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Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices

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Saharon Shelah

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Xiangyu Zhou

Comptes Rendus
Mathématique
Wang Xu and Xiangyu Zhou
Optimal L2 Extensions of Openness Type and Related Topics
Volume 361 (2023), p. 679-683
Published online: 2 March 2023
https://doi.org/10.5802/crmath.437
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
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 679-683
https://doi.org/10.5802/crmath.437
Complex analysis and geometry / Analyse et géométrie complexes
Optimal L2 Extensions of Openness Type and
Related Topics
Wang Xu∗, a and Xiangyu Zhoub
a School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China
b Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese
Academy of Sciences, Beijing, 100190, P. R. China
E-mails: xuwang@amss.ac.cn, xyzhou@math.ac.cn
Abstract. We establish several optimal L2 extension theorems of openness type on weakly pseudoconvex
Kähler manifolds. We prove a product property for certain minimal L2 extensions, which generalizes the
product property of Bergman kernels. We describe a different approach to the Suita conjecture and its
generalizations, which is based on a log-concavity for certain minimal L2 integrals.
Résumé. Nous établissons quelques théorèmes d’extension optimaux L2 pour les formes ouvertes sur les
variété Kähler faiblement pseudoconvexes. Nous prouvons les propriétés de produit de certaines extensions
minimales de L2, qui généralisent les propriétés de produit du noyau Bergman. Sur la base de la concavité
logarithmique de certaines intégrales minimales de L2, nous donnons une méthode différente pour la
conjecture de Suita et son extension.
2020 Mathematics Subject Classification. 32D15, 32A36, 32L05, 32W05, 32E10, 14C30, 30C40.
Funding. The second author is partially supported by the NSFC grant (11688101 and 12288201).
Manuscript received 5 February 2022, accepted 16 October 2022.
1. Backgrounds
In this note, we study the following problem: let S be a closed submanifold of a complex manifold
M and E be a Hermitian holomorphic vector bundle on M ; suppose f is an L2 holomorphic
section of E defined in a neighborhood U of S, find a holomorphic section F ∈ Γ(M ,E) so that
F |S = f |S and the L2-norm ∥F∥M is uniformly and optimally controlled by the L2-norm ∥ f ∥U .
This problem is closely related to but different from the usual L2 extension problem (where f
is defined on S and ∥ f ∥U is replaced by ∥ f ∥S ), which is called optimal L2 extension problem of
openness type, in order to distinguish with the usual one.
The existence part of the L2 extension problem of openness type is well studied, e.g. Jen-
nane [15] and Demailly [4, 5] (see also [6, Chapter VIII.7]). This note is devoted to the optimal
part of the problem and some related topics. Details of the proofs will appear in [17].
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
680 Wang Xu and Xiangyu Zhou
2. Optimal L2 Extension Theorems of Openness Type
Recall that, a complex manifold Ω is said to be weakly pseudoconvex if there exists a smooth
psh exhaustion function for Ω; an upper semi-continuous function ϕ : Ω → [−∞,+∞) is said
to be quasi-psh if ϕ is locally the sum of a psh function and a smooth function. Given a quasi-
psh function ϕ on a complex manifold Ω, we denote by I (ϕ) ⊂ OΩ the multiplier ideal sheaf
associated to ϕ, i.e. I (ϕ)x = { fx ∈OΩ,x : | fx |2e−ϕ is integrable in some neighborhood of x}.
Theorem 1. Let (Ω,ω) be a weakly pseudoconvex Kähler manifold and (E ,h) be a Hermitian
holomorphic vector bundle overΩ. Suppose there are quasi-psh functionsψ< 0 andϕ, continuous
real (1,1)-forms γ≥ 0 and ρ onΩ such thatp−1∂∂ψ≥ γ, p−1∂∂ϕ≥ ρ and p−1Θ(E ,h)+ (γ+ρ)⊗ IdE ≥Nak 0.
Let Ωa := {z ∈Ω : ψ(z) < −a}, where a ∈ R+. Then for any holomorphic section f ∈ Γ(Ωa ,KΩ⊗E)
satisfying
∫
Ωa
| f |2
ω,he
−ϕdVω <+∞, there exists F ∈ Γ(Ω,KΩ⊗E) such that
F |Ωa − f ∈ Γ(Ωa ,O (KΩ⊗E)⊗I (ϕ+ψ)) (1)
and ∫
Ω
|F |2ω,he−ϕdVω ≤ ea
∫
Ωa
| f |2ω,he−ϕdVω. (2)
The proof of Theorem 1 uses the L2 techniques developed by Guan–Zhou [13] and Zhou–
Zhu [18], together with a log-concavity for certain minimal L2 integrals (which is essentially due
to Guan [8]). Since Ω is a weakly pseudoconvex Kähler manifold, we approximate the quasi-psh
functions on relatively compact subdomains by the methods of [7] and then solve ∂-equations
with error terms. Using the L2 techniques, we prove the theorem with the uniform constant
replaced by ea +1. Using the log-concavity, we then obtain the uniform constant ea .
Definition 2. Let U ⊂Cn be an open set and f , g ∈O (U ). Given x ∈U and k ∈N, if ∂α f (x) = ∂αg (x)
for all multi-order α ∈ Nn with |α| ≤ k, then we say “f coincides with g up to order k at x”. In
particular, if f coincides with the zero function up to order k at x, then we say “f vanishes up to
order k at x”. Clearly, f vanishes up to order k at x if and only if [ f ]x ∈mk+1x . These concepts can be
easily extended to holomorphic sections of holomorphic vector bundles.
If there exists a closed subset S ⊂ Ω so that I (ϕ +ψ)x ⊂ mk+1x for any x ∈ S, then the
condition (1) implies that F coincides with f up to order k along S. In particular, we have the
following corollary.
Corollary 3. Let Ω be a bounded pseudoconvex domain in Cn , ϕ be a psh function on Ω, and
GΩ( · , w) be the pluricomplex Green function of Ωwith a pole at w ∈Ω. Let U = {GΩ( · , w) <−a} for
some a ∈ R+. For any f ∈ O (U ) satisfying
∫
U | f |2e−ϕdλ<+∞, there exists a holomorphic function
F ∈O (Ω) such that F coincides with f up to order k ∈N at w and∫
Ω
|F |2e−ϕdλ≤ e2(n+k)a
∫
U
| f |2e−ϕdλ. (3)
The case of k = 0 and ϕ≡ 0 has been obtained by Błocki [2], whose proof used a tensor power
trick. It is clear that the uniform estimates of Theorem 1 and Corollary 3 are optimal.
Using similar arguments as Theorem 1, we prove another L2 extension theorem of openness
type, whose prototype comes from [6, Chapter VIII.7]. Demailly’s theorem corresponds to the
case of m = 0 and the right hand side of (4) is replaced by (1+ (p+1)2ε )
∫
U | f |2ω,he−ϕdVω.
Theorem 4. Let (Ω,ω) be a weakly pseudoconvex Kähler manifold of dimension n and (E ,h) be a
Hermitian holomorphic vector bundle over Ω. Suppose there exists a quasi-psh function ϕ and a
continuous real (1,1)-form ρ onΩ such thatp−1∂∂ϕ≥ ρ and p−1Θ(E ,h)+ρ⊗ IdE ≥Nak 0.
Wang Xu and Xiangyu Zhou 681
Suppose w = (w1, . . . , wp ) is a tuple of holomorphic functions onΩ (1 ≤ p ≤ n), let
S = {x ∈Ω : w(x) = 0} and U = {x ∈Ω : |w(x)| < 1}.
Assume that dw1 ∧ ·· · ∧ dwp ̸= 0 generically on S. Given m ∈ N and 0 < ε < m + p, for any
holomorphic section f ∈ Γ(U ,KΩ⊗E) satisfying
∫
U | f |2ω,he−ϕdVω <+∞, there exists F ∈ Γ(Ω,KΩ⊗E)
such that F coincides with f up to order m along S and∫
Ω
|F |2
ω,he
−ϕ
(1+|w |2)m+p+ε dVω ≤
∫ 1
0 τ
m+p−1(1−τ)ε−1dτ∫ 1/2
0 τ
m+p−1(1−τ)ε−1dτ
∫
U
| f |2
ω,he
−ϕ
(1+|w |2)m+p+ε dVω. (4)
Let Ω = Cn and ω be the standard Kähler form. Let w = (z1, . . . , zp ) be the first p coordinates
of Cn , then S ∼= Cn−p is a linear subspace and U ∼= Bp ×Cn−p . Let (E ,h) = (K −1Ω ,detω) and ϕ ≡ 0.
Let u be a homogeneous polynomial in z1, . . . , zp of degree m and v ∈O (Cn−p ) be a holomorphic
function of zp+1, . . . , zn . Let f := uv ∈ O (U ) and let F be the unique holomorphic function on Cn
such that F coincides with f up to order m along S and the left hand side of (4) is minimized. It is
easy to show that F ≡ uv and the inequality (4) is sharp.
3. Sharper Estimates in Optimal L2 Extension Theorems
Let Ω be a weakly pseudoconvex Kähler manifold of dimension n. Let (E ,h) be a Hermitian
holomorphic vector bundle over Ω whose curvature is Nakano semi-positive. Assume that ψ< 0
is a psh function on Ω having a logarithmic pole at x ∈ Ω: if (U , z) is a coordinate chart so that
z(x) = 0, then ψ− log |z| is bounded near x. By the optimal L2 extension theorem ( [13, 18]):
for any ξ ∈ Ex , there exists a holomorphic section F ∈ Γ(Ω,KΩ ⊗E) such that F (x) = dz ⊗ ξ and∫
Ω(
p−1)n2 F ∧h F ≤ (2π)
n
n! e
−2nc |ξ|2h , where dz := dz1 ∧·· ·∧dzn and c := limz→0(ψ(z)− log |z|).
The uniform estimate in the above statement is optimal: there are admissible situations where
the constant (2π)
n
n! e
−2nc cannot be replaced by a smaller one. Under some additional conditions,
we may obtain a sharper estimate. For example, if the curvature of (E ,h) is Nakano positive
somewhere, then there exists an extension F of dz ⊗ξ so that ∫Ω(p−1)n2 F ∧h F < (2π)nn! e−2nc |ξ|2h .
Hosono’s [14] result is a special case. Using Theorem 1, we have one more example.
Theorem 5. LetΩ ∋ x, (E ,h) and ψ be the same as above. Moreover, we assume that:
(i) the curvature of (E ,h) is Griffiths positive at x;
(ii) (U , z) can be chosen so that z(x) = 0 and ψ(z) = c + log |z|+o(|z|2) as z → 0.
There exists a constant τ ∈ (0,1) depends on h and ψ, for any ξ ∈ Ex , we can find a holomorphic
section F ∈ Γ(Ω,KΩ⊗E) such that F (x) = dz ⊗ξ and∫
Ω
(
p−1)n2 F ∧h F ≤ (1−τ)
(2π)n
n!
e−2nc |ξ|2h . (5)
Since (E ,h) is Griffiths positive at x, we can construct a local L2 holomorphic extension with
sharper estimate, then we apply Theorem 1 to obtain a global extension with sharper estimate.
The technical condition (ii) guarantees that the sublevel set {ψ < −a} (with a ≫ 1) is almost an
Euclidean ball in U . Moreover, ifΩ is a hyperbolic Riemann surface andψ=GΩ( · , x) is the Green
function, then the condition (ii) is automatically satisfied.
4. A Product Property for Minimal L2 Extensions
The tensor power trick of [2] relies on the product property of Bergman kernels. Notice that,
Bergman kernel records the norm of certain minimal L2 extension: letΩ⊂Cn be an open set, then
BΩ(w)−1 = inf{
∫
Ω | f |2dλ : f ∈ O (Ω), f (w) = 1}. As a generalization, we prove a product property
for general minimal L2 extensions.
682 Wang Xu and Xiangyu Zhou
For i = 1 and 2, let (Ωi ,dVi ) be a complex manifold with continuous volume form, (Ei ,hi ) →Ωi
be a holomorphic vector bundle with continuous Hermitian metric, Si ⊂Ωi be an arbitrary closed
subset, and ψi be a measurable function onΩi which is locally bounded above. We define
A2(Ωi ,Ei ) :=
{
fi ∈ Γ(Ωi ,Ei ) : ∥ fi∥2 =
∫
Ωi
| fi |2hi e
−ψi dVi <+∞
}
, i = 1 and 2. (6)
Let Ω :=Ω1 ×Ω2 be the product manifold and let pi :Ω→Ωi be the natural projections. Let
dV := p∗1 dV1×p∗2 dV2, E := p∗1 E1⊗p∗2 E2, h := p∗1 h1⊗p∗2 h2 andψ := p∗1ψ1+p∗2ψ2. Then we define
A2(Ω,E) in a similar way as (6). We can show that A2(Ω,E) = A2(Ω1,E1)⊗̂A2(Ω2,E2).
By constructing a special orthogonal decomposition which is compatible with the Hilbert
tensor product ⊗̂, we can prove a product property for minimal L2 holomorphic extensions. Since
we need to assume the existence of global L2 extensions, we simply assume that fi ∈ A2(Ωi ,Ei ),
and we are interested in high-order minimal L2 extensions of fi |Si .
Theorem 6. Let f1 ∈ A2(Ω1,E1) and f2 ∈ A2(Ω2,E2) be given. For i = 1 and 2, let Fi be the unique
element with minimal norm in A2(Ωi ,Ei ) that coincides with fi up to order mi ∈N along Si . Let F
be the unique element with minimal norm in A2(Ω,E) that coincides with f := f1 ⊗ f1 up to order
m1 +m2 along S := S1 ×S2. Then
∥F∥2A2(Ω,E) ≥ ∥F1∥2A2(Ω1,E1)∥F2∥
2
A2(Ω2,E2)
. (7)
Moreover, if mi = 0 or fi vanishes up to order mi −1 on Si for i = 1 and 2, then
F = F1 ⊗F2 and ∥F∥2A2(Ω,E) = ∥F1∥2A2(Ω1,E1)∥F2∥
2
A2(Ω2,E2)
. (8)
If Si are singleton sets and mi = 0, then (8) corresponds to the product property of Bergman
kernels. Using Theorem 6 and imitating the tensor power trick of [2], one can prove a weak version
of Theorem 1.
5. Generalized Suita Conjecture with Jets and Weights
A motivation for optimal L2 extension theorems is the long-standing Suita conjecture. LetΩ be a
potential-theoretically hyperbolic Riemann surface. Let (U , z) be a coordinate chart ofΩ. Locally,
we denote by κΩ = BΩ|dz|2 the Bergman kernel of Ω and cβ|dz| the logarithmic capacity of Ω.
Suita [16] conjectured that πBΩ ≥ c2β, and πBΩ(w) = cβ(w)2 for some w ∈ Ω if and only if Ω is
conformally equivalent to the unit disc less a possible closed polar set. The inequality part of
Suita’s conjecture was solved by Błocki [1] and Guan–Zhou [11], and the equality part was solved
by Guan–Zhou [13]. Moreover, Guan–Zhou [12, 13] proved the extended Suita conjecture and
Błocki-Zwonek [3] obtained a high-order generalization to the inequality part.
In proving Theorem 1, we use a log-concavity for minimal L2 integrals. Tracing the proof, we
obtain a necessary condition for the concavity degenerating to linearity (see also [9]). Using the
concavity and the necessary condition, we have a different approach to Suita’s conjecture and its
generalizations.
Proposition 7. Let ϕ be a harmonic function onΩ and ψ= 2(m +1)GΩ( · , w). For each t ≥ 0, let
Ωt := {z ∈Ω :ψ(z) <−t } and At :=
{
F ∈ Γ(Ωt ,KΩ) : ∥F∥2 =
∫
Ωt
p−1
2 F ∧F e−ϕ <+∞
}
.
Let f be a holomorphic 1-form defined in a neighborhood of w. For each t ≥ 0, let Ft ∈ At be the
unique element with minimal norm that coincides with f up to order m at w. Let I (t ) := ∥Ft∥2At
and we assume that I (t ) > 0. Then r 7→ I (− logr ) is a concave non-decreasing function on (0,1] and
I (0) ≤ I (t )e t for all t > 0. Moreover, if r 7→ I (− logr ) is linear, then Ft ≡ F0|Ωt for any t > 0.
Wang Xu and Xiangyu Zhou 683
Let D= {z ∈C : |z| < 1} and S= {z ∈C : |z| = 1}. Let p :D→Ω be a universal covering. Then the
fundamental group π1(Ω) is identified with a subgroup of Aut(D), and p ◦σ= p for any σ ∈π1(Ω).
For any w ∈Ω, there exists an fw ∈O (D) so that log | fw | = p∗(GΩ( · , w)). Then there is a multiplier
χw ∈ Hom(π1(Ω),S) so that p∗ fw = χw (σ) fw for any σ ∈ π1(Ω). For any harmonic function η on
Ω, there exists an fη ∈ O (D) so that log | fη| = p∗η. Then there is a multiplier χη ∈ Hom(π1(Ω),S)
so that p∗ fη =χη(σ) fη for any σ ∈π1(Ω).
Theorem 8. Given an integer m ∈N and a harmonic function η onΩ, we define
B (m)
Ω,η (w) := sup
{∣∣∣∣∂m f∂zm (w)
∣∣∣∣2 : F ∈ Γ(Ω,KΩ) with F |U = f dz,∫
Ω
p−1
2 F ∧F e−2η ≤ 1,[ f ]w ∈mmw
}
, w ∈Ω. (9)
Then
πe−2η(w)B (m)
Ω,η (w) ≥ m!(m +1)!cβ(w)2m+2. (10)
Moreover, the equality holds if and only if χ−1η = χm+1w , if and only if there exists a holomorphic
function g ∈O (Ω) such that log |g | = (m +1)GΩ( · , w)+η.
The extended Suita conjecture is the case of m = 0, and Błocki-Zwonek’s generalization is the
inequality (10) with η ≡ 0. Recently, Guan–Mi–Yuan [10] obtain a result generalizing Theorem 8,
but our proof is different.
References
[1] Z. Błocki, “Suita conjecture and the Ohsawa-Takegoshi extension theorem”, Invent. Math. 193 (2013), no. 1, p. 149-
158.
[2] ——— , “A lower bound for the Bergman kernel and the Bourgain-Milman inequality”, in Geometric aspects of
functional analysis, Springer, 2014, p. 53-63.
[3] Z. Błocki, W. Zwonek, “One dimensional estimates for the Bergman kernel and logarithmic capacity”, Proc. Am. Math.
Soc. 146 (2018), no. 6, p. 2489-2495.
[4] J.-P. Demailly, “Estimations L2 pour l’opérateur ∂ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une
variété kählérienne complète”, Ann. Sci. Éc. Norm. Supér. 15 (1982), no. 3, p. 457-511.
[5] ——— , “Scindage holomorphe d’un morphisme de fibrés vectoriels semi-positifs avec estimations L2”, in Seminar
Pierre Lelong-Henri Skoda (Analysis), 1980/1981, Springer, 1982, p. 77-107.
[6] ——— , “Complex analytic and differential geometry”, e-book, 2012.
[7] J.-P. Demailly, T. Peternell, M. Schneider, “Pseudo-effective line bundles on compact Kähler manifolds”, Int. J. Math.
12 (2001), no. 6, p. 689-741.
[8] Q. Guan, “A sharp effectiveness result of Demailly’s strong openness conjecture”, Adv. Math. 348 (2019), p. 51-80.
[9] Q. Guan, Z. Mi, “Concavity of minimal L2 integrals releted to multiplier ideal sheaves”, Peking Math. J. (2022),
https://doi.org/10.1007/s42543-021-00047-5.
[10] Q. Guan, Z. Mi, Z. Yuan, “Concavity property of minimal L2 integrals with Lebesgue measurable gain II”, 2022,
https://arxiv.org/abs/2211.00473v1.
[11] Q. Guan, X. Zhou, “Optimal constant problem in the L2 extension theorem”, C. R. Math. Acad. Sci. Paris 350 (2012),
no. 15-16, p. 753-756.
[12] ——— , “Optimal constant in an L2 extension problem and a proof of a conjecture of Ohsawa”, Sci. China Math. 58
(2015), no. 1, p. 35-59.
[13] ——— , “A solution of an L2 extension problem with an optimal estimate and applications”, Ann. Math. 181 (2015),
no. 3, p. 1139-1208.
[14] G. Hosono, “On sharper estimates of Ohsawa-Takegoshi L2-extension theorem”, J. Math. Soc. Japan 71 (2019), no. 3,
p. 909-914.
[15] B. Jennane, “Extension d’une fonction définie sur une sous-variété avec contrôle de la croissance”, in Séminaire
Pierre Lelong-Henri Skoda (Analyse), Année 1976/77, Springer, 1978, p. 126-133.
[16] N. Suita, “Capacities and kernels on Riemann surfaces”, Arch. Ration. Mech. Anal. 46 (1972), p. 212-217.
[17] W. Xu, X. Zhou, “Optimal L2 extensions of openness type”, 2022, https://arxiv.org/abs/2202.04791v2.
[18] X. Zhou, L. Zhu, “An optimal L2 extension theorem on weakly pseudoconvex Kähler manifolds”, J. Differ. Geom. 110
(2018), no. 1, p. 135-186.


Optimal $L^2$ Extensions of Openness Type and Related Topics

Roland Bacher

Euclid meets Popeye: The Euclidean Algorithm for $2\times 2$ Matrices

Anne Boutet de Monvel

Lech Zielinski

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
www.centre-mersenne.org
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-
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[15] Q. Xie, H. Zhong, M. T. Batchelor, C. Lee, “The quantum Rabi model: solution and dynamics”, J. Phys. A. Math. Theor.
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Asymptotic formula for large eigenvalues of the two-photon quantum Rabi model

Nathan Shourick

Ibrahim Cheddadi

Pierre Saramito

A new framework for shallow approximations of incompressible flows

Marouane Il Idrissi

Nicolas Bousquet

Fabrice Gamboa

Bertrand Iooss

Jean-Michel Loubes

On the coalitional decomposition of parameters of interest

khadijeh Baghaei

Blow-up of nonradial solutions to the hyperbolic-elliptic chemotaxis system with logistic source

Volker Kempf

Comptes Rendus
Mathématique
Volker Kempf
Brezzi–Douglas–Marini interpolation on anisotropic simplices and
prisms
Volume 361 (2023), p. 437-443
https://doi.org/10.5802/crmath.424
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
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 437-443
https://doi.org/10.5802/crmath.424
Numerical analysis / Analyse numérique
Brezzi–Douglas–Marini interpolation on
anisotropic simplices and prisms
Interpolation Brezzi–Douglas–Marini sur les simplexes
et prismes anisotropes
Volker Kempf ∗, a
a Universität der Bundeswehr München, Germany
E-mail: volker.kempf@unibw.de (V. Kempf)
Abstract. The Brezzi–Douglas–Marini interpolation error on anisotropic elements has been analyzed in two
recent publications, the first focusing on simplices with estimates in L2, the other considering parallelotopes
with estimates in terms of Lp -norms. This contribution provides generalized estimates for anisotropic
simplices for the Lp case, 1 ≤ p ≤∞, and shows new estimates for anisotropic prisms with triangular base.
Résumé. L’erreur d’interpolation de Brezzi–Douglas–Marini sur les éléments anisotropes a été analysée dans
deux publications récentes, la première se concentrant sur les simplices avec des estimations dans L2, l’autre
considérant les parallelotopes avec des estimations en termes de normes Lp . Notre contribution fournit des
estimations généralisées pour les simplexes anisotropes pour le cas Lp , 1 ≤ p ≤ ∞, et montre de nouvelles
estimations pour les prismes anisotropes à base triangulaire.
2020 Mathematics Subject Classification. 65D05, 65N30.
Manuscript received 12 May 2022, revised 9 September 2022, accepted 15 September 2022.
1. Introduction
The Brezzi–Douglas–Marini (BDM) finite element [8] was introduced to approximate H div by
polynomials. This proves useful for problems of incompressible fluid flow, where recent ap-
proaches employ H div-conforming discretizations to approximate the velocity solution. The cor-
responding interpolation operator can also be used as reconstruction operator to gain pressure-
robust methods in the spirit of [7]. Boundary layers or edge singularities in these problems re-
quire the use of anisotropic, i.e., highly stretched elements, so interpolation error estimates for
such settings are required, see [3]. In [2], these estimates were shown for simplicial elements in
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
438 Volker Kempf
terms of the L2-norm, and [5] contains estimates for anisotropic parallelotopes in terms of Lp -
norms.
The main focus in [5] is on estimates in the H div-norm and while the technique for these proofs
can also be used for simplices, it is not applicable for the case of prisms as defined in [8], as here
the commuting diagram property [5, (3)] is not satisfied. Interestingly, it is neither satisfied on
cubes as defined in [8]; here [5] uses the original definition of the BDM elements, see, e.g., [4].
In the following sections we provide a generalization of the results from [2] to Lp spaces,
1 ≤ p ≤ ∞, and show new interpolation error estimates for anisotropic triangular prisms. The
results with detailed proofs are contained in the author’s PhD thesis [6] that is to appear.
Notation
Vectors, vector valued functions and the spaces of such functions are set in bold and we use
In = {1, . . . , n} for index sets. The spatial dimension is denoted by d and the expression a . b
means there is a positive constant C so that a ≤ C b. The norms of the Sobolev spaces W m,p (G)
are denoted by ‖·‖m,p,G , multi-indices by α. The directional derivative in direction l is denoted
by ∂∂l .
2. Error estimates on anisotropic simplices
We state the anisotropic interpolation error estimates on simplices in terms of Lp -norms, which
is a generalization of the results from [2], where only p = 2 was considered. As the proofs are
largely analogous to those in [2], we omit them here. The principal idea of the proof is to first
show stability estimates on the reference elements and then transfer them in two steps first to an
element of a reference family using an affine transformation with a diagonal matrix and then to
the general element.
Recall that a simplex is said to satisfy the maximum angle condition if all angles within and
between facets are bounded by a constant φ̄<π. In addition, it satisfies the regular vertex property
if for one vertex there is a constant c̄ > 0 so that |det N | ≥ c̄ holds for the matrix N whose columns
consist of the unit vectors l i , i ∈ Id , along the outgoing edges from this vertex. The lengths of
the edges corresponding to the vectors l i are the element size parameters and are denoted by hi .
The degrees of freedom for the BDM interpolation operator I BDMk of order k on simplices can be
found in [8, p. 59].
The error estimates for the BDM interpolation depending on the geometric regularity are given
by the two following theorems, cf. [2, Theorems 4.3, 4.4] and [1, Theorems 6.2, 6.3].
Theorem 1. Let a simplicial element T satisfy the regular vertex property with constant c̄. Then
for k ≥ 1, 0 ≤ m ≤ k and v ∈W m+1,p (T ), 1 ≤ p ≤∞, the estimate∥∥v − I BDMk v
∥∥
0,p,T .
∑
|α|=m+1
hα
∥∥Dαl v
∥∥
0,p,T +hT
∑
|α|=m
hα
∥∥Dαl div v
∥∥
0,p,T
holds, the constant only depends on c̄ and k. Here
hT = diamT,hα =
∏
j ∈ Id
h
α j
j and D
α
l =
∂|α|
∂
α1
l 1
· · ·∂αdl d
.
Theorem 2. Let a simplicial element T satisfy the maximum angle condition with constant φ̄.
Then for k ≥ 1, 0 ≤ m ≤ k and v ∈W m+1,p (T ), 1 ≤ p ≤∞, the estimate∥∥v − I BDMk v
∥∥
0,p,T . h
m+1
T
∥∥Dm+1v
∥∥
0,p,T
holds, the constant only depends on φ̄ and k. The notation Dn means the sum of the absolute values
of all derivatives of order n.
Volker Kempf 439
1
1
1
p1
p2
p3
p1 p
2
p3
P̂
x̂1
x̂2
x̂3
JP̃
h1
h2
h3
P̃
x̃1
x̃2
x̃3
Figure 1. Reference prism P̂ with vertex numbering, and a transformed prism of the
reference family RP .
3. Error estimates on anisotropic prisms
Similar results can be achieved for triangular prisms with some modifications. The prismatic
reference element P̂ with the notation for the vertices is given on the left hand side of Figure 1.
The element P̂ is transformed to an element P̃ of the reference family RP by the transforma-
tion x̃ = JP̃ x̂ , where
JP̃ =


h1 0 0
0 h2 0
0 0 h3

 , (1)
see Figure 1. The vertical facet of P̂ opposite of the vertices p i and p
i is denoted by ei , the
horizontal facet at x̂3 = 0 by eb and the one at x̂3 = 1 by et . The facet normals of P̂ are thus
given by
nP̂ ,e1 =


−1
0
0

 , nP̂ ,e2 =


0
−1
0

 , nP̂ ,e3 =
1p
2


1
1
0

 , nP̂ ,eb =


0
0
−1

 , nP̂ ,et =


0
0
1

 .
The local BDM interpolation operator I BDMk of order k on a prism P maps into the space
P k,k (P ) of vector valued polynomials of total degree k in x1 and x2 and degree k in x3. It is defined
by the functionals, see [8, p. 64],∫
ei
I BDMk v ·nP,ei z ds =
∫
ei
v ·nP,ei z ds ∀ z ∈ Pk (ei ), i ∈ {b, t }, (2a)
∫
ei
I BDMk v ·nP,ei z ds =
∫
ei
v ·nP,ei z ds ∀ z ∈Qk (ei ), i ∈ I3, (2b)
∫
P
(
I BDMk v
)
3 z3 dx =
∫
P
v3z3 dx ∀ z3 ∈ Pk,k−2(P ), (2c)
∫
P
(
I BDMk v
)
1 z1 +
(
I BDMk v
)
2 z2 dx =
∫
P
v1z1 + v2z2 dx ∀ (z1, z2) ∈Pk−1,k (P ), (2d)
where Pk (e) is the space of polynomials with maximal degree k, Qk (e) is the space of polynomials
with degree k in each of the dimensions of the facet e, Pm,n(P ) is the space of polynomials
with total degree m in x1 and x2, and degree n in x3. The space Pm,n(P ) consists of pairs of
polynomials with degree n in x3, which are for fixed x3, i.e., on the triangle Tx3 , in the space
N m(Tx3 ) = P m−1(Tx3 )⊕Sm(Tx3 ), where Sm(Tx3 ) = {p ∈ P m(Tx3 ) : p(x) · x = 0 ∀ x ∈ Tx3 }.
We first require the analog of [2, Lemma 3.1] for the prism reference element. The proof is
omitted here for brevity since it follows along the same steps as the proof for the analogous lemma
for simplices.
440 Volker Kempf
Lemma 3. Let P̂ be the reference element from Figure 1, f̂ j ∈ Lp (e j ), j ∈ I2, f̂ 3 ∈ Lp (eb), 1 ≤ p ≤∞,
and
û(x̂) = ( f̂ 1(x̂2, x̂3),0,0
)T
, v̂ (x̂) = (0, f̂ 2(x̂1, x̂3),0
)T
, ŵ (x̂) = (0,0, f̂ 3(x̂1, x̂2)
)T
.
Then there are functions q̂ j ∈ Pk (e j ), j ∈ I2, q̂3 ∈ Pk (eb), so that
Î BDMk û(x̂) =
(
q̂1(x̂2, x̂3),0,0
)T
, Î BDMk v̂ (x̂) =
(
0, q̂2(x̂1, x̂3),0
)T
, Î BDMk ŵ (x̂) =
(
0,0, q̂3(x̂1, x̂2)
)T
.
With this lemma, we get the stability estimate on the reference element.
Lemma 4. Let û ∈W 1,p (P̂ ), 1 ≤ p ≤∞. Then we have the estimates
∥∥(Î BDMk û
)
i
∥∥
0,p,P̂
. ‖ûi‖1,p,P̂ +
∥∥∥d̂iv û
∥∥∥
0,p,P̂
+
∥∥∥∥
∂û3
∂x̂3
∥∥∥∥
0,p,P̂
, i ∈ I2,
∥∥(Î BDMk û
)
3
∥∥
0,p,P̂
. ‖û3‖1,p,P̂ +
∥∥∥d̂iv û
∥∥∥
0,p,P̂
.
Proof. The proofs for i ∈ I2 are analogous so we only show the details for the first and third
components, starting with i = 1. Let
û∗ =


0
û2(x̂1,0, x̂3)
û3(x̂1, x̂2,0)

 , v̂ = û − û∗ =


û1
û2 − û2(x̂1,0, x̂3)
û3 − û3(x̂1, x̂2,0)

 .
The previous Lemma 3 thus yields (Î BDMk v̂ )1 = (Î BDMk û)1, and it holds d̂iv v̂ = d̂iv û−d̂iv û∗ = d̂iv û.
We define the two functions
v̂∗ =


0
x̂2q̂2
x̂3q̂3

 , ŵ = v̂ − v̂∗ =


v̂1
v̂2 − x̂2q̂2
v̂3 − x̂3q̂3

 ,
where q̂2 ∈ Pk−1,k (P̂ ), q̂3 ∈ Pk,k−1(P̂ ), so that∫
P̂
ŵ2z dx =
∫
P̂
(v̂2 − x̂2q̂2)z dx = 0 ∀ z ∈ Pk−1,k (P̂ ),
∫
P̂
ŵ3z dx =
∫
P̂
(v̂3 − x̂3q̂3)z dx = 0 ∀ z ∈ Pk,k−1(P̂ ).
This means that the functions q̂2 and q̂3 are well defined. Since v̂∗ ∈ P k,k (P̂ ) it follows that
Î BDMk v̂∗ = v̂∗ and thus
(
Î BDMk ŵ
)
1 =
(
Î BDMk v̂
)
1 =
(
Î BDMk û
)
1 .
The interpolated function Î BDMk ŵ = t̂ = (t̂ 1, t̂ 2, t̂ 3)T is then defined by the relations, see (2),∫
eb
t̂ 3z ds =
∫
eb
ŵ3z ds = 0 ∀ z ∈ Pk (eb),
∫
et
t̂ 3z ds =
∫
et
ŵ3z ds =
∫
et
ŵ3z ds −
∫
eb
ŵ3z ds =
∫
P̂
∂ŵ3
∂x̂3
z dx ∀ z ∈ Pk (et ),
∫
e1
t̂ 1z ds =
∫
e1
ŵ1z ds ∀ z ∈Qk (e1),
∫
e2
t̂ 2z ds =
∫
e2
ŵ2z ds = 0 ∀ z ∈Qk (e2),
∫
e3
(t̂ 1 + t̂ 2)z ds =
∫
e3
(ŵ1 + ŵ2)z ds ∀ z ∈Qk (e3),
∫
P̂
t̂ 3z3 dx =
∫
P̂
ŵ3z3 dx =
∫
P̂
(v̂3 − x̂3q̂3)z3 dx = 0 ∀ z3 ∈ Pk,k−2(P̂ ),
∫
P̂
t̂ 1z1 + t̂ 2z2 dx =
∫
P̂
ŵ1z1 + ŵ2z2 dx =
∫
P̂
ŵ1z1 dx ∀ (z1, z2) ∈Pk−1,k (P̂ ),
Volker Kempf 441
where the definitions of q̂2, q̂3, and that ŵ2|e2 ≡ 0, ŵ3|eb ≡ 0 were used. Additional computations
for the relation on e3 yield
1p
2
∫
e3
(t̂ 1 + t̂ 2)z ds = 1p
2
∫
e3
(ŵ1 + ŵ2)z ds =
∫
e3
ŵ ·ne3 ds
=
∫
P̂
(
∂ŵ1
∂x̂1
+ ∂ŵ2
∂x̂2
)
z dx +
∫
P̂
ŵ1
∂z
∂x̂1
dx −
∫
∂P̂\e3
ŵ ·n∂P̂ z ds
=
∫
P̂
(
d̂iv ŵ −2∂ŵ3
∂x̂3
)
dx +
∫
P̂
ŵ1
∂z
∂x̂1
dx −
∫
e1
ŵ1z ds .
Thus, the terms∫
e1
ŵ1z ds =
∫
e1
û1z ds ,
∫
P̂
ŵ1z dx =
∫
P̂
û1z dx ,
∫
P̂
∂ŵ3
∂x̂3
z dx =
∫
P̂
∂û3 − x̂3q̂3
∂x̂3
z dx ,
∫
P̂
(d̂iv ŵ )z dx =
∫
P̂
d̂iv(û − v̂∗)z dx
define the interpolant. We get the desired estimate
∥∥(Î BDMk û
)
1
∥∥
0,p,P̂
. ‖û1‖1,p,P̂ +
∥∥∥d̂iv û
∥∥∥
0,p,P̂
+
∥∥∥∥
∂û3
∂x̂3
∥∥∥∥
0,p,P̂
using a trace theorem and steps similar to those in the proof of [1, Lemma 3.3] to estimate the
terms involving x̂3q̂3 and d̂iv v̂∗. For the third component, with the definitions
v̂ = û − û∗ =


û1 − û1(0, x̂2, x̂3)
û2 − û2(x̂1,0, x̂3)
û3

 , v̂∗ =


x̂1q̂1
x̂2q̂2
0

 , ŵ = v̂ − v̂∗ =


v̂1 − x̂1q̂1
v̂2 − x̂2q̂2
v̂3

 ,
where the functions q̂ j ∈ Pk−1,k (P̂ ), j ∈ I2, are now defined by∫
P̂
ŵ j z dx =
∫
P̂
(
v̂ j − x̂ j q̂ j
)
z dx = 0 ∀ z ∈ Pk−1,k (P̂ ),
the relevant terms of the interpolation relations are∫
eb
ŵ3z ds =
∫
eb
û3z ds ,
∫
et
ŵ3z ds =
∫
et
û3z ds ,
∫
P̂
ŵ3z3 dx =
∫
P̂
û3z3 dx ,
∫
P̂
d̂iv ŵ z dx =
∫
P̂
d̂iv(û − v̂∗)z dx .
From here the same steps as for the first component yield the desired estimate. 
Using the transformation (1) we bring the stability estimate to an element of the reference
family.
Lemma 5. Let P̃ = JP̃ P̂+x0, x0 ∈R3, and ṽ ∈W 1,p (P̃ ), 1 ≤ p ≤∞. Then on the prism P̃ the estimate
∥∥Ĩ BDMk ṽ
∥∥
0,p,P̃ .
∑
|α|≤1
hα
∥∥Dαṽ
∥∥
0,p,P̃ +hP̃
∥∥∥d̃iv ṽ
∥∥∥
0,p,P̃
+ (h1 +h2)
∥∥∥∥
∂ṽ3
∂x̃3
∥∥∥∥
0,p,P̃
holds, where hP̃ = max{h1,h2,h3}.
Proof. Using Lemma 4 and the relations
‖w̃‖0,p,P̃ =
(∫
P̃
∑
i ∈ Id
|w̃ i |p dx
)1/p
≤ (det JP̃
)1/p ∑
i ∈ Id
i h
−1
(∫
P̂
|ŵ i |p dx
)1/p
= (det JP̃
)1/p ∑
i ∈ Id
i h
−1‖ŵ i‖0,p,P̂ ,
(det JP̃ )
1/p‖v̂ i‖1,p,P̂ = i h
∑
|α|≤1
hα
∥∥Dαṽ i
∥∥
0,p,P̃ ,
442 Volker Kempf
where i h =∏ j ∈ I3\{i } h j , we compute∥∥Ĩ BDMk ṽ
∥∥
0,p,P̃ ≤
(
det JP̃
)1/p ∑
i ∈ I3
i h
−1∥∥(Î BDMk v̂
)
i
∥∥
0,p,P̂
.
(
det JP̃
)1/p
[ ∑
i ∈ I3
i h
−1
(
‖v̂ i‖1,p,P̂ +
∥∥∥d̂iv v̂
∥∥∥
0,p,P̂
)
+ h1 +h2
h1h2h3
∥∥∥∥
∂v̂3
∂x̂3
∥∥∥∥
0,p,P̂
]
= ∑
i ∈ I3
( ∑
|α|≤1
hα
∥∥Dαṽ i
∥∥
0,p,P̃ +hi
∥∥∥d̃iv ṽ
∥∥∥
0,p,P̃
)
+ (h1 +h2)
∥∥∥∥
∂ṽ3
∂x̃3
∥∥∥∥
0,p,P̃
.
∑
|α|≤1
hα
∥∥Dαṽ
∥∥
0,p,P̃ +hP̃
∥∥∥d̃iv ṽ
∥∥∥
0,p,P̃
+ (h1 +h2)
∥∥∥∥
∂ṽ3
∂x̃3
∥∥∥∥
0,p,P̃
. 
The estimate from the previous lemma can be brought to the general prism P where the
transformation is assumed to be reasonable in a certain sense. The stability estimate on P can
be used to get the interpolation error estimate by a Bramble–Hilbert type argument. The proofs
for the stability and interpolation error estimates on the element P follow the same steps as their
analogs on simplices, see [2, Theorems 3.5, 4.3] and cf. [1, Theorems 3.1, 6.2], which is why they
are omitted for brevity.
Theorem 6. Let P be a prism element that emerges by the affine transformation x = JP x̃ , with
‖JP‖∞,
∥∥J−1P
∥∥
∞ ≤C , of the element P̃ ∈RP . Then for v ∈W 1,p (P ), 1 ≤ p ≤∞, the estimate
∥∥I BDMk v
∥∥
0,p,P . ‖v‖0,p,P +
∑
j ∈ I3
h j
∥∥∥∥
∂v
∂l j
∥∥∥∥
0,p,P
+hP‖div v‖0,p,P + (h1 +h2)
∥∥∥∥
∂v3
∂l 3
∥∥∥∥
0,p,P
is satisfied. The vectors l j are the outgoing unit vectors along the edges adjacent to the transformed
vertex p i .
Theorem 7. Let a prism P satisfy the same condition as in Theorem 6. Then for k ≥ 1, 0 ≤ m ≤ k
and v ∈W m+1,p (P ), 1 ≤ p ≤∞, the estimate
∥∥v − I BDMk v
∥∥
0,p,P .
∑
|α|=m+1
hα
∥∥Dαl v
∥∥
0,p,P +hP
∑
|α|=m
hα
∥∥Dαl div v
∥∥
0,p,P
+ (h1 +h2)
∑
|α|=m
hα
∥∥∥∥∥
∂m+1v3
∂lα11 ∂l
α2
2 ∂l
α3+1
3
∥∥∥∥∥
0,p,P
holds and the constant only depends on k.
Remark 8. A limitation of our results is that the transformation from the general element to
the reference element is affine, which excludes settings where, e.g., the rectangular faces are not
flat or the triangular faces are not parallel in the general element. Such settings require further
investigation.
References
[1] G. Acosta, T. Apel, R. G. Durán, A. L. Lombardi, “Error estimates for Raviart–Thomas interpolation of any order on
anisotropic tetrahedra”, Math. Comput. 80 (2011), no. 273, p. 141-163.
[2] T. Apel, V. Kempf, “Brezzi–Douglas–Marini interpolation of any order on anisotropic triangles and tetrahedra”, SIAM
J. Numer. Anal. 58 (2020), no. 3, p. 1696-1718.
[3] ——— , “Pressure-robust error estimate of optimal order for the Stokes equations: domains with re-entrant edges and
anisotropic mesh grading”, Calcolo 58 (2021), no. 2, article no. 15.
[4] D. Boffi, F. Brezzi, M. Fortin, Mixed Finite Element Methods and Applications, Springer Series in Computational
Mathematics, vol. 44, Springer, 2013.
[5] S. Franz, “Anisotropic Hdi v -norm error estimates for rectangular Hdi v -elements”, Appl. Math. Lett. 121 (2021), article
no. 107453.
Volker Kempf 443
[6] V. Kempf, “Pressure-robust discretizations for incompressible flow problems on anisotropic meshes”, Doctoral thesis,
Universität der Bundeswehr München, Deutschland, 2022.
[7] A. Linke, “On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new
variational crime”, Comput. Methods Appl. Mech. Eng. 268 (2014), p. 782-800.
[8] J.-C. Nédélec, “A new family of mixed finite elements in R3”, Numer. Math. 50 (1986), no. 1, p. 57-81.


Brezzi–Douglas–Marini interpolation on anisotropic simplices and prisms

Mamta Balodi

Abhishek Banerjee

Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology

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