Comptes Rendus Mathématique

Mohammad Reza Rismanchian

Group extensions and marginal series of pair of groups

B. Tomas Johansson

On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

Peter Lewintan

Patrizio Neff

Comptes Rendus
Mathématique
Peter Lewintan and Patrizio Neff
Lp -versions of generalized Korn inequalities for incompatible tensor
fields in arbitrary dimensions with p-integrable exterior derivative
Volume 359, issue 6 (2021), p. 749-755
<https://doi.org/10.5802/crmath.216>
© Académie des sciences, Paris and the authors, 2021.
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This article is licensed under the
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Comptes Rendus
Mathématique
2021, 359, n 6, p. 749-755
https://doi.org/10.5802/crmath.216
Partial differential equations / Equations aux dérivées partielles
Lp-versions of generalized Korn inequalities
for incompatible tensor fields in arbitrary
dimensions with p-integrable exterior
derivative
Versions Lp des inégalités généralisées de Korn pour les
champs de tenseurs incompatibles de dimension
quelconque avec dérivée extérieure p-intégrable
Peter Lewintan ∗, a and Patrizio Neff a
a Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127
Essen, Germany
E-mails: peter.lewintan@uni-due.de (P. Lewintan), patrizio.neff@uni-due.de (P. Neff)
Abstract. For n ≥ 2 and 1 < p < ∞ we prove an Lp -version of the generalized Korn-type inequality for
incompatible, p-integrable tensor fields P :Ω→Rn×n having p-integrable generalized Curl and generalized
vanishing tangential trace P τl = 0 on ∂Ω, denoting by {τl }l=1, ...,n−1 a moving tangent frame on ∂Ω, more
precisely we have:
‖P‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥Lp (Ω, (so(n))n )) ,
where the generalized Curl is given by (CurlP )i j k := ∂i Pk j −∂ j Pki and c = c(n, p,Ω) > 0
Résumé. On montre pour n ≥ 2 et 1 < p <∞ une version Lp de l’inégalité généralisée de Korn pour tous les
champs de tenseurs incompatibles et p-intégrables P :Ω→Rn×n , avec rotationnel généralisé p-intégrable et
avec zéro trace tangentielle P τl = 0 sur ∂Ω, où {τl }l=1, ...,n−1 est un repère tangent sur ∂Ω. Plus précisément
on a :
‖P‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥Lp (Ω, (so(n))n )) ,
où les composantes du rotationnel généralisé s’écrivent (CurlP )i j k := ∂i Pk j −∂ j Pki et c = c(n, p,Ω) > 0.
2020 Mathematics Subject Classification. 35A23, 35B45, 35Q74, 46E35.
Funding. This work was initiated in the framework of the Priority Programme SPP 2256 “Variational Meth-
ods for Predicting Complex Phenomena in Engineering Structures and Materials” funded by the Deutsche
Forschungsgemeinschaft (DFG, German research foundation), Project-ID 422730790. The second author was
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
750 Peter Lewintan and Patrizio Neff
supported within the project “A variational scale-dependent transition scheme - from Cauchy elasticity to the
relaxed micromorphic continuum” (Project-ID 440935806). Moreover, both authors were supported in the
Project-ID 415894848 by the Deutsche Forschungsgemeinschaft.
Manuscript received 18th December 2019, revised 8th July 2020 and 16th April 2021, accepted 20th April 2021.
1. Introduction
In [6] we have shown that there exists a constant c = c(p,Ω) > 0 such that
‖P‖Lp (Ω,R3×3) ≤ c
(∥∥symP∥∥Lp (Ω,R3×3) +‖CurlP‖Lp (Ω,R3×3))
holds for all tensor fields P ∈ W 1, p0 (Curl;Ω,R3×3), i.e., for all P ∈ W 1, p (Curl;Ω,R3×3) with vanish-
ing tangential trace P ×ν= 0 (⇔ P τl = 0) on ∂Ω where ν denotes the outward unit normal vec-
tor field and {τl }l=1,2,3 a moving tangent frame on ∂Ω andΩ⊂R3 is a bounded Lipschitz domain.
The crucial ingredients for our proof were the Lions lemma and Nečas estimate, the compactness
of W 1, p0 (Ω) ⊂⊂ Lp (Ω) and an algebraic identity in terms of components of the cross product of a
skew-symmetric matrix with a vector. Recall, that for a bounded Lipschitz domain (i.e. bounded
open connected with Lipschitz boundary) Ω ⊂ Rn , the Lions lemma states that f ∈ Lp (Ω) if and
only if f ∈ W −1, p (Ω) and ∇ f ∈ W −1, p (Ω,Rn), which is equivalently expressed by the Nečas esti-
mate ∥∥ f ∥∥Lp (Ω) ≤ c (∥∥ f ∥∥W −1, p (Ω) +∥∥∇ f ∥∥W −1, p (Ω,Rn )) (1)
with a positive constant c = c(p,n,Ω). In fact, such an argumentation scheme is also used to prove
the classical Korn inequalities, cf. e.g. [1–6] and the discussions contained therein. However, [1–5]
focus on the compatible case, i.e. P = Du, where we deal with general square matrices P ∈ Rn×n ,
thus, the incompatible case.
Here, we extend our results from [6] to the n-dimensional case, hence generalizing the main
result from [8] to the Lp -setting. This is, we prove
‖P‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥Lp (Ω,(so(n))n )) ∀ P ∈W 1, p0 (Curl ;Ω, Rn×n) , (2)
where the generalized Curl is given by (CurlP )i j k := ∂i Pk j −∂ j Pki and the vanishing tangential
trace condition reads P τl = 0 on ∂Ω denoting by {τl }l=1, ...,n−1 a moving tangent frame on ∂Ω.
For a detailed motivation and definitions we refer to [6] and the references contained therein.
Indeed, we follow the argumentation scheme presented in [6] closely, emphasizing only the nec-
essary modifications coming from the generalization of the vector product. The latter then pro-
vides an adequate generalization of the Curl-operator to the n-dimensional setting. Especially,
the generalized curl of vector fields can be seen as their exterior derivative, see also the discus-
sion in [8].
2. Notations
Let n ≥ 2. For vectors a,b ∈ Rn , we consider the scalar product 〈a,b〉 := ∑ni=1 ai bi ∈ R, the
(squared) norm ‖a‖2 := 〈a, a〉 and the dyadic product a ⊗b := (ai b j )i , j=1, ...,n ∈ Rn×n . Similarly,
for matrices P,Q ∈Rn×n we define the scalar product 〈P,Q〉 :=∑ni , j=1 Pi j Qi j ∈R and the (squared)
Frobenius-norm ‖P‖2 := 〈P,P〉. Moreover, P T := (P j i )i , j=1, ...,n denotes the transposition of the
matrix P = (Pi j )i , j=1, ...,n , which decomposes orthogonally into the symmetric part symP :=
1
2 (P +P T ) and the skew-symmetric part skewP := 12 (P −P T ). The Lie-Algebra of skew-symmetric
matrices is denoted by so(n) := {A ∈Rn×n | AT =−A}. The identity matrix is denoted by 1, so that
the trace of a matrix P is given by trP := 〈P,1〉.
C. R. Mathématique — 2021, 359, n 6, 749-755
Peter Lewintan and Patrizio Neff 751
The cross product for vectors a,b ∈Rn generalizes to
a×b := (ai b j −a j bi )i , j=1, ...,n = a ⊗b −b ⊗a = 2 · skew(a ⊗b) ∈ so(n) ∼=R n(n−1)2 . (3)
Using the bijection axl : so(3) →R3 we obtain back the standard cross product for a,b ∈R3:
a ×b =−axl(a×b) (4)
where axl : so(3) →R3 is given in such a way that
A b = axl(A)×b ∀ A ∈ so(3), b ∈R3. (5)
Like in 3-dimensions it holds:
Observation 1. Let n ≥ 2. For non-zero vectors a,b ∈ Rn we have a×b = 0 if and only if a and b
are parallel.
Proof. Since the “if” part is obvious we show the “only if” direction:
a×b = 0 ⇔ skew(a ⊗b) = 0 ⇔ a ⊗b = b ⊗a ⇒ (a ⊗b)b = (b ⊗a)b
⇔ a ‖b‖2 = b 〈a,b〉 . 
As in the 3-dimensional case, we understand the vector product of a square-matrix P ∈ Rn×n
and a vector b ∈Rn row-wise, i.e.
P×b := ((P T ek)×b)k=1, ...,n = (Pki b j −Pk j bi )i , j ,k=1, ...,n ∈ (so(n))n . (6)
For index notations we set: (P×b)i j k := Pki b j −Pk j bi .
Especially, for skew-symmetric matrices A ∈ so(n) we note the following crucial relation for
our considerations:(
A×b)ki j − (A×b)k j i + (A×b) j i k = A j k bi − A j i bk − (Ai k b j − Ai j bk)+ Ak j bi − Aki b j(
Ai j =−A j i
)
= 2Ai j bk ∀ i , j ,k = 1, . . . n
(7)
with the direct consequence
Observation 2. Let n ≥ 2. For A ∈ so(n) and a non-zero vector b ∈Rn we have A×b = 0 if and only
if A = 0.
LetΩ⊂Rn , n ≥ 2, be a domain. As in R3 we formally introduce the generalized curl of a vector
field v ∈D ′(Ω,Rn) via
curl v := v×(−∇) =∇×v =−2 · skew(v ⊗∇) =−2 · skew(Dv) ∈ so(n). (8)
Furthermore, for (n ×n)-square matrix fields we understand this operation row-wise:
CurlP := P×(−∇) = (curl (P T ek))k=1, ...,n = (∂i Pk j −∂ j Pki )i , j ,k=1, ...,n ∈ (so(n))n . (9)
For index notations we define: (CurlP )i j k := ∂i Pk j −∂ j Pki . Of course, Curl Dv ≡ 0.
Moreover, we make use of the generalized divergence Div for matrix fields P ∈ D ′(Ω,Rn×n)
row-wise, via
DivP := (div(P T ek))k=1, ...,n . (10)
In fact, the crucial relation (7) implies that the full gradient of a skew-symmetric matrix is
already determined by its generalized Curl , cf. also [7, p. 155]:
Corollary 3. Let n ≥ 2. For A ∈D ′(Ω,so(n)) the entries of the gradient DA are linear combinations
of the entries from Curl A.
Proof. Replacing b by −∇ in (7) we see that(
Curl A
)
ki j −
(
Curl A
)
k j i +
(
Curl A
)
j i k =−2∂k Ai j . 
C. R. Mathématique — 2021, 359, n 6, 749-755
752 Peter Lewintan and Patrizio Neff
This control of all first partial derivatives of a skew-symmetric matrix field in terms of the
generalized Curl then immediately yields in all dimensions
Corollary 4. Let n ≥ 2. For A ∈ Lp (Ω,so(n)) we have Curl A ≡ 0 in the distributional sense if and
only if A = const almost everywhere inΩ.
2.1. Function spaces
Having above relations at hand we can now catch up the arguments from [6]. For that purpose let
us define for n ≥ 2 and 1 < p <∞ the space
W 1, p
(
Curl ;Ω,Rn×n
)
:= {P ∈ Lp (Ω,Rn×n) | CurlP ∈ Lp (Ω, (so(n))n)} (11a)
equipped with the norm
‖P‖W 1, p (Curl ;Ω,Rn×n) :=
(
‖P‖pLp (Ω,Rn×n ) +
∥∥CurlP∥∥pLp (Ω, (so(n))n )) 1p . (11b)
By definition of the norm in the dual space, we have
P ∈ Lp (Ω,Rn×n) ⇒ CurlP ∈W −1, p (Ω, (so(n))n)
with
∥∥CurlP∥∥W −1, p (Ω,(so(n))n ) ≤ c ‖P‖Lp (Ω,Rn×n ) . (12)
Furthermore, we consider the subspace
W 1, p0
(
Curl ;Ω,Rn×n
)
:= {P ∈W 1, p (Curl ;Ω,Rn×n) | P×ν= 0 on ∂Ω}
= {P ∈W 1, p (Curl ;Ω,Rn×n) | P τl = 0 on ∂Ω for all l = 1, . . . , n −1} , (13)
where ν stands for the outward unit normal vector field and {τl }l=1, ...,n−1 denotes a moving
tangent frame on ∂Ω. Here, the generalized tangential trace P×ν is understood in the sense of
W −
1
p , p (∂Ω,Rn×n) which is justified by partial integration, so that its trace is defined by
∀ k = 1, . . . n, ∀ Q ∈W 1−
1
p′ , p
′ (
∂Ω,Rn×n
)
:〈(
P T ek
)×ν,Q〉∂Ω = ∫
Ω
〈
curl
(
P T ek
)
,Q̃
〉
Rn×n +2
〈
P T ek ,Div
(
skewQ̃
)〉
Rn
dx
having denoted by Q̃ ∈ W 1, p ′ (Ω,Rn×n) any extension of Q in Ω, where, 〈· , ·〉∂Ω indicates the du-
ality pairing between W −
1
p , p (∂Ω,Rn×n) and W 1−
1
p′ , p
′
(∂Ω,Rn×n). Indeed, for P,Q ∈C 1(Ω,Rn×n)∩
C 0(Ω,Rn×n) we have
1
2
〈(
P T ek
)×ν,Q〉
Rn×n =
〈
skew
((
P T ek
)⊗ν) ,Q〉
Rn×n =
〈(
P T ek
)⊗ν, skewQ〉
Rn×n
=
n∑
i , j=1
Pki ν j (skewQ)i j =−
n∑
i , j=1
ν j (skewQ) j i Pki
=−〈ν, (skewQ) (P T ek)〉Rn ,
(14)
C. R. Mathématique — 2021, 359, n 6, 749-755
Peter Lewintan and Patrizio Neff 753
so that using the divergence-theorem, for k = 1, . . . ,n we have1∫
∂Ω
〈(
P T ek
)×ν,Q〉
Rn×n dS
(14)= −2
∫
∂Ω
〈
ν, (skewQ)
(
P T ek
)〉
Rn
dS
=−2
∫
Ω
div
(
(skewQ)
(
P T ek
))
dx
=−2
∫
Ω
〈
Div
[
(skewQ)T
]
,P T ek
〉
Rn
+〈(skewQ) ,D(P T ek)〉Rn×n dx
=
∫
Ω
〈
curl
(
P T ek
)
,Q
〉
Rn×n +2
〈
P T ek ,Div(skewQ)
〉
Rn
dx.
(15)
Further, following [6] we introduce also the space W 1, p
Γ,0 (Curl;Ω,R
n×n) of functions with van-
ishing tangential trace only on a relatively open (non-empty) subset Γ ⊆ ∂Ω of the boundary by
completion of C∞Γ,0(Ω,R
n×n) with respect to the W 1, p (Curl;Ω,Rn×n)-norm.
Remark 5 (Tangential trace condition). Note, that the vanishing of the tangential trace P×ν at
some point is equivalent to P τl = 0 for all l = 1, . . . , n −1, denoting by {τl }l=1, ...,n−1 a frame of the
corresponding tangent space. Indeed, by Observation 1 we have
P×ν= 0
⇔ skew((P T ek)⊗ν)= 0, k = 1, . . . , n, ⇔ (P T ek) parallel to ν for all k = 1, . . . , n
⇔ 〈P T ek ,τl 〉= 0 ∀ l = 1, . . . , n −1, ∀ k = 1, . . . , n ⇔ P τl = 0 ∀ l = 1, . . . , n −1.
3. Main results
We will now state the results from [6] in the n-dimensional case, for details of the proofs we refer
to the corresponding results therein:
Lemma 6. Let n ≥ 2,Ω⊂Rn be a bounded Lipschitz domain and 1 < p <∞. Then P ∈D ′(Ω,Rn×n),
symP ∈ Lp (Ω,Rn×n) and CurlP ∈W −1, p (Ω, (so(n))n) imply P ∈ Lp (Ω,Rn×n). Moreover, we have the
estimate
‖P‖Lp (Ω,Rn×n ) ≤ c
(
‖skewP‖W −1, p (Ω,Rn×n ) +
∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥W −1, p (Ω, (so(n))n )) , (16)
with a constant c = c(n, p,Ω) > 0.
Proof. Use Corollary 3 and apply the Lions lemma and Nečas estimate, [6, Theorem 2.6] to
skewP , cf. [6, proof of Lemma 3.1]. 
The general Korn-type inequalities then follow by eliminating the first term on the right-hand
side of (16):
Theorem 7. Let n ≥ 2, Ω ⊂ Rn be a bounded Lipschitz domain and 1 0, such that for all P ∈ Lp (Ω,Rn×n) we have
inf
A∈so(n)
‖P − A‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥W −1, p (Ω, (so(n))n )) . (17)
1This partial integration formula slightly differs from the situation in R3 since the generalized Curl has image in
(so(n))n which corresponds to Rn×n only for n = 3.
C. R. Mathématique — 2021, 359, n 6, 749-755
754 Peter Lewintan and Patrizio Neff
Proof. By Corollary 4 the kernel of the right-hand side consists only of constant skew-symmetric
matrices:
K :={P ∈ Lp (Ω,Rn×n)∣∣symP = 0 a.e. and CurlP = 0 in the distributional sense}
= {P = A a.e.|A ∈ so(n)} . (18)
Then there exist M := dimK = n(n−1)2 linear forms `α on Lp (Ω,Rn×n) such that P ∈ K is equal
to 0 if and only if `α(P ) = 0 for all α = 1, . . . , M . Exploiting the compactness Lp (Ω,Rn×n) ⊂⊂
W −1, p (Ω,Rn×n) allows us to eliminate the first term on the right-hand side of (16) so that we
arrive at
‖P‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥W −1, p (Ω, (so(n))n ) + M∑
α=1
|`α(P )|
)
. (19)
Considering P − AP in (19), where the skew-symmetric matrix AP ∈ K is chosen in such
a way that `α(P − AP ) = 0 for all α = 1, . . . , M , then yields the conclusion, cf. [6, proof of
Theorem 3.4]. 
Moreover, the kernel is killed by the tangential trace condition P×ν ≡ 0 (or P τl ≡ 0 for all
l = 1, . . . , n −1), cf. (18) together with Observation 2 (and also Remark 5), so that we arrive at
Theorem 8. Let n ≥ 2, Ω ⊂ Rn be a bounded Lipschitz domain and 1 0, such that for all P ∈W 1, p0 (Curl ;Ω,Rn×n) we have
‖P‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥Lp (Ω, (so(n))n )) . (20)
Proof. Having Observation 2 we can closely follow the proof of [6, Theorem 3.5]. 
Similar argumentations show that (20) also holds true for functions with vanishing tangential
trace only on a relatively open (non-empty) subset Γ⊆ ∂Ω of the boundary, namely
Theorem 9. Let n ≥ 2, Ω ⊂ Rn be a bounded Lipschitz domain and 1 0, such that for all P ∈W 1, p
Γ,0 (Curl ;Ω, R
n×n) we have
‖P‖Lp (Ω,Rn×n ) ≤ c
(∥∥symP∥∥Lp (Ω,Rn×n ) +∥∥CurlP∥∥Lp (Ω, (so(n))n )) . (21)
Furthermore, Theorem 9 reduces for compatible P = Du to a tangential Korn inequality
(Corollary 10) and for skew-symmetric P = A to a Poincaré inequality in arbitrary dimensions
(Corollary 12):
Corollary 10. Let n ≥ 2, Ω ⊂ Rn be a bounded Lipschitz domain and 1 0, such that for all u ∈W 1, p
Γ,0 (Ω,R
n) we have
‖Du‖Lp (Ω,Rn×n ) ≤ c
∥∥symDu∥∥Lp (Ω,Rn ) with Du×ν= 0 on Γ. (22)
Remark 11. On Γ the boundary condition Du×ν = 0 is equivalent to Duτl = 0 for all l =
1, . . . , n −1 and is, e.g., fulfilled if u|Γ ≡ const ., see Remark 5.
Corollary 12. Let n ≥ 2, Ω ⊂ Rn be a bounded Lipschitz domain and 1 0, such that for all A ∈W 1, p
Γ,0 (Curl ;Ω,so(n)) =W
1, p
Γ,0 (Ω,so(n)) we have
‖A‖Lp (Ω,so(n)) ≤ c
∥∥Curl A∥∥Lp (Ω, (so(n))n ) with A×ν= 0 ∗⇔ A = 0 on Γ. (23)
Remark 13. The equivalence of condition ∗ is seen in the following way: In any dimension the
rank of the skew-symmetric matrix A is an even number, cf. [9, p. 30], and by Remark 5 the rows
AT ek are all parallel (⇔ Aτl = 0 for all l = 1, . . . , n−1) such that the rank of A is not greater then 1.
C. R. Mathématique — 2021, 359, n 6, 749-755
Peter Lewintan and Patrizio Neff 755
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from the German 5th ed. by John R. Schulenberger. 1st paperback ed.
C. R. Mathématique — 2021, 359, n 6, 749-755

$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative

Sergey A. Denisov

Comptes Rendus
Mathématique
Sergey A. Denisov
Subharmonic functions in scattering theory
Volume 359, issue 6 (2021), p. 757-762
<https://doi.org/10.5802/crmath.217>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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
Comptes Rendus
Mathématique
2021, 359, n 6, p. 757-762
https://doi.org/10.5802/crmath.217
Partial differential equations / Équations aux dérivées partielles
Subharmonic functions in scattering theory
Sergey A. Denisova
a University of Wisconsin–Madison Mathematics Department 480 Lincoln Dr.,
Madison, WI, 53706, USA
E-mail: denissov@wisc.edu
Abstract. We present a method that uses the properties of subharmonic functions to control spatial asymp-
totics of Green’s kernel of multidimensional Schrödinger operator with rough potential.
2020 Mathematics Subject Classification. 35P25, 81Q10.
Funding. The work was supported by NSF DMS-1764245 and NSF DMS-2054465.
Manuscript received 30th March 2021, accepted 21st April 2021.
1. Introduction
In this note, we look at the multidimensional Schrödinger operator H : H = −∆+V , x ∈ R3 in
the case when the real-valued potential V ∈ L∞(R3) does not satisfy the classical assumptions
of the scattering theory [14]. Our motivation is to present a method based on the elementary
properties of subharmonic functions which provides control on spatial asymptotics of Green’s
function for H in the situation when the standard tools of perturbation theory do not work.
In the classical situation, when V is short-range, i.e., |V (x)| < C (|x| + 1)−γ,γ > 1 the results of
Kato [9] and Agmon [1] provide existence of wave operators in Schrödinger dynamics and the
positive spectrum of H is purely absolutely continuous. The case when V is long-range, e.g.,
|D j V | < C (|x| + 1)−γ− j , γ > 0.5, 0 ≤ j ≤ j0 was addressed in the works of Hörmander [7, 8] and,
again, the positive spectrum was shown to be purely a.c. (see also Saito [12] for results with even
weaker conditions on V ). Following the significant progress in one-dimensional scattering theory
(see [13] for the survey), the focus shifted to understanding the spectrum of multidimensional
operator with rough potentials, e.g., V satisfying only an upper bound |V | < C (|x| + 1)−γ with
γ ∈ (0.5,1]. For that case, the existence of a.c. spectrum is not known, but some results were
obtained [4, 5, 11] if V is assumed to oscillate additionally. We notice that for potentials so
rough the singular spectrum can coexist with the absolutely continuous one and the standard
tools of scattering theory, such as absorption principle, has no chance to work. The alternative
method rooted in complex analysis rather than harmonic analysis is based on the study of spatial
asymptotics of Green’s function. Here is its outline. Given z ∈ C+ and F ∈ L2(R3), we define
Rz F = (H − z)−1F and Green’s function G(x, y, z) as (Rz F )(x) =
∫
R3 G(x, y, z)F (y)dy . For the free
Schrödinger equation when V = 0, the formula for Green’s function is known:
G0(x, y,k
2) = e i k|x−y |(4π|x − y |)−1, k =pz, k ∈C+ .
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
758 Sergey A. Denisov
Let F have compact support and F 6≡ 0. If one defines UF (x,k) = Rk2 F , then the central quantity
of interest is an amplitude AF (k,θ) defined by
AF (k,θ) = lim|x|→∞,x/|x|=θ∈S2 |x|e
−i k|x|UF (x,k) . (1)
If one formally takes F = δ0, then AF tells how the perturbed Green’s function deviates from the
unperturbed one. SupposeσF (E) denotes the spectral measure of F relative to H . The importance
of the amplitude in the study of the spectral type is explained by the following Lemma (see [3,
formula (4.2)]).
Lemma 1. Suppose V and F are compactly supported, then
σ′F (k
2) =π−1k‖AF (k,θ)‖2L2(S2), k ∈R+ . (2)
This identity is remarkable for two reasons. First, by the Spectral Theorem, we always have
equality ∫
R
dσF (E) = ‖F‖2 . (3)
Second, the function h(k) := log‖AF (k,θ)‖L2(S2) is subharmonic in k ∈ {ℑk > 0,ℜk > 0}. Thus, if
we can obtain
(A) rough upper bounds for h, e.g., h(k) ≤ c(ℑk)−α for some α> 0,
(B) some lower bound away from the real line, e.g., h(k0) > δ,ℑk0 = ε with some positive ε
and δ
then the mean-value inequality for subharmonic functions written for a suitable isosceles trape-
zoid T ⊂C+ with the base I ⊂R+ gives∫
∂T
h(ξ)dωk0 ≥ h(k0) . (4)
The symbol ωk0 denotes the harmonic measure with the reference point k0 ∈ T . In particular,
(A) and (B) provide the lower estimate for h on interval I . Combined with (3) and Chebyshev’s
inequality, this gives the following bound∣∣{k ∈ I : ‖AF (k,θ)‖2L2(S2) ≤ e−Λ}∣∣+ ∣∣{k ∈ I : ‖AF (k,θ)‖2L2(S2) ≥ eΛ}∣∣<C (I ,c,α,ε,δ)Λ−1 (5)
for everyΛ> 1. Thanks to (2), we also have control on the spectral measure σF :∣∣{k ∈ I :σ′F (k2) ≤ e−Λ}∣∣+ ∣∣{k ∈ I :σ′F (k2) ≥ eΛ}∣∣<C (I ,c,α,ε,δ)Λ−1
These bounds indicate that “scattering happens” for most positive energies E = k2. At the same
time, they are also consistent with the possibility that some energies carry the singular part ofσF .
In the next section, we use that strategy to study potentials V supported on large annuli. In [2],
this technique was applied to show the presence of a.c. spectral type for 1d Schrödinger with
operator-valued rough potential. We expect this approach to be useful for other problems, e.g.,
spectral theory of difference operators on graphs.
Notation.
• Given an interval I ⊂ R+ and positive α, the symbol RI ,α denotes the rectangle I × (0,α)
in complex plane; cI is the center of I .
• For the function g ∈ L2(S2), the symbol ‖g‖ stands for its L2(S2) norm. Br (x) is the ball
with radius r centered at x.
• Symbol B is Laplace-Beltrami operator on the unit sphere. It is nonpositive in L2(S2).
Given a set E ⊂R3, the symbol χE denotes the characteristic function of E .
• We define the angular component of the gradient (in spherical coordinates) as
∇τV =∇V − x|x|
〈
x
|x| ,∇V
〉
.
C. R. Mathématique — 2021, 359, n 6, 757-762
Sergey A. Denisov 759
2. An example: V supported on the large annulus
We are looking for conditions on V that provide estimates (A) and (B) discussed in Introduction.
Suppose supp V ⊂ {x : T /2 < |x| < T }, F =χB1(0), AF is defined as in (1), and T is a large parameter.
Theorem 2. Consider H =−∆+V , x ∈R3 where non-negative real-valued potential V satisfies
suppV ⊂ {x : T /2 < |x| < T }, |V | ≤C T −γ, |∇τV | ≤C T −γ−δ, δ,γ> 0. (6)
If γ ∈ ( 23 ,1] and γ+ δ2 > 1, then
sup
T>1
∫
I
|log‖AF (k,θ)‖|dk <∞
for every segment I ⊂R+, |I | ∼ 1.
Operator H with potential V that satisfies stronger assumptions on angular gradient has been
studied in [10].
Remark 3. As a consequence, we get∣∣{k ∈ I : ‖AF (k,θ)‖2L2(S2) ≤ e−Λ}∣∣+ ∣∣{k ∈ I : ‖AF (k,θ)‖2L2(S2) ≥ eΛ}∣∣<C (γ,δ, I )Λ−1 (7)
for everyΛ> 1 and T > 1.
To get bounds (A) and (B), we will use Schrödinger equation itself. Recall that the operator H
can be written as
H =−∂2r r −
B
r 2
+V (r,θ)
in the spherical coordinates (r,θ) ∈ R+×S2 and the corresponding map F (x) → f (r,θ) = r F (r,θ)
satisfies ‖F‖L2(R3) = ‖ f ‖L2(R+) where f is considered as L2(S2)-valued function in r . If f represents
F and u represents UF = Rk2 F , we have an equation for u:
−u′′− Bu
r 2
+V u = k2u + f , r > 0.
Similarly, if a represents e−i k|x|UF , then a = ue−i kr and
2i ka′ =−a′′− B a
r 2
+V a − f e−i kr (8)
or
a′ = (2i k)−1
(
−a′′− B a
r 2
+V a − f e−i kr
)
. (9)
Recall that supp f ⊂ [0,1]. Then, the last equation gives the following identity if one takes the
inner product and integrates the real part of resulting equation.
Lemma 4. Let 1 < r1 < r2 and k : ℜk > 0,ℑk > 0, then
‖a(r2,k)‖2 + ℑk|k|2
∫ r2
r1
(
‖a′‖2 − 〈B a, a〉
ρ2
)
dρ+ ℑk|k|2
∫ r2
r1
〈V a, a〉dρ = ‖a(r1,k)‖2 +Q(r2)−Q(r1) (10)
where
Q(r ) := i
2k
〈a′(r,k), a(r,k)〉− i
2k̄
〈a(r,k), a′(r,k)〉 .
If V satisfies
V ≥ 0, supp V ∈ [T /2,T ], ‖V ‖L∞(S2) <∞
all terms in the left-hand side of (10) are non-negative and we get the following apriori estimate
after setting r2 =∞ and applying Cauchy-Schwartz to Q(r1):
‖a(∞,k)‖2 <C I
(‖a(r1,k)‖2 +‖a′(r1,k)‖2) (11)
provided that k ∈ RI ,1 and I ⊂ R+. The Spectral Theorem implies that ‖U‖L2(R3) ≤ C I (ℑk)−1.
Thus, from equation HU = k2U + F , we get ‖∆U‖L2(R3) < C I (ℑk)−1. So, the theorem about
C. R. Mathématique — 2021, 359, n 6, 757-762
760 Sergey A. Denisov
traces of functions in Sobolev class implies ‖a(2,k)‖2 + ‖a′(2,k)‖2 < C I (ℑk)−1 and (11) yields
‖a(∞,k)‖2 <C I (ℑk)−1 if we set r1 = 2. Thus,
h = log‖a(∞,k)‖ <C1 +C2|logℑk|, k ∈ RI ,1 (12)
with some positive T -independent C1 and C2. This is a rough bound (A) from the Introduction
that we are looking for and it does not require any smallness assumption on V at all. Our next
immediate goal is to obtain better control on h away from the real line, i.e., an estimate (B). The
idea is to use (10) with r2 =∞, r1 = T /4, and k ∈ RI ,1 to write
|‖a(∞,k)‖2 −‖a(T /4,k)‖2|
≤C (I )
(
|Q(T /4)|+ (ℑk)T −γ
∫ T
T /2
‖a‖2dρ+ (ℑk)
∫ ∞
T /4
(
‖a′‖2 − 〈B a, a〉
ρ2
)
dρ
)
(13)
Here, we choose r1 = T /4 to guarantee enough separation from the support of V . Our goal is to
show that the right-hand side is small if k is separated from the real axis. If
‖V ‖L∞(S2) ≤C T −γ, k ∈ RI ,1, ℑk > cT −γ (14)
with suitable T -independent constant c, the standard perturbation theory based on Combes-
Thomas estimates (see, e.g., [6]) yields
‖a(T /4,k)‖2 ∼ 1, ‖a′(T /4,k)‖2 ≤C I e−T
κ
, 〈B a(T /4,k), a(T /4,k)〉/T 2 ≤C I e−T
κ
, κ> 0. (15)
Hence, |Q(T /4)| < C I e−T κ . In what follows, we will assume that (14) holds. We need to control
the other three terms in the right-hand side of (13). Taking the inner product with a′ in (8) and
integrating the real part from r to ∞ gives the following lemma.
Lemma 5. Let ℜk > 0,ℑk > 0, and r > 1, then
(4ℑk)
∫ ∞
r
〈a′, a′〉dρ+‖a′(r,k)‖2 −2
∫ ∞
r
〈B a(ρ,k), a(ρ,k)〉
ρ3
dρ
=−〈B a(r,k), a(r,k)〉
r 2
−2ℜ
(∫ ∞
r
〈V a, a′〉dρ
)
.
Since the terms in the left-hand side are all non-negative, we obtain another apriori estimate
for k ∈ RI ,1,ℑk >C T −γ:
(ℑk)
∫ ∞
T /4
‖a′(ρ,k)‖2dρ ≤C I
(
e−T
κ +T −γ
(∫ T
T /4
‖a‖2dr
)1/2 (∫ ∞
T /4
‖a′‖2dr
)1/2)
. (16)
Taking r1 = T /4 and integrating (10) in r2 from T /4 to T , we apply Cauchy-Schwartz and (15) to
get ∫ T
T /4
‖a‖2dr ≤C I
(
T +
(∫ T
T /4
‖a‖2dr
)1/2 (∫ T
T /4
‖a′‖2dr
)1/2)
for k ∈ RI ,1,ℑk >C T −γ. This implies∫ T
T /4
‖a‖2dr ≤C I
(
T +
∫ T
T /4
‖a′‖2dr
)
≤C I
(
T +
∫ ∞
T /4
‖a′‖2dr
)
(17)
which, being substituted into (16), yields∫ ∞
T /4
‖a′‖2dr <C I ((ℑk)−1e−T
κ + (ℑk)−2T 1−2γ)
and, from (17),
T −1
∫ T
T /4
‖a‖2dr ≤C I . (18)
C. R. Mathématique — 2021, 359, n 6, 757-762
Sergey A. Denisov 761
These two last bounds control two additional terms in the right-hand side of (13) in the regime
when k ∈ RI ,1 and ℑk >C T −γ. Thus, we are only left with estimating∫ ∞
T /4
−〈B a, a〉
ρ2
dρ .
in (13). To this end, we use an extra smoothness of V , i.e., the last condition in (6). Taking the
gradient of (9) in the angular variables, we rewrite (6) as
(∇θa)′ = (2i k)−1
(
−(∇θa)′′−
B∇θa
r 2
+V ∇θa +a∇θV
)
. (19)
This equation has the same structure as equation for a except for the term a∇θV . The last con-
dition on V in (6) can be written as ‖∇θV ‖L∞(S2) ≤ C T 1−γ−δ. Repeating the previous arguments
and using the bound (18), we obtain∫ T
T /4
−〈B a, a〉
ρ2
dρ ∼ T −2
∫ T
T /4
‖∇θa‖2dρ ≤C I
(
(ℑk)−2T 1−2γ−2δ+T 3−2γ−2δ
)
.
For r > T , one has V = 0 and the bound∫ ∞
T /4
−〈B a, a〉
ρ2
dρ =
∫ ∞
T /4
‖∇θa‖2
ρ2
dρ ≤C I
(
(ℑk)−2T 1−2γ−2δ+T 3−2γ−2δ
)
follows by the similar reasoning. In (13), we apply the obtained estimates to control the right-
hand side. This gives∣∣‖a(∞,k)‖2 −‖a(a,k)‖2∣∣≤C I (T −1 + (ℑk)(T 1−γ+T 3−2γ−2δ)+ (ℑk)−1T 1−2γ) .
Given (6), we can always assume T 3−2γ−2δ > T 1−γ and find a small positive ε such that the set
PI ,T defined by PI ,T =
{
k ∈C+ : ℜk ∈ I , T ε(T 1−2γ) <ℑk < T −ε(T 2γ+2δ−3)} is nonempty. Hence, if
k ∈ PI ,T , then
‖a(∞,k)‖2 = ‖a(T /4,k)‖2 +O(T −ε1 ) (20)
with some ε1 > 0 and thus ‖a(∞,k)‖2 ∼ 1 there. This establishes the required bound (B) and we
are ready to prove Theorem 2.
Proof of Theorem 2. For every I ⊂ R+, consider RI ,δT with δT = T −ε(T 2γ+2δ−3) and write mean-
value inequality for subharmonic function h := log‖a(∞,k)‖:∫
∂RI ,δT
hdωk0 ≥ h(k0)
where k0 = x + iδT /2, x ∈ I ′ ⊂ I and the subinterval I ′ is chosen such that cI ′ = cI , |I ′| > c|I | with
fixed T -independent c.
0.5δT
δT
cI = cI ′ I ′ I
k0
Figure 1. Rectangle RI ,δT
If ℑk > δT /2 and k ∈ ∂RI ,δT , we have ‖a(∞,k)‖ ∼ 1 by (20). On the other hand, (12) guarantees
that
∫
Γ |h|dωk0 <C uniformly in T where Γ are the legs of RI ,δT , i.e., Γ= {k ∈ ∂RI ,δT ,0 <ℑk < δT }.
C. R. Mathématique — 2021, 359, n 6, 757-762
762 Sergey A. Denisov
In the end, if we use the standard estimates on the harmonic measure of the rectangle and
integrate in x ∈ I ′, we obtain the bound ∫I ′ h(k)dk > C uniformly in T . The formula (3) yields∫
I ‖a(∞,k)‖2dk <C uniformly in T and the trivial estimate max{0, log t } ≤ t , t > 0 gives
sup
T>1
∫
I ′
|log‖a(∞,k)‖|dk <∞ .
For an arbitrary I ′, we can find the corresponding I ⊂R+ so the proof is finished. 
References
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(1975), p. 151-218.
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[3] ——— , “Absolutely continuous spectrum of multidimensional Schrödinger operator”, Int. Math. Res. Not. 2004
(2004), no. 74, p. 205-214.
[4] ——— , “Multidimensional L2 conjecture: a survey”, in Recent trends in analysis, Theta Series in Advanced Mathe-
matics, vol. 16, The Theta Foundation, 2013, p. 101-112.
[5] ——— , “Spatial asymptotics of Green’s function for elliptic operators and applications: a.c. spectral type, wave
operators for wave equation”, Trans. Am. Math. Soc. 371 (2019), no. 12, p. 8907-8970.
[6] F. Germinet, A. Klein, “Operator kernel estimates for functions of generalized Schrödinger operators”, Proc. Am.
Math. Soc. 131 (2003), no. 3, p. 911-920.
[7] L. Hörmander, “The existence of wave operators in scattering theory”, Math. Z. 146 (1976), p. 69-91.
[8] ——— , The analysis of linear partial differential operators. IV. Fourier integral operators, Classics in Mathematics,
Springer, 2009.
[9] T. Kato, Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, vol. 132,
Springer, 1966.
[10] G. Perelman, “Stability of the absolutely continuous spectrum for multidimensional Schrödinger operators”, Int.
Math. Res. Not. 2005 (2005), no. 37, p. 2289-2313.
[11] O. Safronov, “Absolutely continuous spectrum of multi-dimensional Schrödinger operators with slowly decaying
potentials”, in Spectral theory of differential operators, Translations. Series 2, vol. 225, American Mathematical
Society, 2008, p. 205-214.
[12] Y. Saito, Spectral representations for Schrödinger operators with long-range potentials, Lecture Notes in Mathematics,
vol. 727, Springer, 1979.
[13] B. Simon, Szegő’s theorem and its descendants. Spectral theory for L2 perturbations of orthogonal polynomials,
Princeton University Press, 2011.
[14] D. R. Yafaev, Mathematical scattering theory, Mathematical Surveys and Monographs, vol. 158, American Mathemat-
ical Society, 1992.
C. R. Mathématique — 2021, 359, n 6, 757-762

Subharmonic functions in scattering theory

Snehajit Misra

Nabanita Ray

On Ampleness of vector bundles

Ilia Smilga

Comptes Rendus
Mathématique
Ilia Smilga
New sequences of non-free rational points
Volume 359, issue 8 (2021), p. 983-989
<https://doi.org/10.5802/crmath.230>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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
Comptes Rendus
Mathématique
2021, 359, n 8, p. 983-989
https://doi.org/10.5802/crmath.230
Group theory, Number theory / Théorie des groupes, Théorie des nombres
New sequences of non-free rational points
Ilia Smilga∗, a
a Institut des Hautes Études Scientifiques et CNRS, 35 route de Chartres, 91440
Bures-sur-Yvette, France.
E-mail: ilia.smilga@normalesup.org
Abstract. We exhibit some new infinite families of rational values of τ, some of them squares of rationals, for
which the group or even the semigroup generated by the matrices ( 1 10 1 ) and (
1 0
τ 1 ) is not free.
Funding. The author is supported by the European Research Council (ERC) under the European Union
Horizon 2020 research and innovation programme (ERC starting grant DiGGeS, grant agreement No. 715982).
Manuscript received 9th February 2021, revised 30th May 2021 and 2nd July 2021, accepted 3rd June 2021.
For each α,β ∈C, we define the group
Γ(α,β) :=
〈(
1 α
0 1
)
,
(
1 0
β 1
)〉
,
and we define S(α,β) to be the semigroup generated by the same two matrices. The problem
of determining values of the product τ := αβ for which the group Γ(α,β) (conjugate by ( 1 00 α )
to Γ(1,τ)) is free has already attracted a considerable amount of attention [3, 5, 6, 8, 10–12]; the
corresponding problem for the semigroup S(α,β) has also been studied [4,13]. In particular, a lot
of people have focused on the special case where τ is the square of a rational [2, 7, 10, 14] (which
is a natural condition as it corresponds to studying Γ(µ,µ) with rational µ), or more generally any
rational number [1, 9].
To avoid confusion, we should warn the reader that previous authors have variously used as
the key parameter either the product αβ (e.g. [8, 9]), its half, usually denoted by λ (focusing on
Γ(2,λ)) (e.g. [1,5,6,10,11,13]), or its square root, usually denoted by µ (e.g. [2–4,12,14]). We make
the former choice, hence the following terminology:
Definition 1. We say that a number τ ∈ C is free (resp. semigroup-free) if the group Γ(1,τ) (resp.
the semigroup S(1,τ)) is free.
An elementary observation (first made by Brenner [3]) is that for τ ≥ 4, the group Γ(1,τ) is
always Schottky, hence free. So for groups, it suffices to study the interval (−4,4) (in fact even
(0,4), given the symmetry Γ(1,τ) = Γ(1,−τ)). Similarly, for τ≥ 1, the semigroup S(1,τ) is Schottky
(see also [4, Theorem 2.6]), so it suffices to study the interval (−4,1).
Beyond this, very little is known. It is conjectured that all rational τ ∈ (−4,4) are non-free,
and that all rational τ ∈ (−4,1) are non-semigroup-free. There are no known counterexamples,
but only a limited number of values for which these two conjectures have been proved. Indeed,
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
984 Ilia Smilga
group relations are easiest to find for τ close to 0. A spectacular development in this direction
is [9], proving that all rational numbers τ ∈ (−4,4) whose numerator is at most 23, or between 25
and 27, are non-free. On the other hand, as τ gets closer to 4, finding relations becomes extremely
hard.
It is even harder to find accumulation points of non-free rational values. It seems that the only
ones known so far (besides 0, which is trivial as 1n is never free) are 1 [7], those of the form
1
n [2]
and those of the form nn±1 [14]. As for accumulation points of non-semigroup-free rational values,
only 0 is known.
Our goal in this paper has been to find accumulation points of non-free and non-semigroup-
free rational values (or squares of rationals) that lie as close as possible to the endpoints of the
corresponding interval. The result that we obtain can be summarised as follows.
Theorem 2. The set of τ that are rational and non-semigroup-free (hence in particular non-free)
has −2±p2, −φ±2, −2 and 1 as accumulation points (where φ=
p
5+1
2 is the golden ratio). The set
of τ that are squares of rationals and non-semigroup-free (hence in particular non-free) has 1 as
an accumulation point.
Proof. This follows from the more precise Theorem 3 below, which gives a concrete description
of the infinite families that accumulate at these points. Note that:
• the values given in (iii) for positive (resp. negative) k are in fact the convergents of the
continued fraction of φ2 (resp. of φ−2);
• the values given in (iv) for positive (resp. negative) k are in fact the convergents of the
continued fraction of 2+p2 (resp. of 2−p2). 
The technique used to prove points (iii) and (iv) is similar to that used by Beardon [2] and Tan
and Tan [14]: a one-parameter family of relations leading to values of τ that are convergents of
some continued fraction. The main difference is that these two papers focused on the case where
τ is the square of a rational, whereas we consider all rational τ. Moreover, with similar methods,
one can also construct many other accumulation points; we have chosen to present only those
that are closest to the endpoints of the interval.
For each member of each of the following five families, we will actually exhibit an explicit
relation for the corresponding group or semigroup (see Proposition 7). We came upon some of
these relations by picking out patterns in the results of a computer search, using the SageMath
software [15].
Theorem 3. All of the following values of τ are non-free. The values given in (i) are also non-
semigroup-free. For τ as in (ii) with k > 0 or as in (iii) or (iv) with (−1)k k > 0, its negative −τ is
non-semigroup-free.
(i) τ = ( n−1n )2, for all n = 6σ0σ1 uσk uσk+1, where σ = (σ0,σ1) is any pair of distinct numbers
among {1,2,3}, and (uσk )k ∈Z is the integer sequence determined by the following recurrence
relations: {
uσ0 = uσ1 = 1;
∀ k ∈Z, uσk−1 −2σ(k mod 2)uσk +uσk+1 = 0
(1)
(see discussion below for explicit list of values).
(ii) τ= 2k±1k , for all integer k 6= 0.
(iii) τ = Fk+2Fk for all integer k 6= 0, where F is the Fibonacci sequence (F1 = F2 = 1). Explicitly,
these are
2,3,
5
2
,
8
3
,
13
5
,
21
8
, . . . (2)
and their reciprocals.
C. R. Mathématique — 2021, 359, n 8, 983-989
Ilia Smilga 985
(iv) τ = Hk+1Pk for all integer k 6= 0, where Pk are the Pell numbers and Hk are the half-
companion Pell numbers, given by(
Hk
Pk
)
:=
(
1 2
1 1
)k (
1
0
)
.
Explicitly, these are
3,
7
2
,
17
5
,
41
12
,
99
29
,
239
70
, . . . (3)
and twice their reciprocals.
Note that in point (i), only non-semigroup-freeness is novel; non-freeness of τ= ( n−1n )2 for all
integer n (positive or negative) is shown in [7].
Discussion of the values given in (i). One readily checks the following symmetry of the se-
quences uσk :
∀σ0 6=σ1 ∈ {1,2,3}, ∀ k ∈Z, u(σ0,σ1)k = u
(σ1,σ0)
1−k , (4)
so it suffices to consider only three of the six sequences. Moreover the sequences u(1,2) and u(1,3)
are symmetric:
∀σ1 ∈ {2,3}, ∀ k ∈Z, u(1,σ1)k = u
(1,σ1)
−k (5)
(indeed observe that in this case uσ−1 = 2uσ0 −uσ1 = 1 = uσ1 , then use induction), so it suffices to
consider their nonnegative terms. Explicitly, we have:(
u(1,2)±k
)
k ≥0 = 1,1,3,5,17,29,99,169, . . . ; (6)(
u(1,3)±k
)
k ≥0 = 1,1,5,9,49,89,485,881, . . . ; (7)(
u(2,3)k
)
k ∈Z = . . . , 1427,373,65,17,3,1,1,5,19,109,417,2393, . . . . (8)
These give rise to the following values of n:(
3u(1,2)k u
(1,2)
k+1
)
k ≥0 = 3,9,45,255,1479,8613,50193, . . . ; (9)(
2u(1,3)k u
(1,3)
k+1
)
k ≥0 = 2,10,90,882,8722,86330, . . . ; (10)(
u(2,3)k u
(2,3)
k+1
)
k ∈Z = . . . , 24245,1105,51,3,1,5,85,2071,45453, . . . . (11)

In fact, instead of relations, we will exhibit so-called “half-relations”, that we will define right
now. We will then see (Proposition 5) that each half-relation easily allows one to write down an
actual relation.
Definition 4. Let τ ∈C, and let us fix the notations g := ( 1 10 1 ) and hτ := ( 1 0τ 1 ). We say that a sequence
(a1, . . . , al ) ∈Zl is a half-relation for τ if:
(i) for odd l , the matrix M = g a1 ha2τ · · · hal−1τ g al satisfies
τc12(M)− c21(M) = 0; (12)
(ii) for even l , the matrix M = g a1 ha2τ · · · g al−1 halτ satisfies
c11(M)− c22(M) = 0, (13)
where ci j denotes the (i , j )th coefficient, so that
∀ M , M =:
(
c11(M) c12(M)
c21(M) c22(M)
)
. (14)
The point of half-relations is that they allow us to construct a special kind of relations for the
group or semigroup corresponding to ±τ:
C. R. Mathématique — 2021, 359, n 8, 983-989
986 Ilia Smilga
Proposition 5. A sequence (a1, . . . , al ) ∈Zl is a half-relation for τ if and only if the identity{
g a1 ha2τ · · · g al = halτ g al−1 · · · ha1τ for odd l
g a1 ha2τ · · · halτ = halτ g al−1 · · · g a1 for even l
(15)
or equivalently {
g a1 ha2τ · · · g al h−a1τ g−a2 · · · h−alτ = Id for odd l
g a1 ha2τ · · · halτ g−a1 h−a2τ · · · h−alτ = Id for even l
(16)
holds. Such an identity provides a nontrivial relation:
• for the group Γ(1,τ) if ∀ i , ai 6= 0;
• for the semigroup S(1,τ) if ∀ i , ai > 0;
• for the semigroup S(1,−τ) if ∀ i , (−1)i ai > 0.
Proof. Let M = g a1 ha2τ · · · denote the left-hand side of (15). Then (a1, . . . , al ) is a half-relation
for τ if and only if M is a fixed point of:
• the involution ( a bc d ) 7→ ( a c/ττb d ), or equivalently M 7→ ( 1 00 τ )M t( 1 00 τ )−1, for odd l ;
• the involution ( a bc d ) 7→ ( d bc a ), or equivalently M 7→ ( 1 00 −1 )M−1( 1 00 −1 )−1, for even l .
Both of them are antimorphisms; moreover the former switches g and hτ, whereas the latter fixes
both of them. In both cases, the involution maps both sides of (15) to each other.
Now assume that this is true. Then (15), usually rewritten as (16) for groups, is a relation in
Γ(1,τ), which is nontrivial if the coefficients are all nonzero. If they are all positive, then (15) is
in fact a relation in S(1,τ). If they have alternating signs (note that whenever (a1, . . . , al ) is a half-
relation, so is (−a1, . . . , −al )), then (15) (for even l ) or (16) (for odd l ) is a relation in S(1,−τ). 
Let us also observe that being a half-relation is a polynomial condition: more precisely, the
left-hand side of (12) (for odd l ) or (13) (for even l ) is a polynomial in a1, . . . , al and τ. One also
easily checks by induction that this polynomial is always divisible by τ, so we can factor τ out:
Definition 6. For each l ≥ 0, we define the polynomial
P lHR (a1, . . . , al ;τ) :=
{(
c12 − 1τc21
)(
g a1 ha2τ · · · g al
)
if l is odd;
1
τ (c11 − c22)
(
g a1 ha2τ · · · halτ
)
if l is even
(17)
(where ci j is the (i , j )th coefficient, as in (14)).
Thus, by construction, whenever τ ∈ C is a root of some polynomial P lHR(a1, . . . , al ) for some
tuple (a1, . . . , al ) of nonzero (resp. positive) integers, it is non-free (resp. non-semigroup-free).
Explicitly, for small l , these polynomials are as follows:
P 1HR (a1;τ) = a1
P 2HR (a1, a2;τ) = a1a2
P 3HR (a1, a2, a3;τ) = a1a2a3τ+a1 −a2 +a3
P 4HR (a1, . . . , a4;τ) = a1a2a3a4τ+a1a2 −a2a3 +a3a4 +a1a4
P 5HR (a1, . . . , a5;τ) = a1a2a3a4a5τ2 + (a1a2a3 −a2a3a4 +a1a2a5 +a1a4a5 +a3a4a5)τ
+a1 −a2 +a3 −a4 +a5.
The observation that non-free values can be obtained as roots of certain polynomials has already
been made previously. Even more specifically, when l = 2k + 1 is odd, the polynomial P 2k+1HR
coincides (up to the substitution τ = −2λ and a sign switch for the odd ai ’s) with Bamberg’s
polynomial Bk [1].
We are now ready to exhibit the half-relations that prove all the statements of Theorem 3. Note
that in point (ii), the subcase (a) suffices by itself to prove the Theorem in the general case; but
C. R. Mathématique — 2021, 359, n 8, 983-989
Ilia Smilga 987
we have also presented two additional special cases, (b) and (c), for which we can shorten the
half-relation by one coefficient.
Proposition 7. Fix some k ∈ Z, assumed to be nonzero except in point (i), and some pair σ =
(σ0,σ1) of distinct coefficients among {1,2,3}.
(i) The value τ= ( n−1n )2, where n = 6σ0σ1 uσk uσk+1, has(
1,
6
σk+1
(
uσk
)2 , 6
σk
(
uσk+1
)2 ,1)
as a half-relation (where σk is a notation shortcut for σ(k mod 2)).
(ii) The value τ= 2k+1k has following half-relations:
(a) All sequences (k,−1,1,−1,k, x), where x is any integer.
(b) If k = 2t for some integer t , the sequence (1,−1,1,−t ,−4t 2 +2t −2).
(c) If k = t (t+1)2 −1 for some integer t , the sequence (1,−1,1,−t +1,−t −2).
(iii) The value τ = Fk+2Fk has (1,−1,1,−1,2(−1)
k Fk−1Fk ) as a half-relation (where F is the
Fibonacci sequence, F1 = F2 = 1).
(iv) The value τ = Hk+1Pk has ((−1)
k Pk−1Pk ,−1,1,−1,1,−1,1,−1,(−1)k Pk−1Pk , x) as a half-
relation for any integer x (where Pk are the Pell numbers and Hk are the half-companion
Pell numbers, given by ( HkPk ) := ( 1 21 1 )k ( 10 )).
A remark about point (i): in fact, one can show that these are the only values of n for which
τ= ( n−1n )2 has a half-relation of length 4 with positive coefficients.
Proof. To prove these statements, we check that all of the listed values make the polynomial PHR
vanish.
• For point (ii), this is a straightforward computation.
• For point (iii), we start by computing
P 5HR(1,−1,1,−1, N ;τ) = Nτ2 − (3N +2)τ+ (N +4). (18)
It remains to verify that for all k 6= 0, the values τ = Fk+2Fk and N = 2(−1)
k Fk−1Fk make
this expression vanish. This is straightforward (if tedious), for example by plugging in the
closed formula
Fk =
φk − (−φ)−k
φ+φ−1
(where φ := 1+
p
5
2 is the golden ratio) and expanding.
• For point (iv), we start by computing
P 10HR(N ,−1,1,−1,1,−1,1,−1, N , x;τ) =
= x
(
N 2τ4 − (6N 2 +2N)τ3 + (10N 2 +10N +1)τ2 − (4N 2 +12N +4)τ+2N +3)
= x (Nτ2 − (2N +1)τ+1)(Nτ2 − (4N +1)τ+ (2N +3)) . (19)
It turns out that for all k 6= 0, the values τ = Hk+1Pk and N = (−1)
k Pk−1Pk make the last
factor vanish (so that the whole expression vanishes regardless of the value of x). This is
straightforward (if tedious) to verify, for example by plugging in the closed formulas
Pk =
αk − (−α)−k
α+α−1 , Hk =
αk + (−α)−k
α−α−1 (20)
(where α= 1+p2) and expanding.
C. R. Mathématique — 2021, 359, n 8, 983-989
988 Ilia Smilga
• Finally, for point (i), we start by computing
P 4HR
(
1,
6
σk+1
(
uσk
)2 , 6
σk
(
uσk+1
)2 ,1;(1− 16
σ0σ1
uσk u
σ
k+1
)2)
= P (σk ,σk+1) (uσk ,uσk+1)
=
{
Pσ
(
uσk ,u
σ
k+1
)
if k is even
Pσ
(
uσk+1,u
σ
k
)
if k is odd,
(21)
where, for all σ= (σ0,σ1), Pσ is the polynomial given by
Pσ(x, y) := 1+σ0σ1 + 6
σ1
x2 + 6
σ0
y2 −12x y. (22)
Recalling the definition (1) of the sequences uσ, to prove (by induction on k) that the
right-hand side of (21) vanishes for all k ∈ Z, it suffices to check the following two
identities. On the one hand, we have
∀σ0 6=σ1 ∈ {1,2,3}, Pσ(1,1) = 0 (23)
(indeed the triplet (σ0σ1,
6
σ0
, 6σ1 ) is always some permutation of (2,3,6)). On the other
hand, we easily check that
∀σ,∀ x, y, Pσ(x, y) = Pσ(2σ1 y −x, y) = Pσ(x,2σ0x − y). (24)

Proof of Theorem 3. Theorem 3 now easily follows by Proposition 5: indeed, one readily checks
that the coefficients of almost all the half-relations listed above are nonzero, and have the
required signs. There are only two exceptions: k = 1 in cases (iii) and (iv), which yields respectively
τ= 2 and τ= 3, which are well-known to be non-free. Also in these cases, one can still write down
the relation (15) (or (16)); some cancellations occur, but one still recovers the nontrivial relation
g 2h−1τ g h−2τ g h−1τ = Id (for τ= 2) and g h−1τ g h−1τ g h−1τ = Id (for τ= 3). 
Acknowledgements
I would like to thank Gregory Margulis for drawing my attention to this problem, and Paul Mercat
for some helpful discussions.
References
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