 < g < 1/2, then the spectrum of H 02 is
explicitly known (see [2]): it is composed of the sequence of eigenvalues
E 0m = mβ+ (β−1)/2, m = 0, 1, 2, . . . (4)
and each eigenvalue E 0m is of multiplicity 2. Thus 0 < g < 1/2 ensures the fact that H 02 is self-
adjoint and has compact resolvent. Since H2−H 02 is bounded, the operator H2 is self-adjoint and
has compact resolvent if 0 < g < 1/2. The explicit values of eigenvalues of H2 are not known when
∆ ̸= 0, but we can described their asymptotic behavior in
Theorem 2. Assume that 0 < g < 1/2. Then one can find {v+m}m∈N ∪ {v−m}m∈N, an orthonormal
basis of C2 ⊗ℓ2(N), such that
H2v
±
m = E±m v±m , m = 0, 1, 2, . . .
and the eigenvalue sequences (E+m)m∈N, (E−m)m∈N, satisfy the large m estimates
E±m = mβ+ (β−1)/2± rm +O(m−1 lnm) (5)
with rm given by the formula
rm =

∆
2
√
β
πg m
cos((2m +1)α) if m is even
∆
2
√
β
πg m
sin((2m +1)α) if m is odd
(6)
where we have denoted
α := arctan
(√
1−2g
1+2g
)
. (7)
Remarks.
(a) One has E±m −E 0m = O(m−1/2) in spite of the fact that H2 − H 02 is not compact. A similar
fact was established in [14] for the one-photon Rabi model.
(b) Following [12], one can prove that the spectrum of H2 is not discrete if g ≥ 1/2.
4. Initial reformulations
It is easy to check that the operator H2 has four closed invariant subspaces:
H −0 spanned by {(1,0)⊗e4k : k ∈N}∪ {(0,1)⊗e4k+2 : k ∈N}
H +0 spanned by {(0,1)⊗e4k : k ∈N}∪ {(1,0)⊗e4k+2 : k ∈N}
H −1 spanned by {(1,0)⊗e4k+1 : k ∈N}∪ {(0,1)⊗e4k+3 : k ∈N}
H +1 spanned by {(0,1)⊗e4k+1 : k ∈N}∪ {(1,0)⊗e4k+3 : k ∈N}
and the matrix of H2 in a suitable basis of H ±µ µ=0, 1, can be written in the form
J±µ =

d±µ (0) bµ(0) 0 0 · · ·
bµ(0) d±µ (1) bµ(1) 0 · · ·
0 bµ(1) d±µ (2) bµ(2) · · ·
0 0 bµ(2) d±µ (3) · · ·
...
...
...
...
. . .
 (8)
where
d±µ (m) := 2m +µ± (−1)m∆/2,
bµ(m) := g
√
(2m +1+µ)(2m +2+µ).
1764 Anne Boutet de Monvel and Lech Zielinski
It therefore remains to investigate the asymptotic behavior of eigenvalues of operators defined
by J−0 , J
+
0 , J
−
1 and J
+
1 . For this purpose, we remark that
bµ(m) = 2g (m +γ)+O(m−1) holds with γ := µ
2
+ 3
4
. (9)
Using the result of Rozenblum stated in Theorem 1.1 of [11], we find that modulo O(m−1),
the asymptotic behaviour of the m-th eigenvalue remains the same if the entries {bµ(m)}m∈N
are replaced by {2g (m +γ)}m∈N with γ := µ2 + 34 . This fact allows us to deduce the assertion of
Theorem 2 from
Theorem 3. Assume that 0 < g < 1/2. Let δ, γ be some real numbers and let Ĵδγ be the linear map
ℓ2,1(N) → ℓ2(N) defined by the formula
Ĵδγem =
(
m + (−1)mδ)em + g (m +γ)em+1 + g (m −1+γ)em−1, m ∈N, (10)
where {em}m∈N is the canonical basis in ℓ2(N) and by convention g (m −1+γ)em−1 = 0 if m = 0.
Then Ĵδγ has discrete spectrum and its non-decreasing eigenvalue sequence satisfies the large n
asymptotic formula
λn( Ĵ
δ
γ) =βn + (γ−1/2)(β−1)+δrγ,n +O(n−1 lnn) (11)
with
rγ,n =
√
β
2πg n
cos
(
4αn + θ̂γ
)
, (12)
where β is given by (3), α by (7) and
θ̂γ = (γ−1/2)(4α−π)+π/4. (13)
In order to prove Theorem 3, we move from ℓ2(N) to ℓ2(Z), the Hilbert space of square
summable sequences x :Z→C. For s > 0 we denote
ℓ2,s (Z) =
{
x ∈ ℓ2(Z) : ∑
k∈Z
(1+k2)s |x(k)|2 <∞
}
and define J̃δγ as the linear map ℓ
2,1(Z) → ℓ2(Z) given by
J̃δγek =
(
k + (−1)kδ)ek + g (k +γ)ek+1 + g (k −1+γ)ek−1, k ∈Z, (14)
where {ek }k∈Z is the canonical basis of ℓ2(Z). We can identify ℓ2(N) with
{x ∈ ℓ2(Z) : x(k) = 0 for k ∈Z\N}
and consider J̃δγ as an extension of Ĵ
δ
γ. Using Theorem 1.1 in [11], we find that the spectrum of J̃
δ
γ
is composed of a non-decreasing sequence of eigenvalues {λ j ( J̃δγ)} j∈Z, which can be labeled so
that for any N > 0 one has the estimate
λn( J̃
δ
γ) =λn( Ĵδγ)+O(n−N ) when n →∞ (15)
5. Diagonalisation of J̃δγ when δ= 0
The operator J̃δγ with δ= 0 was investigated in [1]. Let S be the shift Se j = e j+1 in ℓ2(Z) and denote
Λ := diag( j ) j∈Z. Then (14) gives
J̃ 0γ =Λ+ g
(
S
(
Λ+γ)+ (Λ+γ)S−1)=Λ+ g (S(Λ+γ)+ h.c.)
Let T=R/2πZ and let FT be the isometric isomorphism L2(T) → ℓ2(Z) given by
(FT f )( j ) =
∫ π
−π
f (θ)e−i jθ
dθ
2π
. (16)
Anne Boutet de Monvel and Lech Zielinski 1765
Then Lγ :=F−1T J̃ 0γFT is the first order differential operator
Lγ = 1
2
((
1+2g cosθ)(−i d
dθ
)
+h.c.
)
+ (1+2γ)cosθ
and following [1], we define the diffeomorphism of ]−π,π[ given by
Φ(θ) :=
∫ θ
0
βdθ′
1+2g cosθ′ = 2arctan
(√
1−2g
1+2g tan
(θ
2
))
. (17)
The change of variable η = Φ(θ) defines the unitary operator acting in L2(T) according to the
formula
(UΦ f )(θ) =Φ′(θ)1/2 f (Φ(θ)) (18)
and the direct computation gives
U−1Φ LγUΦ =β
(
−i d
dη
+qγ(η)
)
with
qγ(η) :=β−1(1+2γ)cos(Φ−1(η)).
Let q̃γ be a primitive of qγ. We compute
〈qγ〉 := (q̃γ(π)− q̃γ(−π))/(2π) = (γ−1/2)(1−1/β)
and remark that η→〈qγ〉η− q̃γ(η) is 2π-periodic, hence we can define ( fγ, j ) j∈Z, the orthonormal
basis in L2(T) given by
fγ, j (η) = ei jη ei(〈qγ〉η−q̃γ(η)). (19)
Then (see [1]), for every j ∈Z, one has
β
(
−i d
dη
+qγ
)
fγ, j =β( j +〈qγ〉) fγ, j .
Consequently, for every j ∈Z,
J̃ 0γuγ, j = dγ, j uγ, j (20)
holds with
uγ, j =FTUΦ fγ, j , (21)
dγ, j =β( j +〈qγ〉) =β j + (γ−1/2)(β−1). (22)
6. Generalized rotating-wave approximation (GRWA)
The idea of the GRWA consists in using the diagonal entries of a perturbation as the first
correction for eigenvalues of a perturbed diagonal matrix and we refer to [3] for numerous
examples of this approach. In order to apply this idea, we consider J̃δγ as a perturbation of J̃
0
γ
and use the diagonalisation (20). We remark that
J̃δγ = J̃ 0γ+δV (23)
holds with V := diag((−1) j ) j∈Z. Let Uγ be the unitary operator in ℓ2(Z) defined by Uγe j = uγ, j ,
where {e j } j∈Z is the canonical basis of ℓ2(Z) and {uγ, j } j∈Z is the basis given by (21). Then (23)
gives
U−1γ J̃
δ
γUγ = Dγ+δVγ, (24)
where Dγ = diag(dγ, j ) j∈Z and Vγ = U−1γ V Uγ. We claim that the asymptotic estimate (11)–(13)
follows from
λ j (Dγ+δVγ) = dγ, j +δVγ( j , j )+O( j−1 ln j ) when j →∞ (25)
where
Vγ( j , j ) = 〈e j ,Vγe j 〉ℓ2(Z) = 〈uγ, j ,V uγ, j 〉ℓ2(Z).
1766 Anne Boutet de Monvel and Lech Zielinski
Indeed, since F−1
T
V FT = Tπ is the translation θ+2πZ→ θ+π+2πZ, we find that
Vγ( j , j ) = 〈UΦ fγ, j ,TπUΦ fγ, j 〉L2(T) =
∫ π
−π
ei j (Φ−TπΦ)(θ)pγ(θ) (Tπpγ)(θ)
dθ
2π
(26)
holds with
pγ(θ) := ei〈qγ〉Φ(θ)−iq̃γ(Φ(θ))β1/2(1+2g cosθ)−1/2
and the stationary phase method gives
Vγ( j , j ) = rγ, j +O( j−1) when j →∞ (27)
where rγ, j is given by (12)–(13). Thus (11)–(13) follow from (25), (27) and (15).
The proof of (25) is based on the approach of Yanovich [14] and the explicit expressions of the
entries
Vγ( j ,k) = 〈UΦ fγ, j ,TπUΦ fγ,k〉L2(T). (28)
References
[1] J. Edward, “Spectra of Jacobi matrices, differential equations on the circle, and the su(1,1) Lie algebra”, SIAM J. Math.
Anal. 24 (1993), no. 3, p. 824-831.
[2] C. Emary, R. F. Bishop, “Bogoliubov transformations and exact isolated solutions for simple nonadiabatic Hamilto-
nians”, J. Math. Phys. 43 (2002), no. 8, p. 3916-3926.
[3] I. D. Feranchuk, A. Ivanov, V.-H. Le, A. P. Ulyanenkov, Non-perturbative description of quantum systems, Lecture Notes
in Physics, vol. 894, Springer, 2015.
[4] I. D. Feranchuk, L. I. Komarov, A. P. Ulyanenkov, “Two-level systems in a one-mode quantum field: Numerical
solution on the basis of the operator method”, J. Phys. A. Math. Gen. 29 (1996), no. 14, p. 4035-4047.
[5] C. Gerry, “Two-photon Jaynes–Cummings model interacting with the squeezed vacuum”, Phys. Rev. A 37 (1988), no. 7,
p. 2683-2686.
[6] E. K. Irish, “Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling”, Phys. Rev. Lett. 99 (2007),
article no. 173601.
[7] A. Boutet de Monvel, S. Naboko, L. O. Silva, “Eigenvalue asymptotics of a modified Jaynes–Cummings model with
periodic modulations”, Comptes Rendus Math. Acad. Sci. Paris 338 (2004), no. 1, p. 103-107.
[8] ——— , “The asymptotic behavior of eigenvalues of a modified Jaynes–Cummings model”, Asymptotic Anal. 47
(2006), no. 3-4, p. 291-315.
[9] A. Boutet de Monvel, L. Zielinski, “Oscillatory behavior of large eigenvalues in quantum Rabi models”, Int. Math. Res.
Not. 2021 (2019), no. 7, p. 5155-5213.
[10] ——— , “Asymptotic behavior of large eigenvalues of the two-photon Rabi model”, in Schrödinger operators, spectral
analysis and number theory. In memory of Erik Balslev, Springer Proceedings in Mathematics & Statistics, vol. 348,
Springer, 2021, p. 89-115.
[11] G. V. Rozenblum, “Near-similarity of operators and the spectral asymptotic behavior of pseudodifferential operators
on the circle”, Tr. Mosk. Mat. O.-va 36 (1975), p. 59-84.
[12] J. Sahbani, “Spectral theory of certain unbounded Jacobi matrices”, J. Math. Anal. Appl. 342 (2008), no. 1, p. 663-681.
[13] E. A. Tur, “Jaynes–Cummings model without rotating wave approximation”, https://arxiv.org/abs/math-ph/
0211055, 2002.
[14] E. A. (Tur) Yanovich, “Asymptotics of eigenvalues of an energy operator in a problem of quantum physics”, in
Operator methods in mathematical physics. Conference on operator theory, analysis and mathematical physics
(OTAMP) 2010, Bedlewo, Poland, Operator Theory: Advances and Applications, vol. 227, Birkhäuser, 2013, p. 165-
177.
[15] Q. Xie, H. Zhong, M. T. Batchelor, C. Lee, “The quantum Rabi model: solution and dynamics”, J. Phys. A. Math. Theor.
50 (2017), no. 11, article no. 113001 (40 pages).


Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model

Boscain, Ugo

Cannarsa, Daniele

Franceschi, Valentina

Sigalotti, Mario

Local controllability does imply global controllability

Bouin, Emeric

Journées
ÉQUATIONS AUX DÉRIVÉES PARTIELLES
Aussois, 19–23 juin 2023
Emeric Bouin
Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails
J. É. D. P. (2023), Exposé no III, 7 p.
<https://doi.org/10.5802/jedp.674>
Article mis en ligne dans le cadre du
Centre Mersenne pour l’édition scientifique ouverte
http://www.centre-mersenne.org/
RÉSEAU THÉMATIQUE AEDP DU CNRS
Journées Équations aux dérivées partielles
Aussois, 19–23 juin 2023
RT AEDP (CNRS)
Hydrodynamic limits and hypocoercivity for kinetic equations with
heavy tails
Emeric Bouin
Abstract
This expository article, written for the proceedings of the Journées EDP 2023, presents mainly
joint works with Dolbeault and Laflèche [1] and Mouhot [3]. We will review some results about
long time behaviour of linear kinetic equations for which the microscopic equilibrium (that is, the
kernel of the reorientation operator) is typically a density with polynomial decay. There will be no
space confinement and the reorientation operator could be of scattering, Fokker–Planck or Levy–
Fokker–Planck types. We will first present a spectral approach a la Ellis and Pinsky that yields to a
unified treatment of the macroscopic limits for this kind of equations and then focus on re-shaping
the Dolbeault–Mouhot–Schmeiser L2-hypocoercivity method to get explicit rates of decay to zero in
suitable weighted norms.
In this note, we consider solutions to the kinetic equation
∂tf + v · ∇xf = Lf , f(0, x, v) = f in(x, v), (0.1)
when the local equilibrium M has a fat tail given for some γ > 0 by
∀ v ∈ Rd, M(v) = cα
⌊v⌉d+α
where ⌊v⌉ :=
√
1 + |v|2 . (0.2)
The relevance of such equilibria comes from a full literature in biology and physics, that we shall
not review here, but which is presented in [1, 3].
In (0.1), the distribution function f(t, x, v) depends on a position variable x ∈ Rd, on a velocity
variable v ∈ Rd, and on time t ≥ 0. The collision operator L acts only on the v variable and,
by assumption, its null space is spanned by M. In (0.2) the normalization constant is cα =
π−d/2 Γ
(
(d + α)/2
)
/Γ(α/2) and associated to the measure
dµ = M−1(v) dv ,
we define a scalar product and a norm respectively by
⟨f, g⟩ :=
∫
Rd
f̄ g dµ and ∥f∥2 :=
∫
Rd
|f |2 dµ (0.3)
for functions f and g of the variable v ∈ Rd. Here, f̄ denotes the complex conjugate of f , as we
shall later allow for complex valued functions. For any k ∈ R, we define
∥f∥k := ∥f∥L2(⌊v⌉k dx dµ) ,
and
|||f |||k := ∥f∥L1(dx dv)∩L2(⌊v⌉k dx dµ) :=
(
∥f∥2L1(dx dv) + ∥f∥2L2(⌊v⌉k dx dµ)
)1/2
.
We consider three examples of linear collision operators, that cover a wide class of operators
that conserve mass, used in the literature, and define for each of them an associated parameter β.
1. the Fokker–Planck operator with β := 2 and local equilibrium M
L1f := ∇v ·
(
M ∇v
(
M−1f
))
.
III–1
2. the linear Boltzmann operator, or scattering collision operator
L2f :=
∫
Rd
b( · , v′)
(
f(v′) M( · ) − f( · ) M(v′)
)
dv′ ,
with positive, locally bounded collision frequency
ν(v) :=
∫
Rd
b(v, v′) M(v′) dv′ ∼
|v|→+∞
|v|−β (0.4)
3. the fractional Fokker–Planck operator
L3f := ∆σ/2v f + ∇v · (E f)
with 0 < σ < 2, β := σ − α and a radial friction force E = E(v) as a solution of
L3M = ∆σ/2v M + ∇v · (E M) = 0 . (0.5)
It turns out from a technical result that such a friction force then behaves like ⌊v⌉−βv at
infinity.
Actually, this parameter β can be defined, such that the following microscopic coercivity holds.
Hypothesis 1 (Weighted coercivity). The operator L is linear, independent of time t and space
x, commutes with rotations in v, is closed densely defined on Dom(L) ⊂ L2v(M) and satisfies
L(1) = L∗(1) = 0, where L∗ is the L2v(M)-adjoint. Finally L̃ := ⌊ ·⌉
β
2 L(⌊ ·⌉
β
2 ·) is closed densely
defined on Dom(L̃) ⊂ L2v(M), with the spectral gap estimate
∀ g ∈ Dom(L̃), g ⊥ ⌊·⌉−
β
2 , − Re
〈
L̃g, g
〉
≥ λ ∥g∥2 .
This means, translating back to L,
∀ h ∈ Dom(L), − Re
〈
Lh, h
〉
≥ λ ∥h − Ph∥2−β with Ph :=
(∫
Rd
h(v′)⌊v′⌉−βM(v′)dv′
)
.
Observe that due to total mass conservation, an initial data with finite total mass will necessary
go to zero as t goes to infinity since there is no global equilibrium state with finite mass except
from zero. Our first concern is to derive rates of convergence to zero. We shall use the notation
β+ = max(0, β) and the convention 1/0+ = +∞. Let
ζ = ζ(α, β) :=

2 when α ≥ 2 + β
α + β
1 + β when α < 2 + β,
(0.6)
Theorem 2. Let d ≥ 2, β ∈ R, α > max{0, −β}, ζ given by (0.6) and k ∈ [0, ζ). Under Assump-
tion (H), let us consider a solution f to (0.1) with initial condition f in ∈ L1(dx dv)∩L2
(
⌊v⌉kdx dµ
)
.
If α ̸= 2 + β or if α = 2 + β and kβ+ >
d
2 , then
∀ t ≥ 0 , ∥f(t, · , · )∥2L2(dx dµ) ≤
C
(1 + t)τ
∣∣∣∣∣∣f in∣∣∣∣∣∣2
k
with τ = min
{
d
ζ ,
k
β+
}
.
In the critical case γ = 2 + β, and with either k = 0 if β < 0, or k > 0 if β ≥ 0, and under the
additional condition kβ+ ≤
d
2 if d ≥ 3,
∀ t ≥ 2 , ∥f(t, · , · )∥2L2(dx dµ) ≤
C
(t log t)d/2
∣∣∣∣∣∣f in∣∣∣∣∣∣2
k
.
In the above estimates, C > 0 is a constant which does not depend on f in.
This result is in the spirit of theorems proved by the Dolbeault–Mouhot–Schmeiser method [5].
However major difficulties arise compare to the original matter of [5]. Indeed, in the present context,
no microscopic and macroscopic coercivities hold. We shall present some elements of the strategy
of the proof. Thanks to the translational invariance, we may perform a mode-by-mode analysis
taking the Fourier transform in space.
∂tf̂ + i (v · ξ)f̂ = Lf̂ ,
III–2
We define a new mode-by-mode hypocoercivity functional by
Hξ[f ] :=
1
2∥f̂∥
2 + δ Re⟨Aξ f̂ , f̂⟩, Aξ :=
1
⌊v⌉2
Π (− i v · ξ)⌊v⌉
−β
1 + ⌊v⌉2 |1−β| |ξ|2
.
One key step of the proof is to obtain in a general fashion for all models,
Re⟨Aξ f̂ , f̂⟩ ≥ k1(ξ) ∥Πf̂∥2 − k2(ξ) ∥(1 − Π)f̂∥2η .
To come back to the space variable, one may integrate over ξ,
H[f ] =
∫
Rd
Hξ[f ] dξ.
Then, H[f ] is equivalent to ∥ · ∥ when δ is small and Plancherel identities give
d
dtH[f ] = Re⟨f , Lf⟩ + δ
∫
Rd
Re⟨Aξ f̂ , f̂⟩ dξ.
A crucial ingredient then is to relate norms with losses of weights ∥(1 − Π)f∥η to ∥(1 − Π)f∥, for
this we may prove and use conservation of weighted L2 norms,
∀ t ≥ 0 , ∥f(t, · , · )∥L2(⌊v⌉kdx dµ) ≤ Ck ∥f in∥L2(⌊v⌉kdx dµ) .
All in all, yields an ODE of the form:
H ′ ≤ −ϕ(t)Hr,
which can be integrated to give the result of the theorem. The full details of the proofs are in [1].
We shall end this exposition with a comment about the stability of the rates in the diffusive
scaling. In the range of parameters for which we prove decay at rate t− dα , our previous computations
give that the rate is uniform in the rescaling t → tεα and x →
x
ε . When the rate is given by t
− kβ ,
the way that the rate degenerates into t− dα (the rate of the macroscopic limit, see e.g. [3]) is a bit
more intricate, and we leave this issue for future work.
The second result of this discussion is about scaling limits in such a framework. Consider a
solution f in L∞t ([0, +∞); L2x,v(M−1)) to (0.1) with initial data f
(ε)
in . Note that the initial data
f
(ε)
in , before the rescaling, is allowed to depend on ε. Given ε > 0 and θ(ε), we rescale the solution
and define a weighted rescaled spatial density:
fε(t, x, v) :=
1
εd
f
(
t
θ(ε) ,
x
ε
, v
)
∈ L∞t
(
[0, +∞); L2x,v(M−1)
)
rε(t, x) :=
∫
Rd
fε(t, x, v)⌊v⌉−βdv.
The equation satisfied by fε is
θ(ε)∂tfε + εv · ∇xfε = Lfε. (0.7)
The rescaled initial data is then fε(0, x, v) = ε−df (ε)in (ε−1x, v), and in the following theorem we
assume the original initial data f (ε)in to be well-prepared (see (0.9)–(0.10)–(0.11)): this means that
the fluid limit holds at time zero with fε(0, · ) ∼ rε(0, · )M and rε ∼ r(0, · ) which provides the initial
data for the limit equation; this is standard in the literature. We however note that when (0.9)
is satisfied but (0.10)–(0.11) are not imposed at t = 0, the energy estimate and compactness
arguments on rε would imply that (0.10)–(0.11) are satisfied at any later positive time τ > 0
(without information on the rate though), and our method would prove the fluid approximation
for t ≥ τ . This would allow us for instance to choose f (ε)in = rM independent of ε. We however kept
the assumptions (0.10)–(0.11) in order to precisely track the rate of convergence and the initial
data of the limit equation.
It turns out from a deep understanding that the scaling function should be
θ(ε) :=

εζ when α ∈ (0, +∞] \ {2 + β},
ε2|ln ε| when α = 2 + β,
ε
β
1+β
|ln ε| when α = 0.
(0.8)
III–3
Note that the threshold α = 2+β between standard and fractional diffusion corresponds to whether
or not ⌊v⌉−βM has finite variance.
The main result of [3] is as follows.
Theorem 3 (Unified second fluid approximation). Assume structural hypothesis detailed precisely
in [3], have α ≥ 0, and consider fε ∈ L∞t ([0, +∞); L2x,v(M−1)) solving (0.7) in the weak sense
with initially
∥∥∥∥fε(0, · , · )M
∥∥∥∥
0
= o


θ(ε)− 12 , when α > β,
ε−
β
1+β |ln (ε)|−
1
2 when α = β,
ε−
α
1+β when α ∈ (0, β),
|ln(ε)| 32 when α = 0,
 (0.9)
and
∥∥∥∥ fεM (0, · , · ) − rε(0, · )
∥∥∥∥
−β
= o


1, when α > β,
|ln (ε)|−
1
2 when α = β,
ε
β−α
2(1+β) when α ∈ (0, β),
ε
β
2(1+β) |ln(ε)| when α = 0,
 , (0.10)
and (recalling the definition of ζ in (0.6))
rε(0, · )
H−ζ(Rd)−−−−−−→
ε→0
r(0, · ). (0.11)
Then for any T > 0 ∥∥∥∥ fεM − r
∥∥∥∥
L2t ([0,T ];H
−ζ
x L2v(Mβ))
−−−→
ε→0
0 (0.12)
when α > β and ∥∥∥∥∣∣∣∣ln 2|∇x|1 + |∇x|
∣∣∣∣ ( fεM − r
)∥∥∥∥
L2t ([0,T ];H
−ζ
x L2v(Mβ))
−−−→
ε→0
0
when α = β and ∥∥∥∥|∇x| β−|α|2(1+β) ⌊∇x⌉− β−|α|2(1+β) ( fεM − r
)∥∥∥∥
L2t ([0,T ];H
−ζ
x L2v(Mβ))
−−−→
ε→0
0
when α ∈ [0, β), where r = r(t, x) solves
∂tr = κ ∆
ζ
2
x r, t > 0, with initial data r(0, · ) defined in (0.11).
The coefficient κ is explicit and its derivation is explained below. The rates of convergence are
estimated and are explicit in terms of parameters and known initial error terms. Apart from (0.11),
the errors we obtain are polynomial in ε for α ∈ (−β, +∞) \ {0, 2 + β} and logarithmic for α ∈
{0, 2 + β}.
The main ingredient we need to derive this result is a quantitative construction of a branch
of “fluid eigenmode” in the asymptotic of large time and small spatial frequencies, i.e. a unique
eigenvalue branching from zero for L̃∗ + iη⌊v⌉β(v · σ) for small η.
Lemma 4 (Construction of the fluid mode). Given some structural hypotheses on L as detailed
in [3], there are η0 > 0 and r0 ∈ (0, λ), explicit in terms of the constants in these hypotheses, such
that for any η ∈ (0, η0) and any σ ∈ Sd−1, there is a unique solution ϕη = ϕη(v) ∈ L2v(⌊ ·⌉−βM)
and µ(η) ∈ B(0, r0) to
−L∗ϕη − iη(v · σ)ϕη = µ(η)⌊v⌉−βϕη with
∫
Rd
ϕη(v) M(v)⌊v⌉−βdv = 1.
Moreover, the branch (ϕη, µ(η)) connects to (1, 0) as η → 0, with µ(η) > 0 and the asymptotics
∥ϕη − 1∥−β ≲ µ(η)
1
2 and µ(η) ∈ (r0Θ(η), r1Θ(η)) (0.13)
III–4
for some 0 < r0 < r1, where the function Θ is defined by
Θ(η) :=

η2 when α > 2 + β,
η2|ln(η)| when α = 2 + β,
η
α+β
1+β when − β < α < 2 + β.
(0.14)
Note that Θ is well-defined in the case α ∈ (−β, 2 + β) since (1 + β) > (α + β)/2 > 0. In
this lemma and in the rest of the paper the dependency in σ is kept implicit rather than explicit
in order to lighten notation. In fact, ϕη also depends on σ, but µ(η) does not if L is invariant
by rotations in v. To identify the macroscopic limit with quantitative rates and constants, it is
necessary to estimate the leading order of µ(η). This requires estimates on the eigenvector, that
are quantitative and obtained for each typical operator L. For the sake of brevity, we refer directly
to [3, Hypothesis 4] and state the important result directly.
With these four hypotheses we can characterise the precise scaling of the fluid eigenvalue:
Lemma 5 (Rescaled limit of the fluid eigenvalue). The eigenvalue µ(η) constructed in Lemma 4
satisfies (with convergence rate explicit in terms of the constants, error terms and convergence
rates in the hypotheses)
µ(η) ∼η→0 µ0Θ(η), (0.15)
where the constant µ0 ∈ (r0, r1) is positive and determined as follows:
µ0 :=
∫
Rd
(v · σ) F (v)M(v)dv when α > 2 + β,
where F = lim
η→0
Im ϕη
η
is solution to LF = −(v · σ) and
∫
Rd
F (v) Mβ(v)dv = 0,
µ0 :=
c2+β,β
1 + β
∫
Sd−1
(σ · σ′)Ω(σ′)dσ′ when α = 2 + β,
where Ω(u) = lim
λ→0, λ ̸=0
Im Φ (λu)
λ1+β
and Φ = lim
η→0
Φη = lim
η→0
ϕη
(
η−
1
1+β ·
)
,
µ0 := cα,β
∫
Rd
(u · σ) Im Φ(u)|u|−d−αdu when α ∈ (−β, 2 + β).
Note how, when α > 2 + β, the function F used in the previous works on standard diffusive
limit (usually with β = 0) is recovered here as a limit of our fluid mode; this allows our proof to
track the convergence rate.
Lemma 6 (Diffusion coefficient). Assume structural hypotheses detailed in [3] hold true and α ≥ 0.
Then the following limit holds true with convergence rate explicit in terms of the constants, error
terms and convergence rates in the hypotheses: for any ξ ∈ Rd \ {0},
κ := lim
ε→0
(
µ(ε|ξ|)|ξ|−ζ
θ(ε)
〈
1, ϕε|ξ|
〉) = µ0 ×

∥M∥−1
L1(Rd) when α > 0,
1 + β
|Sd−1|
when α = 0.
(0.16)
The diffusion coefficient thus emerges from ratios between (rescaled) integrals as follows:
κ :=

∫
Rd (v · σ) F (v)M(v)dv
∥M∥L1(Rd)
when α > 2 + β
1
1 + β
∫
Sd−1(σ · σ
′)Ω(σ′)dσ′∫
Rd⌊v⌉−d−αdv
when α = 2 + β
∫
Rd(u · σ) Im Φ(u)|u|
−d−αdu∫
Rd⌊v⌉−d−αdv
when α ∈ (0, 2 + β)
1 + β
|Sd−1|
∫
Rd(u · σ) Im Φ(u)|u|
−d−αdu∫
Rd⌊v⌉−d−α−βdv
when α = 0
(0.17)
III–5
where we recall
F = lim
η→0
Im ϕη
η
, Φ = lim
η→0
Φη = lim
η→0
ϕη
(
η−
1
1+β ·
)
, Ω(u) = lim
λ→0, λ ̸=0
Im Φ (λu)
λ1+β
,
and (when α > 2+β) F is also the unique solution to LF = −(v·σ) with
∫
Rd F (v) ⌊v⌉
−d−α−βdv = 0.
The method presented here extends to the fractional diffusive limit the approach pioneered
in [6, 11] of constructing exact dispersion laws in the regime of parabolic time-space scaling and
small eigenvalues; this extension is inspired by the recent one-dimensional result [9] and in particular
we use and generalise the idea of rescaling velocities to obtain a non-trivial dispersion law in [9].
In comparison with [9], the main novelty of the present paper is a quantitative spectral method
for constructing the branch of fluid eigenvalue: in [9] it was done by a one-dimensional argument
connecting two infinite series on R− and R+ (and it was done by fixed points in the simpler case
of classical diffusive limit in the older works [6, 11]). In terms of results, this recovers and unifies
the fractional diffusive limit results in [4, 7, 8, 9, 10]. Novel contributions include formulas for
the diffusion coefficient in dimension higher than 1, and the quantitative argument providing a
convergence rate.
We shall conclude with perspectives and extensions. An interesting work in progress with Mouhot
and Kanzler [2] is the use of the spectral method in a context where the operator L will have more
than one invariant, namely three invariants that are mass, momentum and energy. Several models
from the literature for which not much is known fall down in this context.
References
[1] E. Bouin, J. Dolbeault & L. Lafleche, “Fractional hypocoercivity”, Commun. Math.
Phys. 390 (2022), no. 3, p. 1369-1411.
[2] E. Bouin, L. Kanzler & C. Mouhot, “Quantitative fluid approximation with more invari-
ants”, in progress, 2023.
[3] E. Bouin & C. Mouhot, “Quantitative fluid approximation in transport theory: a unified
approach”, Probability and Mathematical Physics 3 (2022), no. 3, p. 491-542.
[4] P. Cattiaux, E. Nasreddine & M. Puel, “Diffusion limit for kinetic Fokker-Planck equa-
tion with heavy tails equilibria: the critical case”, Kinet. Relat. Models 12 (2019), no. 4,
p. 727-748.
[5] J. Dolbeault, C. Mouhot & C. Schmeiser, “Hypocoercivity for Linear Kinetic Equations
Conserving Mass”, Trans. Am. Math. Soc. 367 (2015), no. 6, p. 3807-3828.
[6] R. S. Ellis & M. A. Pinsky, “The first and second fluid approximations to the linearized
Boltzmann equation”, J. Math. Pures Appl. 54 (1975), p. 125-156.
[7] N. Fournier & C. Tardif, “Anomalous diffusion for multi-dimensional critical kinetic
Fokker–Planck equations”, Ann. Probab. 48 (2020), no. 5, p. 2359-2403.
[8] ——— , “One dimensional critical kinetic Fokker-Planck equations, Bessel and stable pro-
cesses”, Commun. Math. Phys. 381 (2021), no. 1, p. 143-173.
[9] G. Lebeau & M. Puel, “Diffusion approximation for Fokker Planck with heavy tail equilib-
ria: a spectral method in dimension 1”, Commun. Math. Phys. 366 (2019), no. 2, p. 709-735.
[10] E. Nasreddine & M. Puel, “Diffusion limit of Fokker-Planck equation with heavy tail
equilibria”, ESAIM, Math. Model. Numer. Anal. 49 (2015), no. 1, p. 1-17.
[11] B. Nicolaenko, “Dispersion Laws for plane wave propagation”, in The Boltzmann Equation
(A. Grünbaum, ed.), Courant Institute, 1971.
III–6
Emeric Bouin
CEREMADE - Université Paris-Dauphine,
UMR CNRS 7534,
Place du Maréchal de Lattre de Tassigny,
75775 Paris Cedex 16,
France.
bouin@ceremade.dauphine.fr
III–7


Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails

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Asymptotics and limit theorems for horocycle ergodic integrals à la Ratner (with an appendix by Emilio Corso)

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Hydrodynamic limits and hypocoercivity for kinetic equations with heavy tails

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