David Krejcirik
email: david.krejcirik@fjfi.cvut.cz

Nicolas Raymond
email: nicolas.raymond@univ-rennes1.fr

AND Julien Royer
email: julien.royer@math.univ-toulouse.fr

Petr Siegl
email: petr.siegl@math.unibe.ch

D Krej Či Ří K

Reduction of

Keywords: 2010 Mathematics Subject Classification. Primary: 81Q15; Secondary: 35P05, 35P20, 58J50, 81Q10, 81Q20 reduction of dimension, norm resolvent convergence, Born-Oppenheimer approximation, thin layers, quantum waveguides, effective Hamiltonian

come

1. Introduction 1.1. Motivation and context. In this paper we develop an abstract tool for dimensional reductions via the norm resolvent convergence obtained from variational estimates. The results are relevant in particular for PDE problems, typically Schrödinger-type operators depending on an asymptotic parameter having various interpretations (semiclassical limit, shrinking limits, large coupling limit, etc.). In applications, our resolvent estimates lead to accurate spectral asymptotic results for eigenvalues lying in a suitable region of the complex plane. Moreover, avoiding the traditional min-max approach, with its fundamental limitations to self-adjoint cases, we obtain an effective operator, the spectrum of which determines the spectral asymptotics. The flexibility of the latter is illustrated on a non-self-adjoint example in the second part of the paper.

The power of our approach is demonstrated by a unified treatment of diverse classical as well as latest problems occurring in mathematical physics such as: -semiclassical Born-Oppenheimer approximation, -shrinking tubular neighborhoods of hypersurfaces subject to various boundary conditions, -domains with very attractive Robin boundary conditions. In spite of the variety of operators, asymptotic regimes, and techniques considered in the previous literature, all these results are covered in our general abstract and not only asymptotic setting. Our first result (Theorem 1.1) gives a norm resolvent convergence towards a tensorial operator in a general self-adjoint setting. A remarkable feature is that only two quantities need to be controlled: the size of a commutator of a "longitudinal operator" with spectral projection on low lying "transverse modes" and the size of the "spectral gap" of a "transverse operator", see (1.5) and (1.2), respectively. Although the latter is also very natural it was hardly visible in existing literature due to many seemingly different technical steps as well as various ways how these quantities enter. As particular cases of the application of Theorem 1.1, we recover, in a short manner, known results for quantum waveguides (see for instance [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF], [START_REF] Krejčiřík | The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions[END_REF], [START_REF] Krejčiřík | Magnetic effects in curved quantum waveguides[END_REF] or [START_REF] Krejčiřík | The magnetic Laplacian in shrinking tubular neighborhoods of hypersurfaces[END_REF]) and cast a new light on Born-Oppenheimer type results (see [START_REF] Lampart | The adiabatic limit of the Laplacian on thin fibre bundles[END_REF], [START_REF] Wachsmuth | Effective Hamiltonians for constrained quantum systems[END_REF], [START_REF] Jecko | On the mathematical treatment of the Born-Oppenheimer approximation[END_REF] or [START_REF] Raymond | Bound States of the Magnetic Schrödinger Operator[END_REF]Sec. 6.2]). To keep the presentation short we deliberately do not strive for the weakest possible assumptions in examples, although the abstract setting allows for many further generalizations and it clearly indicates how to proceed.

In the second part of the paper, we prove, in the same spirit as previous results, the norm convergence result for a non-self-adjoint Robin Laplacian, see Theorem 1.5. It will partially generalize previous works in the self-adjoint (see [START_REF] Pankrashkin | An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter[END_REF], [START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF] and [START_REF] Oliveira | Effective operators for Robin Laplacian in thin two-and threedimensional curved waveguides[END_REF]) and in the non-self-adjoint (see [START_REF] Borisov | The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions[END_REF]) cases.

As a matter of fact, the crucial step in all the proofs of the paper is an abstract lemma (see Lemma 1.7) of an independent interest. It provides a norm resolvent estimate from variational estimates, which is particularly suitable for the analysis of operators defined via sesquilinear forms.

1.2. Reduction of dimension in an abstract setting and self-adjoint applications. We first describe the reduction of dimension for an operator of the form

L " S ˚S `T, T " à sPΣ T s , (1.1)
acting on the Hilbert space H " À sPΣ H s . The norm and inner product in H will be denoted by } ¨} and x¨, ¨y, respectively; the latter is assumed to be linear in the second argument.

Here Σ is a measure space and T s is a self-adjoint non-negative operator on a Hilbert space H s for all s P Σ. Precise definitions will be given in Section 2. A typical example is the Schrödinger operator H " p´i B s q 2 `p´iB t q 2 `V ps, tq , acting on L 2 pR s ˆRt q. We consider a function s Þ Ñ γ s such that

γ " inf sPΣ γ s ą 0 . (1.2)
Then we denote by Π s P LpH s q the spectral projection of T s on r0, γ s q, and we set Π K s " Id Hs ´Πs . We denote by Π the bounded operator on H such that for Φ P H and s P Σ we have pΠΦq s " Π s Φ s . We similarly define Π K P LpHq. Our purpose is to compare some spectral properties of the operator L with those of the simpler operator

L eff " ΠL Π. (1.3)
This is an operator on ΠH with domain ΠH X DompL q.

In fact, we will first compare L with

x L " ΠL Π `ΠK L Π K . (1.4)
Then L eff and L K will be defined as the restrictions of x L to ΠH and Π K H, respectively, so that x L " L eff ' L K .

We will give a sufficient condition for z P ρp x L q to be in ρpL q and, in this case, an estimate for the difference of the resolvents. Then, since ΠH and Π K H reduce x L , it is not difficult to check that far from the spectrum of L K the spectral properties of x L are the same as those of L eff , so we can state a similar statement with x L replaced by L eff . In applications, we can for instance prove that the first eigenvalues of L are close to the eigenvalues of the simpler operator L eff .

We assume that DompSq is invariant under Π, that rS, Πs extends to a bounded operator on H, and we set a " }rS, Πs} LpHq ? γ .

(1.5)

For z P C, we also define

η 1 pzq " 3 ? 2 a 2 γ `6a ? 2 p1 `aq|z| `3a γ ? 2 ˆ2 `a ? 2 ˙|z| 2 , η 2 pzq " 3a ? 2 p1 `aq `3a γ ? 2 ˆ2 `a ? 2 ˙|z| , η 3 pzq " 3a ? 2 ˆ1 `a ? 2 ˙`3a γ ? 2 ˆ2 `a ? 2 ˙|z| , η 4 pzq " 3a γ ? 2 ˆ2 `a ? 2 ˙.
(1.6)

Theorem 1.1. Let z P ρp x L q. If 1 ´η1 pzq}p x L ´zq ´1} ´η2 pzq ą 0, then z P ρpL q and }pL ´zq ´1 ´p x L ´zq ´1} ď η 1 pzq}pL ´zq ´1}}p x L ´zq ´1} `η2 pzq}pL ´zq ´1} `η3 pzq}p x L ´zq ´1} `η4 pzq.
In particular, }pL ´zq ´1} ď pη 3 pzq `1q}p x L ´zq ´1} `η4 pzq

1 ´η1 pzq}p x L ´zq ´1} ´η2 pzq .
In order to compare the resolvent of L to the resolvent of L eff , this theorem is completed by the following easy estimate: Proposition 1.2. We have Spp x L q " SppL eff q Y SppL K q and, for z P ρp x L q such that z R rγ, `8q,

› › ›p x L ´zq ´1 ´pL eff ´zq ´1Π › › › ď 1 distpz, rγ, `8qq
.

In this estimate, it is implicit that pL eff ´zq ´1 is composed on the left by the inclusion ΠH Ñ H.

Remark 1.3. These results cover a wide range of situations. In Section 3, we will discuss three paradigmatic applications. The space Σ will be R or a submanifold of R d , d ě 2. The set H s is fixed, but the Hilbert structure thereon may depend on s. In our examples pT s q sPΣ is related to an analytic family of self-adjoint operators which are not necessarily non-negative. Nevertheless, under suitable assumptions, we can reduce ourselves to the non-negative case. Indeed, in our applications, for all s P Σ, T s is bounded from below, independently of s P Σ. Moreover, the bottom of the spectrum of T s will be an isolated simple eigenvalue µ 1 psq. Then, we notice that inf sPΣ µ 1 psq is well-defined and that T s ´inf sPΣ µ 1 psq is non-negative. We denote by u 1 psq a corresponding eigenfunction. We can assume that }u 1 psq} H " 1 for all s P Σ and that u 1 is a smooth function of s. Π s is the projection on u 1 psq and ΠH can be identified with L 2 pΣq via the map ϕ Þ Ñ ps Þ Ñ ϕpsqu 1 psqq. In particular L eff can be seen as an operator on L 2 pΣq, which is what is meant by the "reduction of dimension". Finally, γ s is defined as the bottom µ 2 psq ´inf sPΣ µ 1 psq of the remaining part of the spectrum and

γ " inf s µ 2 psq ´inf s µ 1 psq ď inf SpppL ´inf sPΣ µ 1 psqq K q . (1.7)
We recall that we assume the spectral gap condition γ ą 0, see (1.2).

1.3. The Robin Laplacian in a shrinking layer as a non-self-adjoint application. We now consider a reduction of dimension result in a non-self-adjoint setting, namely the Robin Laplacian in a shrinking layer. Let d ě 2. Here, Σ is an orientable smooth (compact or noncompact) hypersurface in R d without boundary. The orientation can be specified by a globally defined unit normal vector field n : Σ Ñ S d´1 . Moreover Σ is endowed with the Riemannian structure inherited from the Euclidean structure defined on R d . We assume that Σ admits a tubular neighborhood, i.e. for ε ą 0 small enough the map

Θ ε : ps, tq Þ Ñ s `εtnpsq (1.8)
is injective on Σ ˆr´1, 1s and defines a diffeomorphism from Σ ˆp´1, 1q to its image. We set Ω " Σ ˆp´1, 1q and Ω ε " Θ ε pΩq .

(1.9)

Then Ω ε has the geometrical meaning of a non-self-intersecting layer delimited by the hypersurfaces Σ ˘,ε " Θ ε pΣ ˆt˘1uq . Moreover Σ ˘,ε can be identified with Σ via the diffeomorphisms Θ ˘,ε :

" Σ Ñ Σ ˘,ε s Þ Ñ s ˘εnpsq .
Let α : Σ Ñ C be a smooth bounded function. We set α ˘,ε " α ˝Θ´1 ˘,ε : Σ ˘,ε Ñ C and we consider on L 2 pΩ ε q the closed operator P ε,α (or simply P ε if no risk of confusion) defined as the usual Laplace operator on Ω ε subject to the Robin boundary condition Bu Bn `α˘,ε u " 0, on Σ ˘,ε .

(1.10)

Remark 1.4. Note that a very special choice of Robin boundary conditions is considered in this section. Indeed, the boundary-coupling functions considered on Σ `,ε and Σ ´,ε are the same except for a switch of sign, see (4.1). More specifically, α ˘,ε psq " αpsq for every s P Σ and n is an outward normal to Ω ε on one of the connected parts Σ ˘,ε of the boundary BΩ ε , while it is inward pointing on the other boundary. This special choice is motivated by Parity-Time-symmetric waveguides [START_REF] Borisov | PT -symmetric waveguides[END_REF][START_REF] Borisov | The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions[END_REF] as well as by a self-adjoint analogue considered in [START_REF] Oliveira | Effective operators for Robin Laplacian in thin two-and threedimensional curved waveguides[END_REF]. It is straightforward to extend the present procedure to the general situation of two different boundary-coupling functions on Σ `,ε and Σ ´,ε , but then the effective operator will be ε-dependent (in analogy with the Dirichlet boundary conditions, see Proposition 3.4) or a renormalization would be needed (cf. [START_REF] Krejčiřík | The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions[END_REF]).

Our purpose is to prove that, at the limit when ε goes to 0, the operator P ε converges in a norm-resolvent sense to a Schrödinger operator L eff " ´∆Σ `Veff , on Σ. Here ´∆Σ is the Laplace-Beltrami operator on Σ, and the potential V eff depends both on the geometry of Σ and on the boundary condition. More precisely we have

V eff " |α| 2 ´2αRepαq ´αpκ 1 `¨¨¨`κ d´1 q.
(1.11)

Note that the sum of the principal curvatures is proportional to the mean curvature of Σ. Notice also that H eff defines an (unbounded) operator on the Hilbert space L 2 pΣq. In particular P ε and H eff do not act on the same space.

We denote by Π P LpL 2 pΩqq the projection on functions which do not depend on t: for u P L 2 pΩq and ps, tq P Ω we set pΠuqps, tq " 1 2

ż 1
´1 ups, θqdθ . Then we define Π K " Id ´Π.

Theorem 1.5. Let K be a compact subset of ρpH eff q. Then there exists ε 0 ą 0 and C ě 0 such that for z P K and ε P p0, ε 0 q we have z P ρpH ε q and

}pP ε ´zq ´1 ´U ´1 ε pL eff ´zq ´1ΠU ε } LpL 2 pΩεqq ď Cε .
Here U ε is a unitary transformation from L 2 pΩ ε , dxq to L 2 pΩ, w ε pxqdσdtq, where for some C ą 1 we have @ε P p0, ε 0 q, @x P Ω,

1 C ď |w ε pxq| ď C .
As for Theorem 1.1 it is implicit that the resolvent pL eff ´zq ´1 is composed on the left by the inclusion ΠL 2 pΩ ε q Ñ L 2 pΩ ε q. Moreover the operator L eff on L 2 pΣq has been identified with an operator on ΠL 2 pΩ ε q.

Remark 1.6. In the geometrically trivial situation Σ " R d´1 and special choice Repαq " 0, a version of Theorem 1.5 was previously established in [START_REF] Borisov | The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions[END_REF]. At the same time, in the self-adjoint case Impαq " 0 and very special geometric setting d " 1 (Σ being a curve), a version of Theorem 1.5 is due to [START_REF] Oliveira | Effective operators for Robin Laplacian in thin two-and threedimensional curved waveguides[END_REF]. In our general setting, it is interesting to see how the geometry enters the effective dynamics, through the mean curvature of Σ, see (1.11).

1.4. From variational estimates to norm resolvent convergence. All the results of this paper are about estimates of the difference of resolvents of two operators. These estimates will be deduced from the corresponding estimates of the associated quadratic forms by the following general lemma: Lemma 1.7. Let K be a Hilbert space. Let A and p A be two closed densely defined operators on K. Assume that p A is bijective and that there exist η 1 , η 2 , η 3 , η 4 ě 0 such that 1´η 1 } p A ´1}´η 2 ą 0 and

@φ P DompAq, @ψ P Domp p A ˚q , |xAφ, ψy ´xφ, p A ˚ψy| ď η 1 }φ}}ψ} `η2 }φ}} p A ˚ψ} `η3 }Aφ}}ψ} `η4 }Aφ}} p A ˚ψ} .
Then A is injective with closed range. If moreover A ˚is injective, then A is bijective and we have the estimates

}A ´1} ď pη 3 `1q} ´1 } `η4 1 ´η1 } p A ´1} ´η2 (1.12) and › › ›A ´1 ´p A ´1› › › ď η 1 }A ´1}} p A ´1} `η2 }A ´1} `η3 } p A ´1} `η4 .
Since the proof is rather elementary, let us provide it already now.

Proof. Let φ P DompAq and consider ψ " p p A ´1q ˚φ P Domp p A ˚q. We have

|}φ} 2 ´xAφ, p p A ´1q ˚φy| " |xφ, p A ˚ψy ´xAφ, ψy| ď pη 1 } p A ´1} `η2 q}φ} 2 `´η 3 } p A ´1} `η4 ¯}Aφ}}φ} , so }φ} 2 ď ´η1 } ´1 } `η2 ¯}φ} 2 `´pη 3 `1q} ´1 } `η4 ¯}φ}}Aφ}. Then if η 1 } p A ´1} `η2 ă 1 , we get }φ} ď pη 3 `1q} ´1 } `η4 1 ´η1 } p A ´1} ´η2 }Aφ} . (1.13)
In particular, A is injective with closed range. If A ˚is injective, the range of A is dense and thus A is bijective. In particular, with (1.13), we obtain (1.12). Finally for f, g P K, φ " A ´1f and ψ " p p A ´1q ˚g we have

x `A´1 ´p A ´1˘f
, gy " xφ, p A ˚ψy ´xAφ, ψy, and the conclusion follows by easy manipulations.

1.5. Organization of the paper. In Section 2, we prove Theorem 1.1. We first define the operators L , x L and L eff , and then we show how Lemma 1.7 can be applied. In Section 3, we discuss some applications of Theorem 1.1 to the semiclassical Born-Oppenheimer approximation, the Dirichlet Laplacian on a shrinking tubular neighborhood of an hypersurface and the Robin Laplacian in the large coupling limit. Section 4 is devoted to the proof of Theorem 1.5 about the non-self-adjoint Robin Laplacian on a shrinking layer.

Abstract reduction of dimension

In this section we describe more precisely the setting introduced in Section 1.2 and we prove Theorem 1.1. The applications will be given in the following section.

2.1. Definition of the effective operator. Let pΣ, σq be a measure space. For each s P Σ we consider a separable complex Hilbert space H s . Then, on H s we consider a closed symmetric nonnegative sesquilinear form q s with dense domain Dompq s q. We denote by T s the corresponding self-adjoint and non-negative operator, as given by the Representation Theorem. As already said in Section 1.2, we consider a function s P Σ Þ Ñ γ s P R whose infimum is positive, see (1.2). Then we denote by Π s P LpH s q the spectral projection of T s on r0, γ s q, and we set Π K s " Id Hs ´Πs . We denote by H the subset of À sPΣ H s which consists of all Φ " pΦ s q sPΣ such that the functions s Þ Ñ }Φ s } Hs and s Þ Ñ }Π s Φ s } Hs are measurable on Σ and

}Φ} 2 " ż Σ }Φ s } 2
Hs dσpsq ă `8 .

It is endowed with the Hilbert structure given by this norm. We denote by Π the bounded operator on H such that for Φ P H and s P Σ we have pΠΦq s " Π s Φ s . We similarly define Π K P LpHq.

We say that Φ " pΦ s q sPΣ P H belongs to DompQ T q if Φ s belongs to Dompq s q for all s P Σ, the functions s Þ Ñ q s pΦ s q and s Þ Ñ q s pΠ s Φ s q are measurable on Σ and

Q T pΦq " ż Σ q s pΦ s qdσpsq ă `8 .
We consider on H an operator S with dense domain DompSq. We assume that DompSq is invariant under Π, that rS, Πs extends to a bounded operator on H, and we define a as in (1.5). We assume that DompQq " DompSq X DompQ T q is dense in H, and for Φ P DompQq we set QpΦq " }SΦ} 2 `QT pΦq .

(2.1)

We assume that Q defines a closed form on H. The form Q is symmetric and non-negative and the associated operator is the operator L introduced in (1.1).

Then we define the operator x L (see (1.4)) by its form. For this we need to verify that the form domain is left invariant both by Π and Π K . Lemma 2.1. For all Φ P DompQq we have ΠΦ P DompQq and Π K Φ P DompQq.

Proof. Let Φ " pΦ s q sPΣ P DompQq. We have Φ P DompSq, so by assumption we have ΠΦ P DompSq. By assumption again, the function s Þ Ñ q s pΠ s Φ s q " q s pΠ s Π s Φ s q is measurable and we have ż

Σ q s pΠ s Φ s q dσpsq ď sup sPΣ γ s ż Σ }Φ s } 2
Hs dσpsq ă `8. This proves that ΠΦ belongs to DompQ T q, and hence to DompQq. Then the same holds for Π K Φ " Φ ´ΠΦ.

With this lemma we can set, for Φ, Ψ P DompQq, p QpΦ, Ψq " QpΠΦ, ΠΨq `QpΠ K Φ, Π K Ψq .

Lemma 2.2. For all Φ P Domp p Qq we have QpΦq ď 2 p QpΦq .

In particular the form p Q is non-negative, closed, and it determines uniquely a self-adjoint operator x L on H. Moreover we have rΠ, x L s " 0 on Domp x L q.

Proof. We have QpΦq ´p QpΦq " QpΠΦ, Π K Φq `QpΠ K Φ, ΠΦq . Since the form Q is non-negative we can apply the Cauchy-Schwarz inequality to write

QpΠΦ, Π K Φq ď a QpΠΦq a QpΠ K Φq ď 1 2 `QpΠΦq `QpΠ K Φq ˘" 1 2 p QpΦq .
We have the same estimate for QpΠ K Φ, ΠΦq, and the first conclusions follow. We just check the last property about the commutator. Let ψ P Domp x L q. For all φ P Domp x L q we have p Qpφ, Πψq " QpΠφ, Πψq " p QpΠφ, ψq " xΠφ, x L ψy H " xφ, Π x L ψy H .

This proves that Πψ P Domp x L q with x L Πψ " Π x L ψ and the proof is complete.

Then, from p Q it is easy to define the forms corresponding to the operators L eff and L K :

Lemma 2.3. Let Q eff be the restriction of Q to ΠDompQq " RanpΠq X DompQq. Then Q eff is non-negative and closed. The associated operator L eff is self-adjoint, its domain is invariant under Π, and rΠ, L eff s " 0 on DompL eff q. Moreover, we have pDomp x L q X RanpΠq, x L q " pDompL eff q, L eff q.

We have similar statements for the restriction Q K of Q to Π K DompQq " RanpΠ K q X DompQq and the corresponding operator L K .

Proof. The closedness of Q eff comes from the closedness of Q and the continuity of Π. The other properties are proved as for Lemma 2.2. We prove the last assertion. Let ψ P DompL eff q. By definition of this domain we have Πψ " ψ. For φ P Domp p Qq, we have p Qpφ, ψq " QpΠφ, Πψq "

Q eff pΠφ, Πψq " Q eff pΠφ, ψq " Πφ, L eff ψ " φ, L eff ψ .
This proves that ψ P Domp x L q and L eff ψ " x L ψ. Thus DompL eff q Ă Domp x L q X RanpΠq and x L " L eff on DompL eff q. The reverse inclusion Domp x L q X RanpΠq Ă DompL eff q is easy, so the proof is complete.

Finally we have proved that

Domp x

L q " `Domp x L q X RanpΠq ˘' `Domp x L q X RanpΠ K q ˘" DompL eff q ' DompL K q and for ϕ P Domp x L q we have x L ϕ " L eff Πϕ `L K Π K ϕ. From the spectral theorem, we deduce the following lemma.

Lemma 2.4. We have Spp x L q " SppL eff q Y SppL K q and, for z P ρp x L q such that z R rγ, `8q,

› › ›p x L ´zq ´1 ´pL eff ´zq ´1Π › › › ď 1 distpz, rγ, `8qq .

2.2.

Comparison of the resolvents. This section is devoted to the proof of the following theorem that implies Theorem 1.1 via Lemma 1.7.

Theorem 2.5. Let L and x L be as above. Let z P C and η 1 pzq, η 2 pzq, η 3 pzq, η 4 pzq as in (1.6). Then for Φ P DompL q and Ψ P Domp x L ˚q we have ˇˇQpΦ, Ψq ´p QpΦ, Ψq ˇˇď η 1 pzq }Φ} }Ψ} `η2 pzq}Φ}}p x L ´zqΨ} `η3 pzq}pL ´zqΦ}}Ψ} `η4 pzq}pL ´zqΦ}}p x L ´zqΨ} .

Theorem 2.5 is a consequence of the following proposition after inserting z and using the triangular inequality.

Proposition 2.6. For all Φ P DompL q and Ψ P Domp x L q we have

1 γ |QpΦ, Ψq ´p QpΦ, Ψq| ď 3a ? 2 ˆ}Φ} `}L Φ} γ ˙} x L Ψ} γ `3a ? 2 ˆa}Φ} `ˆ1 `a ? 2 ˙}L Φ} γ ˙˜}Ψ} `} x L Ψ} γ ¸.
Proof. Let ν " }rS, Πs}. We have QpΦ, Ψq ´p QpΦ, Ψq " QpΠ K Φ, ΠΨq `QpΠΦ, Π K Ψq .

For the first term we write

QpΠ K Φ, ΠΨq " xSΠ K Φ, SΠΨy " xSΠ K Φ, rS, ΠsΠΨy `xSΠ K Φ, ΠSΠΨy , so that QpΠ K Φ, ΠΨq " xSΠ K Φ, rS, ΠsΠΨy `xrS, Π K sΠ K Φ, ΠSΠΨy . We deduce that |QpΠ K Φ, ΠΨq| ď ν}SΠ K Φ}}Ψ} `ν}Π K Φ}}SΠΨ} . (2.
2) Similarly, we get, by slightly breaking the symmetry, |QpΠΦ, Π K Ψq| ď ν}SΠ K Ψ}}Φ} `ν}Π K Ψ}}SΦ} .

(2.3)

We infer that |QpΦ, Ψq ´p QpΦ, Ψq| ď ν}SΠ K Φ}}Ψ} `ν}Π K Φ}}SΠΨ} `ν}SΠ K Ψ}}Φ} `ν}Π K Ψ}}SΦ} . (2.4)

Since Q T is non-negative we have

}SΦ} 2 ď QpΦq ď }L Φ}}Φ}. (2.5)
Similarly, }SΠΨ} 2 ď p QpΨq ď } x L Ψ}}Ψ} .

(2.6) Then we estimate }Π K Φ} and }SΠ K Φ}. We have

xΠ K Φ, L Φy " QpΠ K Φ, Φq " QpΠ K Φq `QpΠ K Φ, ΠΦq ,

and deduce

QpΠ K Φq ď }L Φ}}Π K Φ} `|QpΠ K Φ, ΠΦq| . From (2.3), we get

QpΠ K Φq ď }L Φ}}Π K Φ} `ν}SΠ K Φ}}Φ} `ν}Π K Φ}}SΦ} .

Moreover, we have

QpΠ K Φq ě }SΠ K Φ} 2 `γ}Π K Φ} 2 . We infer that

}SΠ K Φ} 2 `γ}Π K Φ} 2 ď γ 4 }Π K Φ} 2 `1 γ }L Φ} 2 `1 2 }SΠ K Φ} 2 `ν2 2 }Φ} 2 `γ 4 }Π K Φ} 2 `ν2 γ }SΦ} 2 .
Using (2.5) we deduce that

1 2 `}SΠ K Φ} 2 `γ}Π K Φ} 2 ˘ď 1 γ }L Φ} 2 `ν2 2 }Φ} 2 `ν2 2
ˆ}L Φ} 2 γ 2 `}Φ} 2 ˙,
and thus

}SΠ K Φ} 2 γ `}Π K Φ} 2 ď p2 `a2 q }L Φ} 2 γ 2 `2a 2 }Φ} 2 .
(2.7)

Let us now consider }Π K Ψ} and }SΠ K Ψ}. We have easily that

}SΠ K Ψ} 2 `γ}Π K Ψ} 2 ď QpΠ K Ψq " p QpΨ, Π K Ψq ď } x L Ψ}}Π K Ψ} ,
and thus

}SΠ K Ψ} 2 γ `}Π K Ψ} 2 ď } x L Ψ} 2 γ 2 .
(2.8)

It remains to combine (2.4), (2.5), (2.6), (2.7), (2.8), and use elementary manipulations.

Examples of applications

In this section we discuss three applications of Theorem 1.1 and we recall that we are in the context of Remark 1.3.

3.1. Semiclassical Born-Oppenheimer approximation. In this first example we set pΣ, σq " pR, dsq. We consider a Hilbert space H T and set H " L 2 pR, H T q. Then, for h ą 0, we consider on H the operator S h " hD s , where D s " ´iB s . We also consider an operator T on H such that for Φ " pΦ s q sPR P H we have pT Φq s " T s Φ s , where pT s q is a family of operators on H T which depends analytically on s. Thus the operator L " L h takes the form

L h " h 2 D 2
s `T . This kind of operators appears in [START_REF] Lampart | The adiabatic limit of the Laplacian on thin fibre bundles[END_REF][START_REF] Lampart | The adiabatic limit of Schrödinger operators on fibre bundles[END_REF] their spectral and dynamical behaviors are analyzed. As an example of operator T , the reader can have the Schrödinger operator ´∆t V ps, tq in mind, where the electric potential V is assumed to be real-valued. Here the operator norm of the commutator rhD s , Πs is controlled by the supremum of }B s u 1 psq} H . Assuming that }B s u 1 psq} H is bounded, we have a " aphq " Ophq (see (1.5)). Let us also assume, for our convenience, that µ 1 has a unique minimum, non-degenerate and not attained at infinity. Without loss of generality we can assume that this minimum is 0 and is attained at 0. Thus, here γ just satisfies γ " inf sPR µ 2 psq ą 0.

For k P N ˚we set

λ k phq " sup F ĂDompL h q codimpF q"k´1 inf ϕPF }ϕ}"1 L h ϕ, ϕ . (3.1)
By the min-max principle, the first values of λ k phq are given by the non-decreasing sequence of isolated eigenvalues of L h (counted with multiplicities) below the essential spectrum. If there is a finite number of such eigenvalues, the rest of the sequence is given by the minimum of the essential spectrum. We similarly define the sequence pλ eff ,k phqq corresponding to the operator L h,eff . Note that L h,eff can be identified with the operator

h 2 D 2 s `µ1 psq `h2 }B s u 1 psq} 2 HT .
As a consequence of the harmonic approximation (see for instance [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]Chapter 7] or [16, Section 4.3.1]), we get the following asymptotics. Proposition 3.1. Let k P N ˚. We have

λ eff,k phq " p2k ´1q c µ 2 p0q 2 h `ophq , h Ñ 0 .
From our abstract analysis, we deduce the following result.

Proposition 3.2. Let c 0 , C 0 ą 0. There exist h 0 ą 0 and C ą 0 such that for h P p0, h 0 q and z P Z h " tz P r´C 0 h, C 0 hs : distpz, SppL h,eff qq ě c 0 hu we have z P ρpL h q and }pL h ´zq ´1 ´pL h,eff ´zq ´1} ď C .

Proof. Let h ą 0 and z P Z h . If h is small enough we have C 0 h ă γ so z P ρpL h,eff q X ρpL K h q " ρp x L h q. Moreover, by the Spectral Theorem,

› › ›p x L h ´zq ´1› › › ď › › pL h,eff ´zq ´1› › `› › pL K h ´zq ´1› › ď 1 c 0 h `1 γ ´C0 h .
With the notation (1.6) we have lim inf

hÑ0 sup zPZ h ´1 ´η1,h pzq}p x L h ´zq ´1} ´η2,h pzq ¯ą 0.
From Theorems 1.1 and Proposition 1.2, we deduce that z P ρpL h q, }pL h ´zq ´1} À h ´1 , and the estimate on the difference of the resolvents. Here and occasionally in the sequel, we adopt the notation x À y if there is a positive constant C (independent of x and y) such that x ď Cy.

From this norm resolvent convergence result, we recover a result of [START_REF] Lampart | The adiabatic limit of Schrödinger operators on fibre bundles[END_REF]Section 4.2].

Proposition 3.3. Let k P N ˚. Then λ k phq " λ eff,k phq `Oph 2 q, h Ñ 0.
Proof. Let ε ą 0 be such that λ eff,k`1 phq ´λeff,k phq ą 2εh for all h. We set z h " λ eff,k phq `εh.

The resolvent pL h,eff ´zh q ´1 has k negative eigenvalues

1 λ eff,k phq ´zh ď . . . ď 1 λ eff,1 phq ´zh ,
all smaller than ´α{h for some α ą 0, and the rest of the spectrum is positive. By Proposition 3.2 the resolvent pL h ´zh q ´1 is well defined for h small enough and there exists C ą 0 such that › › pL h ´zh q ´1 ´pL h,eff ´zh q ´1› › ď C.

By the min-max principle applied to these two resolvents, we obtain that for all j P t1, . . . , ku the j-th eigenvalue of pL h ´zh q ´1 is at distance not greater than C from 1{pλ eff,k`1´j ´zh q, and the rest of the spectrum is greater than ´C. In particular, for j " 1, ˇˇˇ1

λ k phq ´zh ´1 λ eff,k phq ´zh ˇˇˇď C.
This gives |λ k phq ´λeff,k phq| ď Cεh |λ k phq ´λeff,k phq `εh| , and the conclusion follows for h small enough.

Shrinking neighborhoods of hypersurfaces.

In this paragraph we consider a submanifold Σ of R d , d ě 2, as in Section 1.3. We choose ε ą 0 and define Θ ε , Ω and Ω ε as in (1.8) and (1.9). For ϕ P H 1 0 pΩ ε q, we set

Q Dir Ωε pϕq " ż Ωε |∇ϕ| 2 dx,
and we denote by ´∆Dir Ωε the associated operator. Then we use the diffeomorphism Θ ε to see ´∆Dir

Ωε as an operator on L 2 pΩq. We set, for ψ P H 1 0 pΩ, dσdtq,

Q Dir ε pψq " Q Dir Ωε pψ ˝Θ´1 ε q .
We need a more explicit expression of Q Dir ε in terms of the variables ps, tq on Ω. For ps, tq P Ω we have on T ps,tq Ω » T s Σ ˆnpsqR

d ps,tq Θ ε " pId TsΣ `εtd s nq b εId npsqR . Hence d Θεps,tq Θ ´1 ε " pId TsΣ `εtd s nq ´1 b ε ´1Id npsqR .
We recall that the Weingarten map ´ds n is a self-adjoint operator on T s Σ (endowed with the metric inherited from the Euclidean structure on R d ). For ψ P H 1 pΩ, dσdtq, x P Ω ε and ps, tq " Θ ´1 ε pxq we get }∇pψ ˝Θ´1 ε qpxq} 2 Tx Ωε " }pd x Θ ´1 ε q ˚∇ψps, tq} 2

TxΩε " }pId TsΣ `εtd s nq ´1∇ s ψps, tq} 2 TsΣ `1 ε 2 |B t ψps, tq| 2 .

The eigenvalues of the Weingarten map are the principal curvatures κ 1 , . . . , κ d´1 . In particular for ps, tq P Ω we have

|d ps,tq Θ ε | " εw ε , where w ε ps, tq " d´1 ź j"1 p1 ´εtκ j psqq . (3.2)
The Riemannian structure on Ω is given by the pullback by Θ ε of the Euclidean structure defined on Ω ε . More explicitly, for ps, tq P Ω the inner product on T ps,tq Ω is given by @X, Y P T ps,tq pΩq, g ε pX, Y q " xd ps,tq Θ ε pXq, d ps,tq Θ ε pY qy R d .

Then the measure corresponding to the metric g ε is given by εw ε dσdt. Thus, if we set

G ε ps, tq " pId TsΣ `εtd s nq ´2, (3.3)
we finally obtain

Q Dir ε pψq " ż Ωε |pId TsΣ `εtd s nq ´1∇ s ψpΘ ´1 ε pxqq| 2 dx `1 ε 2 ż Ωε |B t ψpΘ ´1 ε pxqq| 2 dx " ε ż Ω xG ε ps, tq∇ s ψ, ∇ s ψy T Σ w ε dσdt `1 ε 2 ż Ω |B t ψ| 2 εw ε dσdt .
The transverse operator T s pεq is the Dirichlet realization on L 2 pp´1, 1q, εw ε dtq of the differential operator ´ε´2 w ´1 ε B t w ε B t . We denote by µ 1 ps, εq its first eigenvalue and we set µpεq " inf sPR µ 1 ps, εq. We have, by perturbation theory, as ε Ñ 0,

µ 1 ps, εq " π 2 4ε 2 `V psq `Opεq , µpεq " π 2 4ε 2 `Op1q, where V psq " ´1 2 d´1 ÿ j"1 κ j psq 2 `1 4 ˜d´1 ÿ j"1 κ j psq ¸2 .
We denote by L Dir ε the operator associated to the form Q Dir ε and by L Dir ε,eff the corresponding effective operator as defined in the general context of Section 1.2. It is nothing but the operator associated with the form H 1 pΣq Q ϕ Þ Ñ Q Dir ε pϕu s,ε q where u s,ε is the positive L 2 -normalized groundstate of the transverse operator (and actually depending on the principal curvatures analytically). From perturbation theory, we can easily check that the commutator between the projection on u s,ε and S is bounded (and of order ε). Proposition 3.4. Let c 0 , C 0 ą 0. There exist ε 0 ą 0 and C ą 0 such that for all ε P p0, ε 0 q and

z P Z c0,C0,ε " tz P R : |z ´µpεq| ď C 0 , distpz, SppL Dir ε,eff qq ě c 0 u we have › › › `L Dir ε ´z˘´1 ´`L Dir ε,eff ´z˘´1 › › › ď Cε .
We recover a result of [START_REF] Krejčiřík | The magnetic Laplacian in shrinking tubular neighborhoods of hypersurfaces[END_REF] (when there is no magnetic field).

Proof. We are in the context of Remark 1.3. The form Q ε ´µpεq is non-negative. We denote by L ε the corresponding non-negative self-adjoint operator and define x L ε as in Lemma 2.2. Given ε ą 0 and z P Z c0,C0,ε we write ζ for z ´µpεq. Thus, with the notation of the abstract setting we have γ ε " ε ´2, a ε " Opε 2 q, ζ " Op1q and hence η 1,ε pζq " Opεq, η 2,ε pζq " Opε 2 q, η 3,ε pζq " Opεq and η 4,ε pζq " Opε 2 q. Moreover, by the spectral theorem, we have

› › › `x L ε ´ζ˘´1 › › › " Op1q.
Thus, there exists ε 0 ą 0 such that for ε P p0, ε 0 q, z P Z c0,C0,ε and ζ " z ´µpεq the operator L ε ´ζ is bijective and

› › ›pLε ´ζq ´1› › › " Op1q , › › ›pLε ´ζq ´1 ´`x L ε ´ζ˘´1 › › › " Opεq .
The conclusion easily follows.

Given ε ą 0 we define the sequence pλ Dir k pεqq kPN ˚and pλ Dir k,eff pεqq kPN ˚corresponding to the operators L Dir ε and L Dir ε,eff as in (3.1). By using analytic perturbation theory with respect to the parameters pεκ j q 1ďjďd´1 to treat the commutator, we have, for all k P N ˚, λ Dir k,eff pεq "

π 2 4ε 2 `λΣ k `Opεq , ε Ñ 0,
where λ Σ k is the k-th eigenvalue of ´∆s `V psq. We recover a result in the spirit of [START_REF] Duclos | Curvature-induced bound states in quantum waveguides in two and three dimensions[END_REF][START_REF] Krejčiřík | The magnetic Laplacian in shrinking tubular neighborhoods of hypersurfaces[END_REF]. Proposition 3.5. For all k ě 1 we have

λ Dir k pεq " π 2 4ε 2 `λΣ k `Opεq, ε Ñ 0 Proof. Let k ě 1.
There exist c 0 , c0 , C 0 , ε 0 ą 0 such that for ε P p0, ε 0 q we have λ Dir k,eff pεq `c 0 P Z c0,C0,ε . As in the proof of Proposition 3.3 we obtain from Proposition 3.4 and min-max principle ˇˇ`λ k pεq ´pλ Dir k,eff pεq `c 0 q ˘´1 ´`λ Dir k,eff pεq ´pλ Dir k,eff pεq `c 0 q ˘´1 ˇˇ" Opεq.

We deduce ˇˇλ Dir k pεq ´λDir k,eff pεq ˇˇ" Opεq ˇˇ`λ Dir k pεq ´pλ Dir k,eff pεq `c 0 q ˘ˇ, and the conclusion follows.

3.3. Dirichlet-Robin shell with large coupling constant. In this section, we keep considering the hypersurface Σ of the last paragraph (here ε " 1). Let us now consider the Dirichlet-Robin Laplacian in an annulus. In other words, with w 1 and G 1 as defined by (3.2) and (3.3), we consider on the weighted space L 2 pw 1 dsdtq the quadratic form

Q DR α pψq " ż Σˆp0,1q
`xG 1 ps, tq∇ s ψ, ∇ s ψy T Σ `|B t ψ| 2 ˘w1 ps, tqdsdt ´α ż Σ |ψps, 0q| 2 ds.

It is defined for ψ P DompQ DR α q where DompQ DR α q " tψ P H 1 pΣ ˆp0, 1qq : ψps, 1q " 0 , B t ψps, 0q " ´αψps, 0qu . In these definitions α is real, and we are interested in the strong coupling limit α Ñ `8. This quadratic form is of the form (2.1) with S " G 1 2

1 ∇ s and T s " ´w´1 1 B t w 1 B t acting on H 2 pp0, 1qq and Dirichlet-Robin condition. The spectrum of T s is well-understood in the limit α Ñ `8. Actually, the family pT s q depends analytically on the principal curvatures pκ j psqq 1ďjďd´1 . We can deduce from the previous works [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF][START_REF] Helffer | Tunneling for the Robin Laplacian in smooth planar domains[END_REF][START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF] that, as α Ñ `8, µ 1 ps, αq " ´α2 ´ακpsq `Op1q , µ 2 ps, αq ě c ą 0 , and µpαq " inf sPΣ µ 1 ps, αq " ´α2 ´ακ max `Op1q , with κ " ř d´1 j"1 κ j . Here, for simplicity, we assume that κ has a unique maximum at s " 0 that is not degenerate and not attained at infinity. Moreover, we assume that the eigenvalues of D 2 s `1 2 Hess 0 p´κqps, sq are simple. We let Z C0,c0,α " tz P R : |z ´µpαq| ď c 0 α , distpz, Spp x L DR α qq ě C 0 u . Proposition 3.6. There exist C, α 0 ą 0 such that, for all z P Z C0,c0,α

› › › `L DR α ´z˘´1 ´`L DR α,eff ´z˘´1 › › › ď Cα ´1 .
Proof. Here we have γ " Opα 2 q, ν " Opα ´1q and a " Opα ´2 q. We use again Remark 1.3 and we apply Theorem 1.1 with L " L DR α ´µpαq and z replaced by z ´µpαq. For z P Z c0,C0,α , we get η 1 " Opα ´1q , η 2 " Opα ´2 q , η 3 " Opα ´2q , η 4 " Opα ´3 q .

Moreover, for α large enough, we have, for all z P Z c0,C0,α , z P ρ ´L DR α,eff ¯and }pL DR α,eff ´zq ´1} ď C . Then Theorem 1.1 implies the wished estimate.

We recover, under our simplifying assumptions, a result appearing in [START_REF] Helffer | Eigenvalues for the Robin Laplacian in domains with variable curvature[END_REF][START_REF] Pankrashkin | An effective Hamiltonian for the eigenvalue asymptotics of the Robin Laplacian with a large parameter[END_REF][START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF]. Proposition 3.7. For all j ě 1, we have, as α Ñ `8, λ DR j,eff pαq " ´α2 `νj pαq `Op1q , and λ DR j pαq " ´α2 `νj pαq `Op1q , where ν j pαq is the j-th eigenvalue of D 2 s ´ακpsq. Proof. Let us first discuss the asymptotic behavior of the eigenvalues of the effective operator. Let us recall that it is defined as explained in Section 1.2, and that it can be identified with the operator associated with the form H 1 pΣq Q ϕ Þ Ñ Q DR α pϕu s,α q where u s,α is the positive L 2 -normalized groundstate of the transverse operator T psq. The asymptotic expansion of the effective eigenvalues again follows from perturbation theory and a commutator estimate (see [START_REF] Kachmar | Weyl formulae for the Robin Laplacian in the semiclassical limit[END_REF]Section 3] where it is explained how we can estimate such a commutator).

Then, we proceed as in the previous section. Note that, by the harmonic approximation, for all j ě 1, ν j pαq " ´ακ max `α 1 2 νj `Opα 1 4 q , where pν j q jPN ˚is the non-decreasing sequence of the eigenvalues of D 2 s `1 2 Hess 0 p´κqps, sq. In particular, the asymptotic gap between consecutive eigenvalues is of order α 1 2 . Then, there exist c 0 ą 0, C 0 ą 0 and C ą 0 such that, for α large, z " λ DR j,eff pαq `C P Z c0,C0,α . We use Proposition 3.6 and we get, as in the other examples, |λ DR j,eff pαq ´λDR j pαq| ď Cα ´1 .

The non-self-adjoint Robin Laplacian between hypersurfaces

In this section we prove Theorem 1.5. The proof is split in two main steps. We first transform the problem into an equivalent statement, where P ε is replaced by a unitarily equivalent operator on Ω.

4.1.

A change of variables. The operator P ε is associated to the (coercive) quadratic form defined for φ P H 1 pΩ ε q by

Q 1 ε pφq " Q 1 ε,α pφq " ż Ωε |∇φ| 2 `żΣ`,ε α `,ε |φ| 2 ´żΣ´,ε α ´,ε |φ| 2 . (4.1)
As in Section 3.2, we use the diffeomorphism Θ ε to see P ε as an operator on L 2 pΩq: for ψ P H 1 pΩq we set

Q 2 ε pψq " Q 1 ε pψ ˝Θ´1 ε q . We obtain Q 2 ε pψq " ż Ωε |pId TsΣ `εtd s nq ´1∇ s ψpΘ ´1 ε pxqq| 2 dx `1 ε 2 ż Ωε |B t ψpΘ ´1 ε pxqq| 2 dx `żΣ`α `,ε |ψ ˝pΘ ὲ q ´1| 2 ´żΣ´α ´,ε |ψ ˝pΘ ὲ q ´1| 2 " ż Ω xG ε ps, tq∇ s ψ, ∇ s ψy T Σ ε wε dσdt `1 ε ż Ω |B t ψ| 2 wε dσdt `żΣ α p|ψ| 2 wε q| t"1 dσ ´żΣ α p|ψ| 2 wε q| t"´1 dσ,
where, as in (3.2), wps, tq " ś d´1 j"1 p1 ´εtκ j psqq. Notice that L 2 pΩ, dσdtq and L 2 pΩ, ε wε dσdtq (or their corresponding Sobolev spaces) are equal as sets, but Θ ε induces only a unitary transformation from L 2 pΩ, ε wε dσdtq to L 2 pΩ ε , dxq.

A change of function.

In the next step we make a change of function to turn our problem with Robin boundary conditions into an equivalent problem with Neumann boundary conditions. For this we consider the unitary transform Ũε :

" L 2 pΩ, ε wε dσdtq Ñ L 2 pΩ, e ´2εtRepαq wε dσdtq, u Þ Ñ ? εe αεt u.
We set w ε " e ´2εtRepαq wε . Then on H 1 pΩ, w ε dσdtq we consider the transformed quadratic form given by

Q ε pφq " Q 2 ε p Ũ ´1φq " ż Ω xG ε p∇ s ´εt∇ s αqφ, p∇ s ´εt∇ s αqφyw ε dσdt `1 ε 2 ż Ω |B t φ| 2 w ε dσdt ´1 ε ż Ω `αφB t φ `ᾱ φB t φ ˘wε dσdt `żΩ |α| 2 |φ| 2 w ε dσdt `1 ε ż Σ α|φ|w ε dσ ´1 ε ż Σ α|φ| 2 w ε dσ .
By integration by parts we have

´1 ε ż Ω αφB t φw ε dσdt " ´1 ε ż Σ α|φ|w ε dσ `1 ε ż Σ α|φ| 2 w ε dσ `żΩ ˆ´2αRepαq `αB t wε ε wε ˙|φ| 2 w ε dσdt . Finally, Q ε pφq " ż Ω xG ε p∇ s ´εt∇ s αqφ, p∇ s ´εt∇ s αqφyw ε dσdt `1 ε 2 ż Ω |B t φ| 2 w ε dσdt `2i ε ż Ω ImpαqB t φ φw ε dσdt `żΩ V ε |φ| 2 w ε dσdt , where V ε " |α| 2 ´2αRepαq `α B t wε ε wε .
On H 1 pΩ, w ε dσdtq we can also consider the forms defined by

p Q ε pφq " ż Ω |∇ s φ| 2 dσdt `1 ε 2 ż Ω |B t φ| 2 dσdt `żΩ V eff |φ| 2 dσdt and Q eff pφq " ż Ω |∇ s φ| 2 dσdt `żΩ V eff |φ| 2 dσdt.
We denote by L ε , x L ε and L eff the operators corresponding to the forms Q ε , p Q ε and Q eff , respectively. As P ε , the operator L ε is m-accretive. We have the following accretivity estimate when ε goes to 0.

Lemma 4.1. If ε 0 ą 0 is small enough there exist M 0 ě 0 and c 0 ą 0 such that for ε P p0, ε 0 q, M ě M 0 and φ P H

˙.

Proof. There exists C 1 ě 0 such that for ε P p0, ε 0 q and φ P H The conclusion follows if ε 0 ą 0 was chosen small enough.

A remarkable property of L ε is the following complex symmetry (cf. [START_REF] Borisov | PT -symmetric waveguides[END_REF]).

Lemma 4.2. Let ε ą 0 and z P C. If z P C is an eigenvalue for L ε then z is an eigenvalue for L ε . In particular the operator L ε has no residual spectrum.

Proof. Since L ε is unitarily equivalent to P ε " P ε,α , it is sufficient to prove the result for P ε,α . Notice that DompQ 1 ε,α q " DompQ 1 ε,α q. Moreover for φ, ψ P DompQ 1 ε,α q we have Q 1 ε,α pφ, ψq " Q 1 ε,α pψ, φq, so P ε,α " P ε,α . Now let ψ P DompP ε,α q. For all φ P DompQ 1 ε,α q we have Q 1 ε,α pφ, ψq " Q 1 ε,α pφ, ψq " φ, P ε,α ψ " φ, P ε,α ψ . This proves that ψ P DompP ε,α q and P ε,α ψ " P ε,α ψ. Thus, if we denote by J the complex conjugation, we get that P ε,α is J-self-adjoint P ε,α " JP ε,α J.

The conclusion follows.

4.4. Proof of Theorem 1.5. Theorem 1.5 will be a consequence of the following proposition. Proposition 4.3. There exist ε 0 , C ą 0 such that for all ε P p0, ε 0 q, ϕ P Domp x L ε q and ψ P DompL ε q, |Q ε pϕ, ψq ´p Q ε pϕ, ψq| ď Cε}ϕ} We conclude with Lemma 4.1.

By Proposition 4.3, there exists C ě 0 such that for z P K, ϕ P DomppH eff ε q ˚q and ψ P DompH ε q we have |Q eff ε pϕ, ψq ´zxϕ, ψy ´pQ ε pϕ, ψq ´zxϕ, ψyq | ď Cε}ϕ} pH eff ε ´zq ˚}ψ} Hε´z . Finally, we apply Lemma 2.4 and Lemma 1.7, and Theorem 1.5 follows.

4. 3 .

 3 About the new operator L ε . If U ε denotes the composition of the unitary transform associated with Θ ε and Ũε , we write L ε " U ε P ε U ´1 ε and the estimate of Theorem 1.5 can be rewritten as }pL ε ´zq ´1 ´pL eff ´zq ´1Π} LpL 2 pΩqq À ε . (4.2)

  1 pΩq we haveRe `Qε pφq ˘`M }φ} 2 L 2 pΩq ě c 0 ˆ}∇ s φ} 2 L 2 pΩq `1 ε 2 }B t φ} 2 L 2 pΩq `}φ} 2

	L 2 pΩq

  1 pΩq we have Re `Qε pφq ˘ě p1 ´C1 εq}p∇ s ´εt∇ s αqφ} 2 `1 ´C1 ε ε 2 }B t φ} 2 ´2}Impαq} 8 p1 `C1 εq ε }B t φ}}φ} ´C1 }φ} 2 .For some C 2 ě 0 we also have }p∇ s ´εt∇αqφ} 2 ě p1 ´εq}∇ s φ} 2 ´C2 }φ} 2 .

	and
	1 ´C1 ε ε 2 " p1 ´C1 εq }B t ϕ} 2 ´2 }Impαq} 8 p1 `C1 εq ε ˜}B t ϕ} 2 ε 2 ´}B t ϕ} ε 2 }Impαq} 8 p1 `C1 εq }ϕ} }B t ϕ} }ϕ} 1 ´C1 ε p1 ´C1 εq 2 }B t ϕ} 2ε 2 ´C2 }ϕ} 2 .	ȩ

  Proof. We set D ε pϕ, ψq " Q ε pϕ, ψq ´p Q ε pϕ, ψq . Using the Taylor formula, we get|D ε pϕ, ψq| À ε}ϕ} H 1 s }ψ} H 1 s `1 ε }B t ϕ}}B t ψ} L 2 pΩq `}∇ s ψ} 2 L 2 pΩq .The most delicate term is the last one. We haveˇˇˇQ ε ptImpαqϕ, ψq ´1 ε 2 ż ImpαqB t ptϕqB t ψw ε dσdt ˇˇˇÀ }ϕ} H 1 s }ψ} H 1 s `1 ε }ϕ}}B t ψ}, so ˇˇˇQ ε ptImpαqϕ, ψq ´1 ε 2 ż ImpαqϕB t ψw ε dσdt ˇˇˇÀ }ϕ} H 1 s }ψ} H 1 s `1 ε 2 }B t ϕ}}B t ψ} `1 ε }ϕ}}B t ψ} .

		ż
		`ˇˇˇ2 ε
	where }ψ} 2 H 1 s	" }ψ} 2

x L ε }ψ} Lε . Ω ImpαqB t ψ φw ε dσdt ˇˇˇ, Since Q ε ptImpαqϕ, ψq " tImpαqϕ, L ε ψ , we obtain ˇˇˇ1 ε ż ImpαqB t ψϕw ε dσdt ˇˇˇÀ ε}ϕ} H 1 s }ψ} H 1 s `1 ε }B t ϕ}}B t ψ} `}ϕ}}B t ψ} `ε}L ε ψ}}ϕ} .

Acknowledgment

The authors would like to thank the Centre International de Rencontres Mathématiques in Marseille for the support during two weeks in August and September 2016 where this research was initiated in the framework of a Research-In-Pairs grant. The research of P.S. was supported by the Swiss National Science Foundation, SNF Ambizione grant No. PZ00P2 154786. D.K. and P.S. were partially supported by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014.


Wessel Reijers
email: wreijers@adaptcentre.ie

Bert Gordijn
email: bert.gordijn@dcu.ie

Declan O'sullivan
email: declan.osullivan@cs.tcd.ie

Narrative Ethics of Personalisation Technologies

Keywords: Personalisation, Ethics, Narrative technologies, Ricoeur

Personalisation of digital content is becoming one of the major focus areas of contemporary research in human-computer interaction. Interactions between humans and computer systems such as information retrieval operations, digital learning and self-monitoring are "tailored" to the needs of the human user. In this paper, we aim to increase our philosophical understanding of personalisation and of its ethical implications. We utilise a framework of ethics of narrative technologies that is based on the narrative theory of Paul Ricoeur to explicate how personalisation processes shape the person and interpersonal relations. We argue that personalisation processes can actively configure the narrative understanding of a person they interact withby which they can implicitly change or re-enforce a person's normative worldview. Also, personalisation processes can abstract from the world of action by means of profilingwhich can have significant risks with regards to the consistency of a person's character.

Introduction

Computational processes become increasingly "close" to us: they interact with our bodies, movements and emotions and occupy a central place in our daily lives. In line with this development, "the person" has been brought to the centre of the research on design of computer systems. In the contemporary context of "Big Data" systems [START_REF] Lycett | Datafication": making sense of (Big) data in a complex world[END_REF] and an increasing "information overload" on the web [START_REF] Anand | Intelligent Techniques for Web Personalization[END_REF], it has become progressively important to get the right information in the right way to the right person at the right moment in time [START_REF] Fischer | Context-aware systems[END_REF]. In marketing studies, this has led to questions such as how "to tailor electronic commerce interactions between a business and each individual customer" [START_REF] Vesanen | What is personalization? A conceptual framework[END_REF]. In computer science, it has led to improvements of methods for "user profiling", "behaviour preference" and "personalised search engines and recommendation systems" [START_REF] Gao | Personalisation in web computing and informatics: Theories, techniques, applications, and future research[END_REF]. Basically, research in personalisation aims at adjusting digital content to the needs, preferences, desires and whims of the person interacting with it.

The process of personalisation invariably integrates underlying assumptions about what a person is andmore importantlyabout what we can understand as the right information for the right person. It is an automated process of categorisation, and therefore of inclusion and exclusion of both digital content and of "digital personae" or profiles from the information we access. As a result, personalisation processes can influence what type of information we retrieve from our search engine, what kinds of products are recommended to us when browsing online, and what kind of feedback we receive about our daily activities. Also, personalisation processes are used by intelligence agencies to identify suspicious individuals and by insurance companies to establish people's personal credit ratings. Accordingly, these increasingly ubiquitous computational processes that directly influence many aspects of our everyday lives can have significant ethical implications. For instance, de Vries argues that ethical concerns arise from cases of implicit discrimination based on profiling [START_REF] De Vries | Identity, profiling algorithms and a world of ambient intelligence[END_REF]. Moreover, Bohn et al. argue that the practice of matching digital persona with profiles that present a public security concern has ethical implications because it can lead to mass-surveillance practices [START_REF] Bohn | Social, economic, and ethical implications of ambient intelligence and ubiquitous computing[END_REF]. In a different vein, Schubert argues that the use of personalised technologies that nudge their users in certain directions can lead to a reduction of personal autonomy and agency [START_REF] Schubert | On the ethics of public nudging: Autonomy and Agency[END_REF].

Most of these ethical analyses of personalisation processes seem to focus directly on their effects, without explicating how we can actually understand the process of personalisation. In this paper, in contrast, we aim to contribute to the existing ethical reflections by engaging in a discussion of how personalisation processes mediate "the person": showing that the person is often wrongfully understood as a collection of static attributes and offering an alternative understanding, in which the person and the technology are co-shaped, or rather configured, through interaction with personalisation processes. We argue -to use the terms of Verbeek -that in order to understand the ethical implications of personalisation processes we need to understand what kind of "I-technology-world" relationship they constitute [START_REF] Verbeek | What things do; philosophical reflections on technology, agency, and design[END_REF], that we should make explicit how they interact with a human and how the human is transformed in this process. However, we also aim to overcome some problems that theories of technological "mediation" such as Verbeek's confront us with in understand-ing personalisation processes. As an alternative approach, we use a framework of narrative technologies, which is based on Paul Ricoeur's major work on narrativity: Time and Narrative [START_REF] Ricoeur | Time and Narrative[END_REF]. With this framework, we can better understand how human understanding of our technological world changes through interaction with textual technologies such as computer systems. We will first investigate the concept of personhood in the context of personalisation technologies. Second, we will present the framework of narrative technologies that is based on Ricoeur's narrative theory. Third, we will apply this framework in order to identify and understand the ethical implications of personalisation technologies.

Personalisation Technologies as Adaptive Mirrors of Personhood

The process of personalisation is applied to a wide range of different technologies; but it generally revolves around notions of "adapting", "fitting" or "tailoring" digital content to the human being(s) interacting with it. A central term in personalisation research is said to be the "adaptability" of a system [START_REF] Yalcinalp | Ontology and taxonomy design and development for personalised web-based learning systems[END_REF]. Adaptive systems include three basic models in their design: a "user model" that is a structured model for the collection and categorisation of personal data belonging to a user, "an application model", which is a description of relevant features of the application, and an "interaction model" that is meant to structure the organisation of interactions between a user and a system [START_REF] Yalcinalp | Ontology and taxonomy design and development for personalised web-based learning systems[END_REF]. Asif and Krogstie argue that multiple personalisation approaches exist that can be based on "machine-learning algorithms, agent technology and ubiquitous and context-aware computing". They identify a "basic level" of personalisation, at which a user selects a certain configuration of a computing device or interface, which subsequently remains the same [START_REF] Asif | Taxonomy of Personalization in Mobile Services[END_REF]. This basic level corresponds with the conventional, instrumental view of a personalisation technology: of the human user actively configuring its settings. Then there is a "second level", at which the configuration of a system is based on a "profile" of the user, and a "third level" at which both the profile of the user and his or her "context" (mostly comprised of meta-data such as location, time of the activity, type of activity) are used as the basis for the configuration of the system. Bouzeghoub and Kostadinov make a distinction between profiles and queries: a profile being a "user model" "defined by a set of attributes" and a query being an "on-demand user need" [START_REF] Bouzeghoub | Data Personalization : a Taxonomy of User Profiles Knowledge and a Profile Management Tool[END_REF]. Roosendaal offers an additional, distinction, namely one between "digital personae", which are representations of known individuals in the real world and "digital profiles" which are sets of characteristics about persons that can be used as inputs for algorithmic decision making [START_REF] Roosendaal | Digital Personae and Profiles as Representations of Individuals[END_REF].

A clear example of personalisation based on a digital persona and profiles is an automated passport check at an airport [START_REF] Juels | Security and Privacy Issues in E-passports[END_REF]. When someone's passport chip is scanned, the retrieved data is compared with a data entry containing the document number belonging to the respective person, accompanied by her picture, biometric information and information about country of origin, age, and so on. Based on an algorithmic assessment of this personal data, the person can either pass through or will be held and interrogated by the border police. In this process, the digital persona can be compared with and transformed into a digital profile, for instance by linking it with a certain profile containing passport features that are deemed "suspicious", or by using it to add to the digital profile of people originating from the same country. What all processes of personalisation such as the automated passport control have in common is that they use interactions with humans to gain knowledge about them and create a representation that changes the behaviour of a system in order to fit, adapt to or be tailored to this representation. The purpose of many instances of personalisation research is to make sure that the behavioural changes of the system approximate the expectations, wishes and/or needs of the human interacting with it. Terms like tailoring suggest that the userthe human agentis seen as a given, as a static point to which the personalisation processes need to adjust. Just like a tailor adjusts the sizes and shapes of a piece of clothing to the human body that remains the same (or rather, that is defined by fixed measurements), personalisation processes are supposed to be tailored to users who are presumed to remain the same. Accordingly, user needs or preferences are supposed to be fixed. For instance, it might be assumed that the user of a weather application has the fixed need of knowing what the weather will be the next day in her city and on request a personalisation process will link the data of her location and weather forecasts to provide for the desired information.

However, scholars in the area of philosophy of technology have claimed that such an instrumental approach does not adequately capture how humans and technologies interrelate. Ihde explains that "technologies transform our experience of the world and our perceptions and interpretations of our world, and we in turn become transformed in this process" [START_REF] Ihde | Postphenomenology and Technoscience[END_REF]. Accordingly, we argue, personalisation processes could be seen as "mirroring" processes, in a similar sense as Hegel and Lacan gave to them, rather than as mere adjustments of computational outcomes to the static user needs [START_REF] Gasparyan | Mirror for the Other: Problem of the Self in Continental Philosophy (from Hegel to Lacan)[END_REF]. By looking into the mirror, the perception of the self changes because the subject suddenly becomes aware of herself as an object as well: she engages in a process of self-reflection. Dennett argues that the consequent idea of "selfconsciousness"or the ability of the self to reflect on the selfis one of the necessary conditions for personhood [START_REF] Dennett | Conditions of Personhood[END_REF]. However, the "mirror" that we engage with in the sense of personalisation processes differs from the static idea of a glass mirror for it actively adjusts to the user interacting with it. Here, we juxtapose the conception of a static person "subjected to a set of rules in whose making he or she had no part" [START_REF] Kitwood | The concept of personhood and its relevance for a new culture of dementia care[END_REF], which supports conventional views of personalisation, with a mediated conception of the person as changing during interaction with personalisation technologies. This confronts us with the question: if interaction between a human and a personalisation technology results in a constant "mediation" [START_REF] Verbeek | What things do; philosophical reflections on technology, agency, and design[END_REF][START_REF] Ihde | Postphenomenology and Technoscience[END_REF] of the "I-technology-world" relationship, and consequently in a mediation of the personhood of a user of the technology, how can we make sense of this process? To answer this question, we turn to the work of Paul Ricoeur.

A Framework of Narrative Personalisation Technologies

Current theories of technological mediation have certain drawbacks in accounting for the kinds of mediations (the kinds of human-technology relations) personalisation processes constitute. First of all, after the so-called "empirical turn" [START_REF] Brey | Philosophy of Technology after the Empirical Turn[END_REF], theories of technological mediation have been drawn away from analysing the linguistic aspects of technologies, as they were present in classical analyses of technology (Heidegger; Ellul), because they became preoccupied with technologies' material dimensions [START_REF] Coeckelbergh | Language and technology: Maps, bridges and pathways[END_REF]. This disregards the highly linguistic, or textual character of many of the mediations that personalisation technologies bring about. This problem not only relates to the linguistic nature of software in general but also, and perhaps more importantly, to the linguistic nature of the information with which the user engages (search results, nudges for information input, etc.). Secondly, these theories often display a highly individualised approach (focusing on the "I" in technological mediation) and therefore can less adequately account for the mediation of what van den Eede designates as "beingwith-each-other" relations [START_REF] Van Den Eede | In Between Us: On the Transparency and Opacity of Technological Mediation[END_REF], in the context of personalisation. As Kitwood argues, personhood needs to be understood as requiring a "living relationship with at least one other" person [START_REF] Kitwood | The concept of personhood and its relevance for a new culture of dementia care[END_REF]. We cannot talk about an "I-technology-world" relation when the technological mediation of a "person" essentially revolves around a "wetechnology-world" relation. For instance, consider a personalisation process that tells an intelligence agency something about a designated category of individuals. In this case, a mediation of an inter-subjective relation is at play -between an institution and a socially constructed group of people.

Mark Coeckelbergh and the first author of this paper have started developing a "theory of narrative technologies" that tries to account for the abovementioned inadequacies of current theories of technological mediation [START_REF] Coeckelbergh | Crypto currencies as narrative technologies[END_REF][START_REF] Coeckelbergh | Forthcoming. Narrative Technologies: a philosophical investigation of the narrative capacities of technologies[END_REF]. This was done by taking the work of Ricoeur as a valuable starting point for thinking about technological mediation, as David Kaplan already suggested [START_REF] Kaplan | Paul Ricoeur and the Philosophy of Technology[END_REF]. The suggestion is that the model of mediation of human experience by means of a text, as explicated by Ricoeur, can be used as a model for explicating the mediation of human experience by technologies. In his seminal work Time and Narrative, Ricoeur develops a theory that draws from the writings of Augustine and Aristotle to provide a structural account of how narrative understanding, through the engagement with texts, mediates human experience [START_REF] Ricoeur | Time and Narrative[END_REF]. He aims to ground what happens once a human interacts with a text, by conceptualising it as a process consisting of three distinct moments: the prefigured time at the initiation of the interaction, the configured time during the interaction and the refigured time after the interaction has taken place, at which the world of the text merges with the world of the reader. Ricoeur argues that texts mediate our narrative understanding, which is always a public understanding [START_REF] Ricoeur | Narrative Time[END_REF] and therefore deals with inter-subjective relations. We argue that the process of configuration, as explicated by Ricoeur, offers an adequate model for understanding mediation of information and communication technologies (ICTs), especially in the case of personalisation technologies.

Even though a theory of narrative technologies might be less suitable for understanding the individual, material co-shaping of humans and technologies, such as is at stake with wearing classes or a prosthesis, we argue it can much more adequately account for text-like technological mediations that revolve around inter-subjective relations, which are at stake in this paper. Based on Ricoeur's theory, we established two dimensions of technological mediation that characterise the process of configuration: the dimensions of activity and of abstraction [START_REF] Coeckelbergh | Crypto currencies as narrative technologies[END_REF]. Activity refers to the extent to which technologies actively configure human narrative understanding. This can be explained by drawing an analogy between "reading" of personalisation processes by a user and the "reading" of data by a computer. Notably, we do not want to argue that interpreting of information by a human is in any way the exact same process as symbolic manipulation by a computational system (see e.g. Searle, 1980 [27]). Rather, we want to draw the attention to the simultaneity of the activities we usually designate separately as reading and writing. Whenever a computer reads out certain data, simultaneously data are written, which implies that the processes of reading and writing interrelate. Similarly, whenever a person "reads" the output of a personalisation process, her narrative understanding is "re-written", which means that her experience of the world changes. At the same time, interactions with personalisation processes "rewrite" those processes, which leads us to the claim that the technological mediation of personalisation processes can be characterised as a process of active configuration. Arguably then, the three levels of personalisation identified by Asif and Krogstie signify an increase with regards to this dimension [START_REF] Asif | Taxonomy of Personalization in Mobile Services[END_REF]. This means that the more personalisation technologies are able to interact with the context of a user (location, time, personal network), the more that they actively engage in a process of changing her narrative understanding, and consequently in configuring her experience of the world. To illustrate this difference: a modern technology such as a microwave might be personalised, in the sense that a user can configure its settings according to her personal preferences, but such a basic, non-contextual level of narrative configuration is still fairly passive. In contrast, the earlier mentioned automatic passport check can be regarded as an active personalisation technology for its configuration changes according to its interaction with a human and simultaneously the human understanding of the technology is configured (as a result of displayed information and nudges to engage in certain actions).

The second dimension of abstraction can be understood in line with Heidegger's notion of modern technologies transforming aspects of the human life world into "gestell", or standing reserve [START_REF] Heidegger | The Question Concerning Technologies & Other Essays[END_REF]. The collection and processing of personal data transforms aspects of our personhood (e.g. age, gender, occupation) into standing reserve, the raw material of the personalisation process. Abstraction, in Ricoeur's work, is the result of narrative structures in a text that enable it to configure socalled "secondand third-order entities", or higher-order entities [START_REF] Ricoeur | Time and Narrative[END_REF], which are entities that abstract from the world of human action. For instance, although somebody engaging with his phone to buy and sell derivatives on the stock market is a first order entity directly engaged in the world of action, the "derivatives" and "stock market" she engages with are abstracted from the worlds of action they mediate. That is, a derivative trade configures a "distance" between its interaction with a user (the person initiating the trade) and the effects it has on the world of action, in which for instance people are forced to sell their house because the derivatives drop in value [START_REF] Coeckelbergh | Money Machines: Electronic Financial Technologies, Distancing, and Responsibility in Global Finance[END_REF]. The closer mediated interactions with ICTs stay to the world of action, as for instance in a game in which the characters and the plot are configured in a meaningful narrative whole for the player who "acts" in it, the less abstraction a narrative configuration brings about. In that sense, the construction of a "digital persona" as described by Roosendaal leaves the user relatively close to the world of action though the "profiling" of a user [START_REF] Roosendaal | Digital Personae and Profiles as Representations of Individuals[END_REF], in which types of persons defined by measurable variables invoke certain responses of a system abstracted from the world of action. For instance, the automated adjustment of credit ratings based on certain profiles (containing for example gender, ethnicity and occupation), is an example of personalised abstraction for it detaches the generated process from the world of action of a particular person.

4

The Narrative Ethics of Personalisation Technologies

By using the framework of narrative technologies, we have established two claims: (i) that personalisation processes actively configure our narrative understandingprogressively so the more they interact with the context of a user -and that theydepending on the design of the technology -(ii) are capable of abstracting from the world of action. As such, we contend, personalisation technologies are highly similar to the paradigm of the text as discussed by Ricoeur (reference): they are very textual technologies. This means that, just as a reader's experience of the world might change by engaging with a piece of literature or by watching a stage play, his experience of the world might change by interacting with digital personalisation processes. Such a process of personalisation is perhaps best described by Needham, who argues that personalisation in a context of the design of public services "can best be characterised as a 'story-line'" [START_REF] Needham | Personalization: From Story-line to Practice[END_REF]. In other words, personalisation processes configure the narrative structures, or "stories" in an abstract sense, that underlie what Dennett described as one of the necessary conditions for personhood.

This view firmly opposes the idea that personalisation processes in some way get tailored to static user needs. Rather, they in turn configure these "needs": these needs change according to the changes in the narrative understanding of a person. The idea of the person that underlies this argument ties in with the ideas of the 'narrative self' as put forward by scholars such as McIntyre and Taylor, who argue for the "narrative character of human life" [START_REF] Macintyre | After Virtue: a Study in Moral Theory[END_REF] and that we "grasp our lives in a narrative" [START_REF] Taylor | Sources of the Self: The Making of the Modern Identity[END_REF]. According to this idea of personhood, a person's character is shaped according to the narrative structures (the "stories") that configure her narrative understanding through interaction with her life world, which in our time is highly technological. According to Kamtekar, we can understand the notion of character as: "a more-or-less consistent, more-or-less integrated, set of motivations, including the person's desires, beliefs about the world, and ultimate goals and values" [START_REF] Kamtekar | Situationism and Virtue Ethics on the Content of Our Character[END_REF]. When engaging in an ethical analysis of such a notion of configuration of a person's character, we are not merely focussing on evaluating the direct consequences of actions or the design of rule-based patterns of behaviour that can be either right or wrong. The focus on configuration of a person's character instead of consequences (which would lead us to consequentialism) or rule-based systems (which would lead us to deonto-logical ethics), leads us to consider virtue ethics -which takes character as a central notion -as an adequate basis for evaluating personalisation processes in the framework of narrative technologies. Indeed, as van Hooft shows [START_REF] Van Hooft | Understanding Virtue Ethics[END_REF], Ricoeur's hermeneutics is strongly related to the tradition of virtue ethics and adds to it in important ways. In virtue ethics, the notion of a "virtuous" character, or virtuous person, depends on the consistency of motivations and on the extent to which these are nonconflictual. Thus, in order to evaluate the configuration of the person through personalisation processes, we should inquire how the narrative structures configured by these processes influence a person's motivations: the extent to which those are configured in a consistent and non-conflictual manner. Notably, we are therefore not only interested in the ethical impacts of a personalisation process on the user, but also on other people affected by it, such as care-givers in the case of assistive technologies or on insurance agents in the case of personalised credit ratings.

First, we consider the ethical implications of active configuration. This characteristic of personalisation processes implies that they are very powerful tools for either re-enforcing the world-view of the user or for refiguring it. A strong example of this is what Introna and Nissenbaum designate as "the political effects" of search engines as they exclude certain political sources and include others, based on a user's profile [START_REF] Introna | Shaping the Web: why the politics of search engines matters[END_REF]. A democratic voter in the United States might therefore be confronted with search results that exclusively link to media that favour the views of the Democratic Party; based on her "personal needs" and re-enforcing her narrative understanding. However, this process applies also to the more ordinary, everyday activities of users of personalisation technologies. For instance, wearable personalisation technologies such as assistive technology devices constantly monitor the location and bodily state of a person who is in need of care and interact with this person or with her caregivers [START_REF] Teipel | Forthcoming: Information and communication technology solutions for outdoor navigation in dementia[END_REF]. Similar wearable devices can also be used to interpret a user's behaviour and bodily processes according to profiles of "productive" and "non-productive" workers, and can nudge a user to engage in daily exercise in order to be more productive at work. Such technologies are less explicitly political, but can have an even more pervasive influence on the "character" of the human interacting with them, for certain values like a "work ethic" and a preferred "life style" can be embedded in the personalisation processes. We argue that these effects show that we should consider personalisation processes as comparable to the conventional human processes in which media are produced: as automated journalists and writers who constantly confront us with information that configures our narrative understanding. Therefore, the ethics of personalisation processes should mirror an ethics of the public sphere, improving the means for a user to engage in a deliberative process with the technology and ensuring a pluralist character of information that can be accessed through the technology. For instance, this could imply that the underlying values of nudging technologies such as wearable devices on the work floor should be made explicit and subjected to a democratic process of deliberation between the workers. As such, design of the nudges configured by devices could change according to the agreed-upon purpose that is assigned to the technology by the workers.

Second, we consider the dimension of abstraction from the world of action. This implies that a user's narrative understanding engages with "secondand third-level"or higher order -entities as conceptualised by Ricoeur that abstract from the world of action they mediate (reference). This process reflects what Coeckelbergh designates as the "distancing" effect of technologies; the capacity of technologies to constitute (moral) distances between humans who engage with them and the reality they mediate [START_REF] Coeckelbergh | Money Machines: Electronic Financial Technologies, Distancing, and Responsibility in Global Finance[END_REF]. We argue that abstraction by means of personalisation technologies carries with it the risk that a person's motivations are made inconsistent and brought into conflict with one-another. Kitwood discusses this risk in the context of care relations, arguing that the normalisation that is implied in many personalisation processes (for instance, "profiling" a patient to fit care services to needs of her patient type), goes fundamentally against the idea of being a person [START_REF] Kitwood | The concept of personhood and its relevance for a new culture of dementia care[END_REF]. Personalisation, he argues, ought to account for the uniqueness of the person. As Ricoeur [START_REF] Ricoeur | Time and Narrative[END_REF] also argues: narrative structures help to understand the particular, the situated, rather than the general, the universal. The risk for inconsistent or conflicting motivations especially persists when personalisation processes are used to confer indirect, technologically mediated judgement on a person based on information generated by profiling. For instance, an insurance agent might reject a person's request for an insurance contract, based on a digital profile that itself is a higher-order entity. Inconsistency can arise because even though the insurance agent might be motivated to provide people with the best insurance contracts, his judgement can be misguided for it is not based interaction in the world of action but on an abstract representation; by for instance including variables like a person's music taste or ethnicity. This process simultaneously mediates the world of the user who tries to obtain her insurance contract; for decisions based on the digital profile she is related with can influence her senses of financial freedom, status or self-respect. As such, even abstract personalisation processes can function as the earlier mentioned adaptive "mirror" that changes the self-perception of the persons interacting with them.

Concluding Remarks

In this paper, we discuss the narrative ethics of personalisation processes that increasingly influence our daily lives. Adding to existing ethical analyses, we base our investigation on an understanding of technological mediation of the "person". To do so, we depart from established theories of technological mediation and utilised a framework of narrative technologies that is inspired by Ricoeur's work. We first argue that personalisation processes are powerful tools for re-enforcing or configuring the narrative understanding of the people interacting with them. This makes them into technological agents that can influence people's political or even everyday worldviews. Secondly, we argue that personalisation processes configure narrative structures that abstract from the world of action. This carries with it the risk that they configure inconsistencies and conflicting motivations with for the virtuous character of the persons interacting with them. Our analysis contributes in different ways to existing debates on the ethics of personalisation technologies. First, we provide a philosophical understanding of personalisation that goes against instrumental the idea of "tailoring" digital contents to the static needs of a user and instead shows how a "person" also changes in the process of interaction. Second, we link the structured understanding of technological mediation that we gained from Ricoeur's narrative theory to ethical theories of person's character, which enables us to provide a normative account of "personalisation". Third, our analysis goes beyond explicating ethical implications for only the users of personalisation processes. For instance, not only the person whose interaction with personalisation processes results in a credit rating is affected, but also the person consequently utilising this rating.

An initial step in dealing with the ethical implications of our analysis would be to include considerations of narrativity in the design process of personalisation processes. This approach is currently gaining momentum, being referred to as the "narrative approach to personalisation" [START_REF] Conlan | The narrative approach to personalisation[END_REF]. Especially in game-oriented designs of digital education environments, designers focus on the "personalisation and adaptation of Story-based Digital Educational Games" [START_REF] Göbel | Personalized, Adaptive Digital Educational Games Using Narrative Game-Based Learning Objects[END_REF]. For instance, by using technologies based on these design principles, a user can "co-author" the narrative structures she engages with. These design practices could deal with the issues of abstraction, for users would be drawn nearer to the world of action. Eventually, a broader ethical program would be needed to ground the effects of personalisation technologies that will become increasingly pervasive in our lives. One of the key issues will be to discuss the impact of these technologies on the mediation of our public sphere: how we want them to configure our public deliberations in the political and cultural realms.

Acknowledgements:

The ADAPT Centre for Digital Content Technology is funded under the SFI Research Centres Programme (Grant 13/RC/2106) and is co-funded under the European Regional Development Fund.