Holomorphic extension of the de Gennes function

BLAISE PASCAL

Virginie Bonnaillie-Noël,

Frédéric Hérau & Nicolas Raymond

Holomorphic extension of the de Gennes function Volume 24, n^o2 (2017), p. 225-234.

<http://ambp.cedram.org/item?id=AMBP_2017__24_2_225_0>

© Annales mathématiques Blaise Pascal, 2017, Certains droits réservés.

Cet article est mis à disposition selon les termes de la licence Creative Commons attribution – pas de modification 3.0 France.

http://creativecommons.org/licenses/by-nd/3.0/fr/

L’accès aux articles de la revue « Annales mathématiques Blaise Pascal » (http://ambp.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://ambp.cedram.org/legal/).

Publication éditée par le laboratoire de mathématiques Blaise Pascal de l’université Clermont Auvergne, UMR 6620 du CNRS

Clermont-Ferrand — France

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

Annales mathématiques Blaise Pascal24, 225-234 (2017)

Holomorphic extension of the de Gennes function

Virginie Bonnaillie-Noël Frédéric Hérau Nicolas Raymond

Abstract

This note is devoted to prove that the de Gennes function has a holomorphic extension on a half strip containingR+.

1. Introduction

1.1. About the de Gennes operator

The de Gennes operator plays an important role in the investigation of the magnetic Schrödinger operator. Consider the half-plane

R²+={(s, t)∈R² :t >0},

and a magnetic field B(s, t) = 1. We also introduce an associated vector potentialA(s, t) = (−t,0) such thatB(s, t) =∇×A(s, t). Then, we define L_Athe Neumann realization of the differential operator (−i∇+A)²acting on L²(R²+). By using the partial Fourier transform in s, we obtain the direct integral:

F L_AF⁻¹ = Z ⊕

R

L_ξdξ ,

where, forξ ∈R,L_ξ denotes the Neumann realization onR+ of the differ- ential operator −∂_t²+ (ξ−t)². The self-adjoint operator L_ξ is called the de Gennes operator with parameter ξ∈R. Let us give a precise definition of this operator. For ξ∈R, we introduce the sesquilinear form

∀ψ∈B¹(R+), q_ξ(ϕ, ψ) = Z

R+

ϕ⁰ψ⁰+ (t−ξ)²ϕψdt , where

B¹(R+) =ⁿψ∈H¹(R+) :tψ∈L²(R+)^o.

Keywords: de Gennes operator, holomorphic extension, holomorphic perturbation theory.

2010Mathematics Subject Classification:81Q15, 32A10.

This form is continuous on the Hilbert space B¹(R+) and Hermitian. Up to the addition of a constant, q_ξ is coercive on B¹(R+). Therefore, in virtue of the Lax–Milgram representation theorem (see for instance [4, Theorem 3.4]), we can consider the associated closed (and self-adjoint, since the form is Hermitian) operator L_ξ and its domain is

Dom(L_ξ) =ⁿψ∈B²(R+) :ψ⁰(0) = 0^o⊂B¹(R+), where

B²(R+) =ⁿψ∈H²(R+) :t²ψ∈L²(R+)^o.

Since B¹(R+) is compactly embedded in L²(R+), we deduce that L_ξ has compact resolvent and we can consider the non-decreasing sequence of its eigenvalues (µ_k(ξ))k∈N^∗. Each eigenspace is of dimension one (due to the Neumann condition for instance).

Definition 1.1. We call the functionR3ξ 7→ µ₁(ξ)∈R the de Gennes function. For shortness, we let µ=µ1.

In the terminology of Kato’s perturbation theory (see [6, Section VII.2]), the family of self-adjoint operators (L_ξ)_ξ∈Ris analytic of type (A). In other words, for all ξ0 ∈R, the family (L_ξ)_ξ∈R can be extended into a family of closed operators (L_ξ)_ξ∈V with V a complex open ball containing ξ₀ such that

• the domainDom(L_ξ) does not depend on ξ∈ V,

• for all ψ∈Dom(L_ξ0), the map V 3ξ 7→L_ξψ is holomorphic.

By using Kato’s theory and sinceµis a simple eigenvalue,µisrealanalytic, or equivalently, for all given ξ0 ∈R,µ has a local holomorphic extension in a neighborhood of ξ₀. We refer to [8, Section 2.4] for a direct proof. We also recall that µhas a minimum (see [2] or [3, Proposition 3.2.2]). We let Θ₀ = min

ξ∈R

µ(ξ).

This note answers the following question: Can the de Gennes function µ be holomorphically extended on a strip about R+?

1.2. Motivations and result

The aim of this note is to prove the following theorem.

Holomorphic extension

Theorem 1.2. There exist ε > 0 and F a holomorphic function on the half strip

S_ε:={ξ∈C:Reξ >0, |Imξ|< ε},

such that, for all ξ ∈R, µ(ξ) =F(ξ). Moreover, for allξ ∈S_ε, we have ReF(ξ)≥µ(Reξ)−(Imξ)²,

and F(ξ) belongs to the discrete spectrum ofL_ξ.

Remark 1.3. As we can see from the proof, for all k ∈ N^∗, µk has a holomorphic extension on a half strip aboutR+. Of course, the size of this strip depends on kand is expected to shrink when k becomes large.

Remark 1.4. The method used in this note can be applied to the family of Montgomery operators

−∂_t²+ ξ− tⁿ⁺¹ n+ 1

!2

,

acting on L²(R), with n ∈ N^∗ and ξ ∈ R. Their eigenvalues have holo- morphic extensions on a half strip about R+. The Montgomery operators appear in the case of vanishing magnetic fields.

This theorem is motivated by various spectral questions. Firstly, it plays an important role in the investigation of the resonances induced by local perturbations of L_A. Indeed, these resonances can often be defined by analytic dilations (see for instance [5, Chapter 16]) and the existence of a holomorphic extension of the band functions µk could be useful to reveal new magnetic spectral phenomena. Secondly, it is also strongly related to complex WKB analysis (as we have shown in [1]). In particular, the WKB constructions in [1, Section 1.2.2] area priori local and an accurate knowledge of the holomorphy strip would allow to extend the domain of validity of these constructions (see [1, Section 4.2] where the size of the holomorphy strip is involved). Moreover, this holomorphic extension is crucial in the study of the semiclassical magnetic tunneling effect when there are symmetries. Actually, this effect can only be fully understood in the complexified phase space.

2. Proof of the theorem

2.1. Preliminary considerations Let us prove the following separation lemma.

Lemma 2.1. For allk∈N^∗, there exists c_k>0 such that

∀ξ ∈R, µ_k+1(ξ)−µ_k(ξ)≥c_k.

Proof. We recall that the functions µk are real analytic and that, for all integerk, we haveµ_k< µ_k+1. As a consequence of the harmonic approxi- mation (see for instance [3, Section 7.1] or [8, Section 3.2]), we have

∀k∈N^∗, lim

ξ→+∞µ_k(ξ) = 2k−1. (2.1) Therefore, for all k∈N^∗, there exist Ξ_k, d_k>0 such that, for all ξ≥Ξ_k,

µ_k+1(ξ)−µ_k(ξ)≥d_k.

For α > 0, let us consider L_−α. By dilation, this operator is unitarily equivalent to the Neumann realization on R+ of the differential operator

α²(α⁻⁴D²_τ+ (τ + 1)²),

where we denoteDt=−i∂_t. The potential R+3τ 7→(τ + 1)² is minimal at τ = 0 and thus a variant of the harmonic approximation near τ = 0 shows that (see for instance [7, Théorème 1.7])

∀k∈N^∗, µ_k(−α) =

α→+∞α²+ν_kα²³ +o(α²³), (2.2) where (νk)_k∈N^∗ is the increasing sequence of the eigenvalues of the Neu- mann realization onR+ ofD_τ²+ 2τ. Let us briefly recall the main steps of the proof of (2.2). The regime α→+∞ is equivalent to the semiclassical regime h = α⁻² → 0 and one knows that the eigenfunctions associated with the low lying eigenvalues are concentrated near the minimum of the potential τ = 0. Near this point, we can perform a Taylor expansion so that the operator asymptotically becomes

α²(α⁻⁴D²_τ+ 1 + 2τ).

Then, we homogenize this operator with the rescaling τ = h²³τ˜ and the conclusion follows.

We deduce that

α→+∞lim (µ_k+1(−α)−µ_k(−α)) = +∞. (2.3)

Holomorphic extension

Combining (2.1), (2.3), the simplicity of the µ_k and their continuity, the

result follows.

2.2. The family (L_ξ)_ξ∈Sε

Let us fixε >0 and explain how the operatorL_ξ is defined forξ∈Sε. We let, for allξ ∈S_ε,

∀ϕ, ψ∈B¹(R+), qξ(ϕ, ψ) = Z

R+

ϕ⁰ψ⁰+ (t−ξ)²ϕψdt .

The sesquilinear form q_ξ is well defined and continuous on B¹(R+). In order to apply the Lax–Milgram representation theorem, let us consider the quadratic form B¹(R+)3ψ7→Req_ξ(ψ, ψ). We have

Reqξ(ψ, ψ) =kψ⁰k²+ Z

R+

Re(t−ξ)²|ψ|²dt , and

Re(t−ξ)²= (t−Reξ)²−(Imξ)². In particular, we have

Reqξ(ψ, ψ)≥qReξ(ψ, ψ)−(Imξ)²kψk² ≥(µ(Reξ)−(Imξ)²)kψk²

≥(Θ₀−(Imξ)²)kψk². (2.4) Thus, up to the addition of constant, Req_ξ is coercive on B¹(R+). We deduce that, for all ε > 0 and all ξ ∈S_ε, there exist C, c >0 such that, for all ψ∈B¹(R+),

q_ξ(ψ, ψ) +Ckψk²≥ckψk²_B1(R+).

Therefore, we can apply the Lax–Milgram theorem and define a closed operatorL_ξ. It satisfies

∀ψ∈B¹(R+),∀ϕ∈Dom(L_ξ), q_ξ(ϕ, ψ) =hL_ξϕ, ψi. We can easily show that

Dom(L_ξ) =ⁿψ∈B²(R+) :ψ⁰(0) = 0^o,

so that L_ξ has compact resolvent and (L_ξ)_ξ∈Sε is a holomorphic family of type (A). If ε∈ (0,√

Θ₀), as a consequence of (2.4), we get that, for all ξ ∈S_ε,L_ξ is bijective. Indeed, we have, for all ψ∈Dom(L_ξ),

(Θ₀−ε²)kψk ≤ kL_ξψk.

From this, the operator L_ξ is injective with closed range. Since L^∗_ξ =L_ξ, we deduce that L^∗_ξ is also injective and thusL_ξhas a dense image. We get that, for all ξ ∈Sε,

kL⁻¹_ξ k ≤(Θ₀−(Imξ)²)⁻¹. 2.3. Resolvent and projection estimates Let us now prove the main result of this note.

2.3.1. Difference of resolvents

For 0< r₁ < r₂, andz₀ ∈C, we define the annulus A_r1,r2(z₀) ={z∈C:r₁<|z−z₀|< r₂}. We consider, for all r >0 and all ξ∈Sε,

A_r,ξ :=Ar,2r(µ(Reξ)).

Using Lemma 2.1 withk= 1, we know that there exists r₀ >0 such that

• for all ξ∈Sε,

A_r0,ξ ⊂ρ(LReξ), dist(A_r0,ξ,sp(LReξ))≥r0 >0, whereρ(L_Reξ) denotes the resolvent set ofL_Reξ,

• the disk of centerµ(Reξ) with radiusr₀, denoted byD(µ(Reξ), r₀), contains only one eigenvalue of L_Reξ.

The following proposition states an approximation of the resolvent of L_ξ by the one of L_Reξ when εgoes to 0.

Proposition 2.2. There exist C, ε₀ >0 such that for all ε∈(0, ε₀) and all ξ ∈ Sε, we have A_r0,ξ ⊂ ρ(Lξ). In addition, we have, for all ξ ∈ Sε

and z∈ A_r0,ξ,

(L_ξ−z)⁻¹−(L_Reξ−z)⁻¹≤Cε . (2.5) Proof. For allz∈ A_r0,ξ, we have

L_ξ−z=L_Reξ−2iImξ(t−Reξ)−(Imξ)²−z , or, equivalently,

L_ξ−z= (Id−2iImξ(t−Reξ)(LReξ−z)⁻¹−(Imξ)²(L_Reξ−z)⁻¹)(L_Reξ−z).

Holomorphic extension

Thus, we have to show that

Id−2iImξ(t−Reξ)(LReξ−z)⁻¹−(Imξ)²(L_Reξ−z)⁻¹ is bijective. By the spectral theorem, we get

k(L_Reξ−z)⁻¹k ≤ 1

dist(z,sp(LReξ)) ≤ 1 r0

. (2.6)

Let us also estimate (t−Reξ)(L_Reξ−z)⁻¹. Take u∈L²(R+) and let v= (L_Reξ−z)⁻¹u .

We have

(D_t²+ (t−Reξ)²−z)v= (L_Reξ−z)v=u ,

and we must estimate (t−Reξ)v. Taking the scalar product withv, we get

k(t−Reξ)vk²−Rezkvk² ≤ hv, ui ≤ kukkvk ≤r₀⁻¹kuk².

Since the de Gennes function satisfies µ(Reξ)≤1 when Reξ ≥0 (see [3, Proposition 3.2.2]), we get, for all z∈ A_r0,ξ,

Rez≤µ(Reξ) + 2r0≤1 + 2r0. We find

k(t−Reξ)(LReξ−z)⁻¹k ≤ q

3r⁻¹₀ +r⁻²₀ =:r1. (2.7) From (2.6) and (2.7), it follows that

k −2iImξ(t−Reξ)(L_Reξ−z)⁻¹−(Imξ)²(L_Reξ−z)⁻¹k

≤2|Imξ|r₁+|Imξ|²r⁻¹₀ ≤2εr₁+ (εr⁻

1 2

0 )². Thus, there exists ε0 > 0 such that for all ε ∈ (0, ε0), we have, for all ξ ∈S_ε andz∈ A_r0,ξ,

k −2iImξ(t−Reξ)(L_Reξ−z)⁻¹−(Imξ)²(L_Reξ−z)⁻¹k<1. This shows that Id−2iImξ(t−Reξ)(LReξ−z)⁻¹−(Imξ)²(L_Reξ−z)⁻¹ is bijective and thus L_ξ−z is bijective. The estimate of the difference of

the resolvents immediately follows.

2.3.2. Projections Let us now introduce

P_ξ = 1 2iπ

Z

Γξ

(z−L_ξ)⁻¹dz ,

where Γ_ξ is the circle of center µ(Reξ) and radius ³₂r0. It is classical to show, by using the resolvent and Cauchy formulas, thatP_ξis the projection on the characteristic space associated with the eigenvalues of L_ξ enlaced by Γ_ξ.

Lemma 2.3. There exists ε0 > 0 such that for all ε ∈ (0, ε0) and all ξ ∈S_ε, the rank of P_ξ is1 and P_ξ is holomorphic on S_ε.

Proof. Let us fixξ₀ ∈S_ε. There existsδ >0 such that, for allξ∈ D(ξ₀, δ), we have ξ ∈Sε, Γ_ξ0 ⊂ A_r0,ξ and

P_ξ = 1 2iπ

Z

Γξ0

(z−L_ξ)⁻¹dz .

This comes from the continuity of the family of circles Γ_ξ and the holo- morphy of the resolvent z7→(z−L_ξ)⁻¹. From this, we now see thatP_ξis holomorphic on the diskD(ξ₀, δ). Indeed, since (L_ξ)_ξ∈Sε is a holomorphic family of type (A),D(ξ₀, δ)3ξ7→(z−L_ξ)⁻¹ is holomorphic, uniformly for z∈Γ_ξ0. Finally, with (2.5), there existsε0 >0 such that, for allε∈(0, ε0) and all ξ∈S_ε,

kP_ξ−P_Reξk ≤ 3r₀

2 Cε <1.

By a classical lemma about pairs of projections (see [6, Section I.4.6]) and since the rank of P_Reξ is 1, we deduce that the rank ofP_ξ is 1.

Remark 2.4. We can also choose ε₀ > 0 so that, for all ε ∈ (0, ε₀), all ξ ∈S_ε and allψ∈L²(R+),

Z

R+

(P_ξψ)²dt− Z

R+

(P_Reξψ)²dt

≤ 1 2kψk². 2.3.3. Conclusion

Since P_ξ commutes with L_ξ, we can consider the restriction of L_ξ to the range of Pξ. This restriction is a linear map in dimension one. Therefore,

Holomorphic extension for all ξ ∈Sε, there exists a unique F(ξ)∈Csuch that

∀ψ∈L²(R+), L_ξ(P_ξψ) =F(ξ)P_ξψ .

Taking the scalar product with P_ξψ, we get, for all ψ ∈ L²(R+) and all ξ ∈Sε,

F(ξ) Z

R+

(Pξψ)²dt= Z

R+

L_ξ(Pξψ)Pξψdt .

Let ξ₀ ∈Sε. By Remark 2.4, there exists ψ₀ ∈L²(R+) (a real normalized eigenfunction of L_Reξ0 associated with µ(Reξ₀)) such that

Z

R+

(P_ξ0ψ0)²dt6= 0.

Thus there exists a neighborhood V of ξ₀ such that, for allξ ∈ V, F(ξ) =

R

R+L_ξ(Pξψ0)Pξψ0dt R

R+(P_ξψ0)²dt .

ThusF is holomorphic nearξ0. Whenξ∈R, we have clearlyF(ξ) =µ(ξ).

This, with (2.4), terminates the proof of Theorem 1.2.

References

[1] V. Bonnaillie-Noël, F. Hérau&N. Raymond– “Magnetic WKB constructions”, Arch. Ration. Mech. Anal. 221 (2016), no. 2, p. 817–

891.

[2] M. Dauge&B. Helffer– “Eigenvalues variation. I. Neumann prob- lem for Sturm-Liouville operators”, J. Differ. Equations 104 (1993), no. 2, p. 243–262.

[3] S. Fournais & B. Helffer – Spectral methods in surface super- conductivity, Progress in Nonlinear Differential Equations and their Applications, vol. 77, Birkhäuser, Boston, MA, 2010.

[4] B. Helffer – Spectral theory and its applications, Cambridge Stud- ies in Advanced Mathematics, vol. 139, Cambridge University Press, Cambridge, 2013.

[5] P. D. Hislop&I. M. Sigal–Introduction to spectral theory, Applied Mathematical Sciences, vol. 113, Springer, 1996, With applications to Schrödinger operators.

[6] T. Kato– Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, vol. 132, Springer, 1966.

[7] N. Popoff – “Sur l’opérateur de schrödinger magnétique dans un domaine diédral”, Thèse, Université de Rennes 1 (France), 2012.

[8] N. Raymond – Bound states of the magnetic schrödinger operator, EMS Tracts in Mathematics, vol. 27, European Mathematical Society, 2017.

Virginie Bonnaillie-Noël Département de mathématiques et applications,

École normale supérieure, CNRS PSL Research University 75005 Paris, France bonnaillie@math.cnrs.fr

Frédéric Hérau LMJL - UMR6629

Université de Nantes, CNRS 2 rue de la Houssinière, BP 92208 44322 Nantes cedex 3, France herau@univ-nantes.fr

Nicolas Raymond

IRMAR, Univ. Rennes 1, CNRS Campus de Beaulieu

35042 Rennes cedex, France nicolas.raymond@univ-rennes1.fr