Philippe Briet
email: briet@univ-tln.fr

Hamza Abdou

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

Hamza Abdou Soimadou
email: abdou-soimadou-hamza@etud.univ-tln.fr

David Krejčiřík

Spectral analysis of sheared nanoribbons

Keywords: sheared strips, quantum waveguides, Hardy inequality, Dirichlet Laplacian 35R45, 81Q10, Secondary: 35J10, 58J50, 78A50

Introduction

With advances in nanofabrication and measurement science, waveguide-shaped nanostructures have reached the point at which the electron transport can be strongly affected by quantum effects. Among the most influential theoretical results, let us quote the existence of quantum bound states due to bending in curved strips, firstly observed by Exner and Šeba [START_REF] Exner | Bound states in curved quantum waveguides[END_REF] in 1989. The pioneering paper has been followed by a huge number of works demonstrating the robustness of the effect in various geometric settings including higher dimensions, and the research field is still active these days (see [START_REF] Krejčiřík | Ruled strips with asymptotically diverging twisting[END_REF] for a recent paper with a brief overview in the introduction).

In 2008 Ekholm, Kovařík and one of the present authors [START_REF] Ekholm | A Hardy inequality in twisted waveguides[END_REF] observed that the geometric deformation of twisting has a quite opposite effect on the energy spectrum of an electron confined to three-dimensional tubes, for it creates an effectively repulsive interaction (see [START_REF] Krejčiřík | Twisting versus bending in quantum waveguides[END_REF] for an overview of the two reciprocal effects). More specifically, twisting the tube locally gives rise to Hardy-type inequalities and a stability of quantum transport, the effect becomes stronger in globally twisted tubes [START_REF] Ph | Hardy inequalities in globally twisted waveguides[END_REF] and in extreme situations it may even annihilate the essential spectrum completely [START_REF]Waveguides with asymptotically diverging twisting[END_REF] (see also [START_REF] Barseghyan | Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity[END_REF]). The repulsive effect remains effective even under modification of the boundary conditions [START_REF] Ph | Twisted waveguide with a Neumann window, Functional analysis and operator theory for quantum physics[END_REF].

The objective of this paper is to introduce a new, two-dimensional model exhibiting a previously unobserved geometric effect of shearing. Mathematically, it is reminiscent of the effect of twisting in the three-dimensional tubes (in some aspects it also recalls the geometric setting of curved wedges studied in [START_REF]The Hardy inequality and the heat flow in curved wedges[END_REF]), but the lower dimensional simplicity enables us to get an insight into analogous problems left open in [START_REF] Ph | Hardy inequalities in globally twisted waveguides[END_REF] and actually provide a complete spectral picture now. The richness of the toy model is reflected in covering very distinct regimes, ranging from purely essential to purely discrete spectra or a combination of both. We believe that the present study will stimulate further interest in sheared nanostructures.

The model that we consider in this paper is characterised by a positive number d (the transverse width of the waveguide) and a differentiable function f : R → R (the boundary profile of the waveguide). The waveguide Ω is introduced as the set of points in R 2 delimited by the curve x → (x, f (x)) and its vertical translation x → (x, f (x) + d), namely (see Figure 1), Ω := (x, y) ∈ R 2 : f (x) < y < f (x) + d .

(

) 1
We stress that the geometry of Ω differs from the curved strips intensively studied in the literature for the last thirty years (see [START_REF] Krejčiřík | On the spectrum of curved quantum waveguides[END_REF] for a review). In the latter case, the segment (0, d) is translated along the curve x → (x, f (x)) with respect to its normal vector field (so that the waveguide is delimited by two parallel curves), while in the present model the translation is with respect to the constant basis vector in the y-direction. Clearly, it is rather the derivative f that determines the shear deformation of the straight waveguide Ω 0 := R × (0, d). Our standing assumption is that the derivative f is locally bounded and that it admits a (possibly infinite) limit at infinity:

lim |x|→∞ f (x) =: β ∈ R ∪ {±∞}. ( 2
)
If β is finite, we often denote the deficit

ε(x) := f (x) -β , (3)
but notice that ε(x) is not necessarily small for finite x.

We consider an effectively free electron constrained to the nanostructure Ω by hardwall boundaries. Disregarding physical constants, the quantum Hamiltonian of the system can be identified with the Dirichlet Laplacian -∆ Ω D in L 2 (Ω). The spectrum of the straight waveguide Ω 0 (which can be identified with f = 0 in our model) is well known; by separation of variables, one easily conclude that σ(-

∆ Ω 0 D ) = [( π d ) 2 , ∞
) and the spectrum is purely absolutely continuous. The main spectral properties of -∆ Ω D under the shear deformation obtained in this paper are summarised in the following theorems.

First, we locate the possible range of energies of propagating states.

Theorem 1 (Essential spectrum). Let f ∈ L ∞ loc (R) satisfy (2). Then σ ess (-∆ Ω D ) = [E 1 (β), ∞) , where E 1 (β) := (1 + β 2 ) π d 2 . (4)
If β = ±∞ (see Figure 2), the result means σ ess (-∆ Ω D ) = ∅, so the spectrum is purely discrete and there are no propagating states. In this case, the distance of points x ∈ Ω to the boundary ∂Ω tends to zero as |x| → ∞, so Ω is a quasi-bounded domain (cf. Remark 2). This phenomenon is reminiscent of three-dimensional waveguides with asymptotically diverging twisting [START_REF]Waveguides with asymptotically diverging twisting[END_REF][START_REF] Barseghyan | Spectral estimates for Dirichlet Laplacian on tubes with exploding twisting velocity[END_REF].

Our next concern is about the possible existence or absence of discrete eigenvalues below E 1 (β) if β is finite. The following theorem together with Theorem 1 provides a sufficient condition for the existence.

Theorem 2 (Attractive shearing). Let f ∈ L ∞ loc (R) be such that f 2 -β 2 ∈ L 1 (R).
Then either of the following conditions

(i) R (f 2 (x) -β 2 ) dx < 0 ; (ii) R (f 2 (x) -β 2 ) dx = 0, f is not constant and f ∈ L 1 loc (R) ; implies inf σ(-∆ Ω D ) < E 1 (β) . (5)
The theorem indeed implies the existence of discrete spectra under the additional hypothesis (2), because the inequality (5) and the fact that, due to Theorem 1, the essential spectrum of -∆ Ω D starts by E 1 (β) ensure that the lowest point in the spectrum of -∆ Ω D is an isolated eigenvalue of finite multiplicity. Theorem 2 is an analogy of [START_REF] Exner | Spectrum of the Schrödinger operator in a perturbed periodically twisted tube[END_REF], where it was shown that a local "slow down" of twisting in periodically twisted tubes generates discrete eigenvalues (see also [START_REF] Ph | Eigenvalue asymptotics in a twisted waveguide[END_REF]). It is not clear from our proof whether the extra regularity assumption in the condition (ii) is necessary.

Recalling (3), notice that f 2 -β 2 = ε 2 +2βε, so a necessary condition to satisfy either of the conditions (i) or (ii) of Theorem 2 is that the function βε is not non-negative. This motivates us to use the terminology that the shear is attractive (respectively, repulsive) if βε is non-positive (respectively, non-negative) and ε is non-trivial. In analogy with the conjectures stated in [START_REF] Ph | Hardy inequalities in globally twisted waveguides[END_REF], it is expected that the discrete spectrum of -∆ Ω D is empty provided that the shear is either repulsive or possibly attractive but ε 2 -βε. The latter is confirmed in the following special setting (see Figure 3). Theorem 3 (Strong shearing). Let f (x) = β + αε(x), where α, β ∈ R and ε : R → R is a function such that supp ε ⊂ [0, 1] and c 1 ≤ ε(x) ≤ c 2 for all x ∈ [0, 1] with some positive constants c 1 , c 2 . Then there exists a positive number α 0 such that, for all |α| ≥ α 0 ,

σ(-∆ Ω D ) = [E 1 (β), ∞) . (6)
Finally, we establish Hardy-type inequalities in the case of repulsive shear.

Theorem 4 (Repulsive shearing). Let f ∈ L ∞ loc (R).
If βε ≥ 0 and ε is non-trivial, then there exists a positive constant c such that the inequality

-∆ Ω D -E 1 (β) ≥ c 1 + x 2 (7)
holds in the sense of quadratic forms in L 2 (Ω).

Consequently, if the shear satisfies (2) in addition to the repulsiveness, then [START_REF] Dermenjian | Spectral analysis of an acoustic multistratified perturbed cylinder[END_REF] and Theorem 1 imply that the stability result (6) holds, so in particular there are no discrete eigenvalues. Moreover, the spectrum is stable against small perturbations (the smallness being quantified by the Hardy weight on the right-hand side of ( 7)). From Theorem 4 it is also clear that α does not need to be large if αβ ≥ 0 in the setting of Theorem 3.

The organisation of the paper is as follows. In Section 2 we introduce natural curvilinear coordinates to parameterise Ω and express the Dirichlet Laplacian in them. Hardytype inequalities (and in particular Theorem 4) are established in Section 3. The essential spectrum (Theorem 1) is l in Section 4. The sufficient conditions of Theorem 2 (which in particular imply the existence of discrete eigenvalues in the regime (4)) are established in Section 5. Finally, Theorem 3 is proved in Section 6.

Preliminaries

Our strategy to deal with the Laplacian in the sheared geometry Ω is to express it in natural curvilinear coordinates. It employs the identification Ω = L (Ω 0 ), where L : R 2 → R 2 is the shear mapping defined by

L (s, t) := s, f (s) + t . (8)
In this paper we consistently use the notations s and t to denote the "longitudinal" and "transversal" variables in the straight waveguide Ω 0 , respectively. The corresponding metric reads

G := ∇L • (∇L ) T = 1 + f 2 f f 1 , |G| := det(G) = 1, (9)
where the dot denotes the scalar product in R 2 . Recall our standing assumption that f ∈ L ∞ loc (R). It follows that L : Ω 0 → Ω is a local diffeomorphism. In fact, it is a global diffeomorphism, because L is injective. In this way, we have identified Ω with the Riemannian manifold (Ω 0 , G).

The Dirichlet Laplacian -∆ Ω D is introduced standardly as the self-adjoint operator in the Hilbert space L 2 (Ω) associated with the quadratic form

Q Ω D [u] := Ω |∇u| 2 , dom Q Ω D := H 1 0 (Ω).
Employing the diffeomorphism L : Ω 0 → Ω, we have the unitary transform

U : L 2 (Ω) → L 2 (Ω 0 ) : {u → u • L } .
Consequently, -∆ Ω D is unitarily equivalent (and therefore isospectral) to the operator H := U(-∆ Ω D )U -1 in the Hilbert space L 2 (Ω 0 ). By definition, the operator H is associated with the quadratic form

h[ψ] := Q Ω D [U -1 ψ] , dom h := U dom Q Ω D .
Occasionally, we shall emphasise the dependence of h and H on f by writing f as the subscript, i.e. h = h f and H = H f . Given u ∈ C ∞ 0 (Ω), a core of Q Ω D , it follows that ψ := u•L is also compactly supported and ψ ∈ H 1 0 (Ω 0 ). Then it is straightforward to verify that

h[ψ] = ∂ s ψ -f ∂ t ψ 2 + ∂ t ψ 2 , ( 10
)
where • denotes the norm of L 2 (Ω 0 ) and, with an abuse of notation, we denote by the same symbol f the function f ⊗ 1 on R × (0, d). We have the elementary bounds

δ ∂ s ψ 2 + 1 - δ 1 -δ f 2 L ∞ (supp ψ) ∂ t ψ 2 ≤ h[ψ] ≤ 2 ∂ s ψ 2 + 1 + 2 f 2 L ∞ (supp ψ) ∂ t ψ 2 (11)
valid for every δ ∈ (0, 1). The domain of h coincides with the closure of the set of the functions ψ specified above with respect to the graph topology of h. By a standard mollification argument using [START_REF] Exner | Bound states in curved quantum waveguides[END_REF], it follows that C ∞ 0 (Ω 0 ) is a core of h. Moreover, it follows from (11) that if f is bounded, then the graph topology of h is equivalent to the topology of H 1 (Ω 0 ), and therefore dom h = H 1 0 (Ω 0 ) in this case. In general (and in particular if (2) holds with β = ±∞), however, the domain of h is not necessarily equal to H 1 0 (Ω 0 ). Let us summarise the preceding observations into the following proposition.

Proposition 1. Let f ∈ L ∞ loc (R). Then C ∞ 0 (Ω 0 ) is a core of h. If moreover f ∈ L ∞ (R), then dom h = H 1 0 (Ω 0 ). In a distributional sense, we have H = -(∂ s -f ∂ t ) 2 -∂ 2
t , but we shall use neither this formula nor any information on the operator domain of H. Our spectral analysis of H will be exclusively based on the associated quadratic form [START_REF] Exner | Spectrum of the Schrödinger operator in a perturbed periodically twisted tube[END_REF]. Notice that the structure of H is similar to twisted three-dimensional tubes [START_REF] Ekholm | A Hardy inequality in twisted waveguides[END_REF] as well as curved two-dimensional wedges [START_REF]The Hardy inequality and the heat flow in curved wedges[END_REF].

Hardy-type inequalities

In this section we prove Theorem 4. The strategy is to first establish a "local" Hardytype inequality, for which the Hardy weight might not be everywhere positive, and then "smeared it out" to the "global" inequality [START_REF] Dermenjian | Spectral analysis of an acoustic multistratified perturbed cylinder[END_REF] with help of a variant of the classical one-dimensional Hardy inequality. Throughout this section, we assume that β ∈ R and

ε ∈ L ∞ loc (R). Let E 1 := π d 2 and χ 1 (t) := 2 d sin(E 1 t)
denote respectively the lowest eigenvalue and the corresponding normalised eigenfunction of the Dirichlet Laplacian in L 2 ((0, d)). By the variational definition of E 1 , we have the Poincaré-type inequality

∀χ ∈ H 1 0 ((0, d)) , d 0 |χ (t)| 2 dt ≥ E 1 d 0 |χ(t)| 2 dt . ( 12
)
Notice that E 1 = E 1 (0), where the latter is introduced in (4). Neglecting the first term on the right-hand side of (10) and using [START_REF] Krejčiřík | Twisting versus bending in quantum waveguides[END_REF] for the second term together with Fubini's theorem, we immediately get the lower bound

H ≥ E 1 , (13)
which is independent of f .

The main ingredient in our approach is the following valuable lemma which reveals a finer structure of the form h. With an abuse of notation, we use the same symbol χ 1 for the function 1 ⊗ χ 1 on R × (0, d), and similarly for its derivative χ 1 , while ε also denotes the function ε ⊗ 1 on R × (0, d).

Lemma 1 (Ground-state decomposition). For every

ψ ∈ C ∞ 0 (Ω 0 ), we have h[ψ] -E 1 (β) ψ 2 = ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 + χ 1 ∂ t (χ -1 1 ψ) 2 + Ω 0 β ε(s) E 1 + χ 1 (t) χ 1 (t) 2 |ψ(s, t)| 2 ds dt. ( 14
) Proof. Let φ ∈ C ∞ 0 (Ω 0 ) be defined by the decomposition ψ(s, t) = χ 1 (t)φ(s, t). Writ- ing 2 (φ∂ t φ) = ∂ t |φ| 2 ,
integrating by parts and using the differential equation that χ 1 satisfies, we have

∂ t ψ 2 = χ 1 φ 2 + χ 1 ∂ t φ 2 + 2 (χ 1 φ, χ 1 ∂ t φ) = χ 1 ∂ t φ 2 + E 1 ψ 2 ,
where (•, •) denotes the inner product of L 2 (Ω 0 ). Similarly,

∂ s ψ -f ∂ t ψ 2 = ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t φ 2 + β 2 χ 1 φ 2 -2β (χ 1 ∂ s φ -εχ 1 ∂ t φ -εχ 1 φ -βχ 1 ∂ t φ, χ 1 φ) = ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t φ 2 + βE 1 (εχ 1 φ, χ 1 φ) + β(εχ 1 φ, χ 1 φ) + β 2 E 1 (χ 1 φ, χ 1 φ) = ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t φ 2 + βE 1 (εψ, ψ) + β(εχ 1 φ, χ 1 φ) + β 2 E 1 ψ 2 ,
where the second equality follows by integrations by parts in the cross term. Summing up the results of the two computations and subtracting E 1 (β) ψ 2 , we arrive at the desired identity.

From now on, let us assume that the shear is repulsive, i.e. βε ≥ 0 and ε = 0. If β = 0, then Lemma 1 immediately gives the following local Hardy-type inequality

H -E 1 (β) ≥ βε E 1 + χ 1 χ 1 2 . ( 15
)
Recall that E 1 (β) corresponds to the threshold of the essential spectrum if (2) holds (cf. Theorem 1), so [START_REF] Krejčiřík | Ruled strips with asymptotically diverging twisting[END_REF] in particular ensures that there is no (discrete) spectrum below E 1 (β) (even if β = 0). Note that the Hardy weight on the right-hand side of ( 15) diverges on ∂Ω 0 . The terminology "local" comes from the fact that the Hardy weight might not be everywhere positive (e.g. if βε is compactly supported).

In order to obtain a non-trivial non-negative lower bound including the case β = 0, we have to exploit the positive terms that we have neglected when coming from [START_REF]The Hardy inequality and the heat flow in curved wedges[END_REF] to [START_REF] Krejčiřík | Ruled strips with asymptotically diverging twisting[END_REF]. To do so, let I ⊂ R be any bounded open interval and set Ω I 0 := I × (0, d). In the Hilbert space L 2 (Ω I 0 ) let us consider the quadratic form

q I [ψ] := ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 L 2 (Ω I 0 ) + χ 1 ∂ t (χ -1 1 ψ) 2 L 2 (Ω I 0 ) , dom q I := {ψ Ω I 0 : ψ ∈ H 1 0 (Ω 0 )} .
The form corresponds to the sum of the two first terms on the right-hand side of ( 14) with the integrals restricted to Ω I 0 . Lemma 2. The form q I is closed.

Proof. First of all, notice that the form q I is obviously closed if β = 0 (one can proceed as in [START_REF] Exner | Bound states in curved quantum waveguides[END_REF]). In general, given ψ ∈ dom q I , we employ the estimates

δ ∂ s ψ -ε∂ t ψ 2 L 2 (Ω I 0 ) + 1 - δ 1 -δ β 2 χ 1 ∂ t (χ -1 1 ψ) 2 L 2 (Ω I 0 ) ≤ q I [ψ] ≤ 2 ∂ s ψ -ε∂ t ψ 2 L 2 (Ω I 0 ) + (1 + 2β 2 ) χ 1 ∂ t (χ -1 1 ψ) 2 L 2 (Ω I 0 )
valid for every δ ∈ (0, 1) and the identity (easily checked by employing the density of

C ∞ 0 (Ω 0 ) in H 1 0 (Ω 0 )) χ 1 ∂ t (χ -1 1 ψ) 2 L 2 (Ω I 0 ) = ∂ t ψ 2 L 2 (Ω I 0 ) -E 1 ψ 2 L 2 (Ω I 0 ) . (16)
Instead of the latter, we could also use the fact that χ -1

1 ψ ∈ L 2 (Ω 0 ) whenever ψ ∈ H 1 0 (Ω 0 ), cf.
[6, Thm. 1.5.6]. Using in addition the aforementioned fact that q I is closed if β = 0, we conclude that q I is closed for any value β ∈ R.

Consequently, the form q I is associated to a self-adjoint operator Q I with compact resolvent. Let us denote by λ I its lowest eigenvalue, i.e.,

λ I := inf ψ∈dom q I ψ =0 q I [ψ] ψ 2 L 2 (Ω I 0 ) . (17)
The eigenvalue λ I is obviously non-negative. The following lemma shows that λ I is actually positive whenever ε is non-trivial on I.

Lemma 3. λ I = 0 if, and only if, ε = 0 on I.

Proof. If ε I = 0, then the function ψ(s, t) := χ 1 (t) minimises [START_REF] Krejčiřík | Location of the essential spectrum in curved quantum layers[END_REF] with λ I = 0. Conversely, let us assume that λ I = 0 and denote by ψ 1 the corresponding eigenfunction of Q I , i.e. the minimiser of [START_REF] Krejčiřík | Location of the essential spectrum in curved quantum layers[END_REF]. It follows from [START_REF] Krejčiřík | Location of the essential spectrum in curved quantum layers[END_REF] together with (16) that

∂ s ψ 1 -ε∂ t ψ 1 -βχ 1 ∂ t (χ -1 1 ψ 1 ) 2 L 2 (Ω I 0 ) = 0 , ∂ t ψ 1 2 L 2 (Ω I 0 ) -E 1 ψ 1 2 L 2 (Ω I 0 ) = 0 . (18) Writing ψ 1 (s, t) = ϕ(s)χ 1 (t) + φ(s, t), where d 0 χ 1 (t)φ(s, t) dt = 0
for almost every s ∈ I, we have

∂ t ψ 1 2 L 2 (Ω I 0 ) -E 1 ψ 1 2 L 2 (Ω I 0 ) = ∂ t φ 2 L 2 (Ω I 0 ) -E 1 φ 2 L 2 (Ω I 0 ) ≥ (E 2 -E 1 ) φ 2 L 2 (Ω I 0 )
, where E 2 = 4E 1 is the second eigenvalue of the Dirichlet Laplacian in L 2 ((0, d)). Since the gap E 2 -E 1 is positive, it follows from the second identity in (18) that φ = 0. Plugging now the separated-eigenfunction Ansatz ψ 1 (s, t) = ϕ(s)χ 1 (t) into the first identity in [START_REF] Reed | Methods of modern mathematical physics, IV. Analysis of operators[END_REF] and integrating by parts with respect to t in the cross term, we obtain

0 = ∂ s ψ 1 -ε∂ t ψ 1 2 L 2 (Ω I 0 ) = ϕ χ 1 2 L 2 (Ω I 0 ) + εϕχ 1 2 L 2 (Ω I 0 ) . Consequently, ϕ is necessarily constant and ε L 2 (I) = 0.
Using [START_REF] Krejčiřík | Location of the essential spectrum in curved quantum layers[END_REF], we can improve [START_REF] Krejčiřík | Ruled strips with asymptotically diverging twisting[END_REF] to the following local Hardy-type inequality

H -E 1 (β) ≥ βε E 1 + χ 1 χ 1 2 + λ I 1 I×(0,d) , (19)
where 1 M denotes the characteristic function of a set M . The inequality is valid with any bounded interval I ⊂ R, but recall that λ I is positive if, and only if, I is chosen in such a way that ε is not identically equal to zero on I, cf. Lemma 3.

The passage from the local Hardy-type inequality (19) to the global Hardy-type inequality of Theorem 4 will be enabled by means of the following crucial lemma, which is essentially a variant of the classical one-dimensional Hardy inequality.

Lemma 4. Let s 0 ∈ R and ψ ∈ C ∞ 0 (Ω 0 \ {s 0 } × (0, d)). Then ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 + χ 1 ∂ t (χ -1 1 ψ) 2 ≥ 1 4(1 + β 2 ) Ω 0 |ψ(s, t)| 2 (s -s 0 ) 2 ds dt . (20)
Proof. Denote ρ(s, t) := (s -s 0 ) -1 . Given any real number α, we write

∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) -αρψ 2 = ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 + α 2 ρψ 2 -2α (∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ), ρψ) = ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 + (α 2 -α) ρψ 2 + 2αβ (χ 1 ∂ t (χ -1 1 ψ), ρψ) ≤ ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 + (α 2 -α + α 2 β 2 ) ρψ 2 + χ 1 ∂ t (χ -1 1 ψ) 2 ,
where the second equality follows by integrations by parts and the estimate is due to the Schwarz inequality. Since the first line is non-negative, we arrive at the desired inequality by choosing the optimal α :=

[2(1 + β 2 )] -1 .
Now we are in a position to prove the "global" Hardy inequality of Theorem 4.

Proof of Theorem 4. Let ψ ∈ C ∞ 0 (Ω 0 ). First of all, since ε is assumed to be non-trivial, there exists a real number s 0 and positive b such that ε restricted to the interval I := (s 0 -b, s 0 + b) is non-trivial. By Lemma 17, it follows that λ I is positive. Then the local Hardy-type inequality (19) implies the estimate

h[ψ] -E 1 (β) ψ 2 ≥ λ I ψ 2 L 2 (Ω I 0 ) . (21)
Second, let η ∈ C ∞ (R) be such that 0 ≤ η ≤ 1, η = 0 in a neighbourhood of s 0 and η = 1 outside I. Let us denote by the same symbol η the function η ⊗ 1 on R × (0, d), and similarly for its derivative η . Writing ψ = ηψ + (1 -η)ψ, we have

Ω 0 |ψ(s, t)| 2 1 + (s -s 0 ) 2 ds dt ≤ 2 Ω 0 |(ηψ)(s, t)| 2 (s -s 0 ) 2 ds dt + 2 Ω 0 |((1 -η)ψ)(s, t)| 2 ds dt . ( 22
)
Since ηψ satisfies the hypothesis of Lemma 4, we have

Ω 0 |(ηψ)(s, t)| 2 (s -s 0 ) 2 ds dt ≤ 4(1 + β 2 ) ∂ s (ηψ) -ηε∂ t ψ -ηβχ 1 ∂ t (χ -1 1 ψ) 2 + ηχ 1 ∂ t (χ -1 1 ψ) 2 ≤ 8(1 + β 2 ) ∂ s ψ -ε∂ t ψ -βχ 1 ∂ t (χ -1 1 ψ) 2 + 8(1 + β 2 ) η ψ + 4(1 + β 2 ) χ 1 ∂ t (χ -1 1 ψ) 2 ≤ 8(1 + β 2 )(h[ψ] -E 1 (β) ψ 2 ) + 8(1 + β 2 ) η ∞ ψ 2 L 2 (Ω I 0 )
where η ∞ denotes the supremum norm of η . Here the last inequality employs Lemma 1 (recall that we assume that the shear is repulsive). At the same time,

Ω 0 |((1 -η)ψ)(s, t)| 2 ds dt ≤ ψ 2 L 2 (Ω I 0 ) . From (22) we therefore deduce h[ψ]-E 1 (β) ψ 2 ≥ 1 16(1 + β 2 ) Ω 0 |ψ(s, t)| 2 1 + (s -s 0 ) 2 ds dt-η 2 ∞ + 1 8(1 + β 2 ) ψ 2 L 2 (Ω I 0 ) . (23)
Finally, interpolating between ( 21) and ( 23), we obtain

h[ψ] -E 1 (β) ψ 2 ≥ δ 16(1 + β 2 ) Ω 0 |ψ(s, t)| 2 1 + (s -s 0 ) 2 ds dt + (1 -δ)λ I -δ η 2 ∞ - δ 8(1 + β 2 ) ψ 2 L 2 (Ω I 0 )
for every real δ. Choosing (positive) δ in such a way that the square bracket vanishes, we arrive at the global Hardy-type inequality

h[ψ] -E 1 (β) ψ 2 ≥ c Ω 0 |ψ(s, t)| 2 1 + (s -s 0 ) 2 ds dt with c := λ I 16(1 + β 2 )(λ I + η 2 ∞ ) + 2
From this inequality we also deduce

h[ψ] -E 1 (β) ψ 2 ≥ c Ω 0 |ψ(s, t)| 2 1 + s 2 ds dt (24)
with

c := c inf s∈R 1 + s 2 1 + (s -s 0 ) 2 ,
where the infimum is positive. The desired inequality [START_REF] Dermenjian | Spectral analysis of an acoustic multistratified perturbed cylinder[END_REF] follows by the unitary equivalence between H and -∆ Ω D together with the fact that the the coordinates s and x are equivalent through this transformation, cf. (8).

Remark 1. The present proof does not employ the presence of the first term on the right-hand side of (19). The inequality (24) can be consequently improved to

H -E 1 (β) ≥ δ βε(s) E 1 + χ 1 (t) χ 1 (t) 2 + (1 -δ) c 1 + s 2 with any δ ∈ [0, 1].

Location of the essential spectrum

In this section we prove Theorem 1. Since the technical approaches for finite and infinite β are quite different, we accordingly split the section into two subsections. In either case, we always assume f ∈ L ∞ loc (R).

Finite limits

We employ the following characterisation of the essential spectrum, for which we are inspired in [START_REF] Dermenjian | Spectral analysis of an acoustic multistratified perturbed cylinder[END_REF]Lem. 4.2].

Lemma 5. A real number λ belongs to the essential spectrum of H if, and only if, there exists a sequence {ψ n } ∞ n=1 ⊂ dom h such that the following three conditions hold:

(i) ψ n = 1 for every n ≥ 1, (ii) (H -λ)ψ n → 0 as n → ∞ in the norm of the dual space (dom h) * . (iii) supp ψ n ⊂ Ω 0 \ (-n, n) × (0, d) for every n ≥ 1.
Proof. By the general Weyl criterion modified to quadratic forms (cf. [START_REF] Dermenjian | Spectral analysis of an acoustic multistratified perturbed cylinder[END_REF]Lem. 4.1] or [START_REF] Krejčiřík | Location of the essential spectrum in curved quantum layers[END_REF]Thm. 5] where the latter contains a proof), λ belongs to the essential spectrum of H if, and only if, there exists a sequence {φ n } ∞ n=1 ⊂ dom h such that (i) and (ii) hold but (iii) is replaced by

(iii') φ n → 0 as n → ∞ weakly in L 2 (Ω 0 ).
The sequence {ψ n } ∞ n=1 satisfying (i) and (iii) is clearly weakly converging to zero. Hence, one implication of the lemma is obvious. Conversely, let us assume that there exists a sequence {φ n } ∞ n=1 ⊂ dom h satisfying (i), (ii) and (iii') and let us construct from it a sequence {ψ n } ∞ n=1 ⊂ dom h satisfying (i), (ii) and (iii). Since C ∞ 0 (Ω 0 ) is the form core of H (cf. Proposition 1), we may assume that {φ n } n∈N ⊂ C ∞ 0 (Ω 0 ). Let us denote the norms of dom h and (dom h) * by • +1 and • -1 , respectively. One has • ±1 = (H + 1) ±1/2 • . By writing

φ n = (H + 1) -1 (H -λ)φ n + (λ + 1)(H + 1) -1 φ n (25)
and using (ii), we see that the sequence

{φ n } ∞ n=1 is bounded in dom h. Let η ∈ C ∞ (R) be such that 0 ≤ η ≤ 1, η = 0 on [-1, 1] and η = 1 on R \ (-2, 2).
For every k ≥ 1, we set η k (s) := η(s/k) and we keep the same notation for the function η k ⊗ 1 on R × (0, d), and similarly for its derivatives η k and η k . Clearly, supp

η k ⊂ Ω 0 \ (-k, k) × (0, d). For every k ≥ 1, the operator (1 -η k )(H + 1) -1 is compact in L 2 (Ω 0 ). By virtue of the weak convergence (iii'), it follows that, for every k ≥ 1, (1 -η k )(H + 1) -1 φ n → 0 as n → ∞ in L 2 (Ω 0 ). Then there exists a subsequence {φ n k } ∞ k=1 of {φ n } ∞ n=1 such that (1 -η k )(H + 1) -1 φ n k → 0 as k → ∞ in L 2 (Ω 0 ). Consequently, the identity (25) together with (ii) implies that (1 -η k )φ n k → 0 as k → ∞ in L 2 (Ω 0 ).
It follows that η k φ n k can be normalised for all sufficiently large k. More specifically, redefining the subsequence if necessary, we may assume that η k φ n k ≥ 1/2 for all k ≥ 1. We set

ψ k := η k φ n k η k φ n k
and observe that {ψ k } ∞ k=1 satisfies (i) and (iii). It remains to verify (ii). To this purpose, we notice that, using the duality, (ii) means

(H -λ)φ n -1 = sup ϕ∈dom h ϕ =0 |h(ϕ, φ n ) -λ(ϕ, φ n )| ϕ +1 ---→ n→∞ 0 , (26)
where ϕ 2 +1 = h[ϕ] + ϕ 2 . By the direct computation employing integrations by parts, one can check the identity

h(ϕ, η k φ n k ) -λ(ϕ, η k φ n k ) = h(η k ϕ, φ n k ) -λ(η k ϕ, φ n k ) + 2(∂ s ϕ -f ∂ t ϕ, η k φ n k ) + (ϕ, η k φ n k )
for every test function ϕ ∈ C ∞ 0 (Ω 0 ), a core of dom h. Using (26), we have sup

ϕ∈C ∞ 0 (Ω 0 ) ϕ =0 |h(η k ϕ, φ n k ) -λ(η k ϕ, φ n k )| ϕ +1 ≤ sup ϕ∈C ∞ 0 (Ω 0 ) η k ϕ =0 |h(η k ϕ, φ n k ) -λ(η k ϕ, φ n k )| η k ϕ +1 = (H -λ)φ n k -1 ---→ k→∞ 0 .
At the same time, using the Schwarz inequality and the estimates

∂ s ϕ -f ∂ t ϕ 2 ≤ h[ϕ] ≤ ϕ 2 +1 , we get sup ϕ∈C ∞ 0 (Ω 0 ) ϕ =0 |2(∂ s ϕ -f ∂ t ϕ, η k φ n k )| ϕ +1 ≤ η k ∞ φ n k ,
where η k ∞ denotes the supremum norm of η k . In view of the normalisation (i) and since η k ∞ = k -1 η ∞ , we see that also this term tends to zero as k → ∞. Finally, using the Schwarz inequality and the estimate ϕ ≤ ϕ +1 , we have sup

ϕ∈C ∞ 0 (Ω 0 ) ϕ =0 |(ϕ, η k φ n k )| ϕ +1 ≤ η k ∞ φ n k = k -2 η ∞ ---→ k→∞ 0 .
Summing up, we have just checked

(H -λ)(η k φ n k ) -1 → 0 as k → ∞. Recalling η k φ n k ≥ 1/2
, the desired property (ii) for {ψ k } ∞ k=1 follows. Using this lemma, we immediately arrive at the following "decomposition principle" (saying that the essential spectrum is determined by the behaviour at infinity only).

Proposition 2. If (2) holds with β ∈ R, then σ ess (H β+ε ) = σ ess (H β ).
Proof. Let λ ∈ σ ess (H β ). By Lemma 5, there exists a sequence {ψ n } ∞ n=1 satisfying the properties (i)-(iii) with H being replaced by H β . One easily checks the identity

h β+ε (ϕ, ψ n ) = h β (ϕ, ψ n ) -(∂ s ϕ -β∂ t ϕ, ε∂ t ψ n ) -(ε∂ t ϕ, ∂ s ψ n -(β + ε)∂ t ψ n ) ( 27
)
for every test function ϕ ∈ dom h β = H 1 0 (Ω 0 ) = dom h β+ε (cf. Proposition 1). Now we proceed similarly as in the end of the proof of Lemma 5. Using the estimates ∂ s ϕ -β∂ t ϕ 2 ≤ h[ϕ] ≤ ϕ +1 and the Schwarz inequality, we have sup

ϕ∈H 1 0 (Ω 0 ) ϕ =0 |(∂ s ϕ -β∂ t ϕ, ε∂ t ψ n )| ϕ +1 ≤ ε∂ t ψ n ≤ ε L ∞ (R\(-n,n)) ∂ t ψ n ---→ n→∞ 0 .
Recall that {ψ n } ∞ n=1 is bounded in H 1 0 (Ω 0 ), cf. (25) and Proposition 1. Similarly, using ∂ t ϕ ≤ ϕ +1 and the Schwarz inequality, we obtain sup

ϕ∈H 1 0 (Ω 0 ) ϕ =0 |(ε∂ t ϕ, ∂ s ψ n -(β + ε)∂ t ψ n )| ϕ +1 ≤ ε∂ s ψ n + β + ε ∞ ε∂ t ψ n ≤ ε L ∞ (R\(-n,n)) ( ∂ s ψ n + β + ε ∞ ∂ t ψ n ) ---→ n→∞ 0 .
Consequently, since (H β -λ)ψ n -1 → 0 as n → ∞ due to (ii), it follows from ( 27) and ( 26) that also (H β+ε -λ)ψ n -1 → 0 as n → ∞. Hence λ ∈ σ ess (H β+ε ).

The opposite inclusion σ ess (H β+ε ) ⊂ σ ess (H β ) is proved analogously.

It remains to determine the (essential) spectrum for the constant shear.

Proposition 3. One has σ(H β ) = σ ess (H β ) = [E 1 (β), ∞).
Proof. By performing the partial Fourier transform in the longitudinal variable s ∈ R, one has the unitary equivalence

H β = -(∂ s -β∂ t ) 2 -∂ 2 t ∼ = -(iξ -β∂ t ) 2 -∂ 2 t , where ξ ∈ R is the dual variable in the Fourier image. Consequently, σ(H β ) = ξ∈R σ(T β (ξ)) , (28)
where, for each fixed ξ ∈ R, T β (ξ) is the operator in L 2 ((0, d)) that acts as -(iξ-β∂ t ) 2 -∂ 2 t and satisfies Dirichlet boundary conditions. The spectral problem for T β (ξ) can be solved explicitly. Alternatively, one can proceed by "completing the square" and "gauging out a constant magnetic field" as follows:

-(iξ -β∂ t ) 2 -∂ 2 t = (1 + β 2 )(-∂ 2 t ) + 2iξβ∂ t + ξ 2 = -(1 + β 2 ) ∂ t -i ξβ 1 + β 2 2 - ξ 2 β 2 1 + β 2 + ξ 2 = -(1 + β 2 ) ∂ t -i ξβ 1 + β 2 2 + ξ 2 1 + β 2 = e iF t -(1 + β 2 )∂ 2 t + ξ 2 1 + β 2 e -iF t ,
where

F := ξβ 1 + β 2 . Consequently, σ(T β (ξ)) = (1 + β 2 )E n + ξ 2 1 + β 2 ∞ n=1 , (29)
where E n = n 2 E 1 denotes the n th eigenvalue of the Dirichlet Laplacian in L 2 ((0, d)). As a consequence of ( 28) and (29), we get the desired claim.

For finite β, Theorem 1 follows as a direct consequence of Propositions 2 and 3.

Infinite limits

Let us now prove Theorem 1 in the case β = ±∞. Here the proof is based on the following purely geometric fact.

Proposition 4. Let f ∈ L ∞ loc (R) satisfy (2) with β = ±∞. Then lim sup |x|→∞ x∈Ω |B 1 (x) ∩ Ω| = 0 , (30)
where B 1 (x) denotes the disk of radius 1 centred at x ∈ R 2 and | • | stands for the Lebesgue measure.

Proof. Recall the diffeomorphism L : Ω 0 → Ω given by [START_REF] Edmunds | Spectral theory and differential operators[END_REF]. For every number η ∈ R, let Σ(η) denote the set of points intersecting the boundary ∂Ω with the straight horizontal half-line (0, ∞)×{η}. Since f (s) → ±∞ as |s| → ∞, it follows that there exists a positive constant s 0 such that f is either strictly increasing or strictly decreasing on (s 0 , ∞). Let us consider the former situation, the latter can be treated analogously. It follows that there exists a positive constant η 0 such that, for all η ≥ η 0 , the set Σ(η) consists of just two points x 1 (η) = L (s 1 (η), d) and x 2 (η) = L (s 2 (η), 0) with 1 < s 1 (η) < s 2 (η). Let x = L (s, t) be a point in Ω such that s is positive and the vertical component satisfies the inequality η := f (s) + t ≥ 1 + η 0 (notice that |x| → ∞ if, and only if, |s| → ∞). Then we have the crude estimate

|B 1 (x) ∩ Ω| ≤ 2 |s 2 (η + 1) -s 1 (η -1)| .
By the mean value theorem,

2 = f (s 2 (η + 1)) -f (s 1 (η -1)) = f (ξ η ) |s 2 (η + 1) -s 1 (η -1)| ,
where ξ η ∈ (s 1 (η-1), s 2 (η+1)). Since s 1 (η-1) → ∞ as s → ∞, we also have ξ η → ∞, and therefore |B 1 (x) ∩ Ω| → 0 as s → ∞. The case s → -∞ can be treated analogously.

By the Berger-Schechter criterion (cf. [8, Thm. V.5.17 & Rem. V.5.18.4]), it follows from (30) that the embedding W 1,2 0 (Ω) → L 2 (Ω) is compact. Consequently, the Dirichlet Laplacian -∆ Ω D has no essential spectrum if β = ±∞ and Theorem 1 is proved.

Remark 2. A necessary condition for (30) is that lim

|x|→∞ x∈Ω dist(x, ∂Ω) = 0 ,
which means that Ω is a quasi-bounded domain ([8, Sec. X.6.1]) in the regime β = ±∞.

Existence of discrete spectrum

In view of Theorem 1, the conclusion (5) of Theorem 2 guarantees that any of the conditions (i) or (ii) implies that -∆ Ω D possesses at least one isolated eigenvalue of finite multiplicity below E 1 (β). Let us establish these sufficient conditions.

Proof of Theorem 2. Given any positive n, let ϕ n : R → [0, 1] be the continuous function such that ϕ n = 1 on [-n, n], ϕ n = 0 on R\(-2n, 2n) and linear in the remaining intervals. Set ψ n (s, t) := ϕ n (s)χ 1 (t) and observe that ψ n ∈ dom(h)∩H 1 0 (Ω 0 ). Moreover, the identity of Lemma 1 remains valid for ψ n by approximation. Indeed, it is only important to notice that χ -1 1 ψ n ∈ L 2 (Ω 0 ) as a consequence of [START_REF] Davies | Heat kernels and spectral theory[END_REF]Thm. 1.5.6]. Consequently,

h[ψ n ] -E 1 (β) ψ n 2 = ∂ s ψ n -ε∂ t ψ n 2 + Ω 0 β ε(s) E 1 + χ 1 (t) χ 1 (t) 2 |ψ n (s, t)| 2 ds dt = ϕ n 2 L 2 (R) + E 1 R ε(s) 2 + 2βε(s) |ϕ n (s)| 2 ds ,
where the second equality follows by the special form of ψ n (the cross term vanishes due to an integration by parts with respect to t) and the formula χ 1 2 L 2 ((0,d)) = E 1 together with the normalisation of χ 1 . Noticing that ϕ n 2 L 2 (R) = 2n -1 and using the dominated convergence theorem, we get

h[ψ n ] -E 1 (β) ψ n 2 ---→ n→∞ E 1 R ε(s) 2 + 2βε(s) ds ,
where the right-hand side is negative by the hypothesis (i). Consequently, there exists a positive n such that h[ψ n ] -E 1 (β) ψ n 2 < 0, so the desired result [START_REF] Ph | Eigenvalue asymptotics in a twisted waveguide[END_REF] follows by the variational characterisation of the lowest point in the spectrum of H. Now assume (ii), in which case the shifted form h 1 [ψ n ] := h[ψ n ]-E 1 (β) ψ n 2 converges to zero as n → ∞. Then we modify the test function ψ n by adding a small perturbation:

ψ n,δ (s, t) := ψ n (s, t) + δ φ(s, t) with φ(s, t) := ξ(s) t χ 1 (t) ,
where δ is a real number and ξ ∈ C ∞ 0 (R) is a real-valued function to be determined later. Writing lim

n→∞ h 1 [ψ n,δ ] = lim n→∞ h 1 [ψ n ] + 2δ h 1 (ψ n , φ) + δ 2 h 1 [φ] = 2δ lim n→∞ h 1 (ψ n , φ) + δ 2 h 1 [φ] , (31)
it is enough to show, in order to establish [START_REF] Ph | Eigenvalue asymptotics in a twisted waveguide[END_REF], that the limit of h 1 (ψ n , φ) as n → ∞ is non-zero for a suitable choice of ξ. Indeed, it then suffices to choose δ sufficiently small and of suitable sign to make the second line of (31) negative, and subsequently choose n sufficiently large to make the whole expression h 1 [ψ n,δ ] negative. Employing Lemma 1 and the fact that ϕ n = 1 on supp ξ for all sufficiently large n, we have

lim n→∞ h 1 (ψ n , φ) = Ω 0 χ 1 (t)χ 1 (t) [ε(s) 2 ξ(s) + βε(s)ξ(s)] -χ 1 (t)χ 1 (t)t ε(s)ξ (s) + χ (t) 2 t [ε(s) 2 ξ(s) + βε(s)ξ(s)] + χ 1 (t) 2 t E 1 βε(s)ξ(s) ds dt = 1 2 R ε(s)ξ (s) + E 1 d [ε(s) 2 + 2βε(s)] ξ(s) ds = 1 2 R -ε (s) + E 1 d [ε(s) 2 + 2βε(s)] ξ(s) ds .
Here the last equality follows by an integration by parts employing the extra hypothesis that the derivative ε exists as a locally integrable function. Now let assume, by contradiction, that the last integral equals zero for all possible choices of ξ. Then ε must solve the differential equation -ε (s) + E 1 d [ε(s) 2 + 2βε(s)] = 0, which admits the explicit one-parametric class of solutions

ε c (s) = 2β c e -2E 1 dβs -1 , c ∈ R .
The solutions ε c for non-zero c are not admissible because f 2 -β is either a positive (if c > 0) or negative (if c < 0) function, so its integral cannot be equal to zero (for positive c the function ε c additionally admits a singularity at s = -(2E 1 dβ) -1 log c), while ε 0 is excluded by the requirement that f is not constant. Hence, there exists a compactly supported ξ such that the limit of h 1 (ψ n , φ) as n → ∞ is non-zero and the proof is concluded by the argument described below (31).

Strong shearing

In this last section we prove Theorem 3. Accordingly, let us assume that the shear admits the special form f (x) = β + αε(x), where α, β ∈ R and ε : R → R is a function such that supp ε ⊂ [0, 1] and c 1 ≤ ε(x) ≤ c 2 for all x ∈ [0, 1] with some positive constants c 1 , c 2 .

We divide the proof into several subsections.

Preliminaries

If β is non-negative (respectively, non-positive), then the desired stability result (6) obviously holds for all α ≥ 0 (respectively, α ≤ 0) due to the repulsiveness of the shear (cf. Theorem 4). On the other hand, the shear becomes attractive if β is positive (respectively, negative) and α is small in absolute value and negative (respectively, positive), so (6) cannot hold in these cases (cf. Theorem 2). The non-trivial part of Theorem 3 therefore consists in the statement that (6) holds again in the previous regimes provided that α becomes large in absolute value. Without loss of generality, it is enough to prove the remaining claim for β positive, α negative and the shear function

f (x) :=      βx if x < 0 , βx + α x 0 ε(s) ds if 0 ≤ x ≤ 1 , βx + α 1 0 ε(s) ds if 1 < x . (32)
As above, we define Ω by (1) and do not highlight the dependence on α. We use the notation (x, y) ∈ R 2 for the Cartesian coordinates in which we describe Ω.

Subdomains decomposition

Our strategy is to decompose Ω into a union of three open subsets Ω ext , Ω ver , Ω int and of two connecting one-dimensional interfaces Σ ext , Σ int on which we impose extra Neumann

Neumann bracketing

Notice that the majority of the sets introduced above depend on α, although it is not explicitly highlighted by the notation. In particular, the parallelogram-like domain determined by the vertices OCO C (subset of Ω ver ∪ Ω int ) converges in a sense to the half-line {0} × (-∞, 0) as α → -∞. Hence it is expected that this set is spectrally negligible in the limit and the spectrum of -∆ Ω D converges as α → -∞ to the spectrum of the Dirichlet Laplacian in {(x, y) ∈ Ω : x < 0}.

Since we only need to show that the spectral threshold of -∆

σ(-∆ Ω D ) ≥ min inf σ(-∆ Ωext DN ), inf σ(-∆ Ωver DN ), inf σ(-∆ Ω int DN ) ,
where -∆ Ωext DN (respectively, -∆ Ωver DN ; respectively, -∆ Ω int DN ) is the operator in L 2 (Ω ext ) (respectively, L 2 (Ω ver ); respectively, L 2 (Ω int )) that acts as the Laplacian and satisfies Neumann boundary conditions on Σ ext (respectively, on Σ ext ∪ Σ int ; respectively, on Σ int ) and Dirichlet boundary conditions on the remaining parts of the boundary. It remains to study the spectral thresholds of the individual operators.

6.4 Spectral threshold of the decomposed subsets 6.4.1 Exterior sets Notice that Ω ext consists of two connected components, each of them being congruent to the half-strip (0, ∞) × (0, d/ 1 + β 2 ). Consequently, -∆ Ωext DN is isospectral to the Laplacian in the half-strip, subject to Neumann boundary conditions on {0} × (0, d/ 1 + β 2 ) and Dirichlet boundary conditions on the remaining parts of the boundary. By a separation of variables, it is straightforward to see that σ(-∆ Ωext DN ) = [E 1 (β), ∞). In particular, the spectral threshold of -∆ Ωext DN equals E 1 (β) for all α.

Interior set

Given any y ∈ [f (1) + d, 0], let x 1 (y) (respectively, x 2 (y)) be the unique solution of the equation f (x 1 (y)) = y (respectively, f (x 2 (y)) + d = y). Notice that x 1 (y) < x 2 (y). In particular, x 1 (0) = 0 and x 2 (0) = x 0 , while x 1 (f (1) + d) = x 0 and x 2 (f (1) + d) = f (1) + d.

To estimate the spectral threshold of -∆ Ω int DN , we write where the second inequality follows by a Poincaré inequality of the type (12) and Fubini's theorem. Subtracting the equations that x 1 (y) and x 2 (y) satisfy, the mean value theorem yields d = f (x 1 (y)) -f (x 2 (y)) = -f (ξ)[x 2 (y) -x 1 (y)] with 0 < ξ < x 0 < 1. Since f (x) = β + αε(x) ≥ β -c 2 |α| for x ∈ [0, 1], it follows that, for all α < β/c 2 , we have the uniform bound x 2 (y) -x 1 (y) ≥ d/(c 2 |α| -β). Consequently, inf σ(-∆ Ω int DN ) → ∞ as α → -∞. In particular, the spectral threshold of -∆ Ω int DN is greater than E 1 (β) for all negative α with sufficiently large |α|.

Verge sets

The decisive set Ω ver consists of two connected components Ω ± ver . We consider Ω - ver , the argument for Ω + ver being analogous. Let us thus study the spectral threshold of the operator -∆ Ω - Consequently, there exists a negative α 0 such that, for every α ≤ α 0 , the spectral threshold of -∆ Ωver DN is also bounded from below by E 1 (β). This concludes the proof of Theorem 3.

Figure 1 :

 1 Figure 1: The geometry of a sheared waveguide.

Figure 2 :

 2 Figure 2: A sheared waveguide with β = +∞.

Figure 3 :

 3 Figure 3: The subdomains decomposition for Ω corresponding to (32).

Ω

  int |∇ψ(x, y)| 2 dx dy ≥ Ω int |∂ x ψ(x, y)| 2 dx dy ≥ Ω int π x 2 (y) -x 1 (y) 2 |ψ(x, y)| 2 dx dy ,

  ver DN in L 2 (Ω - ver ) that acts as the Laplacian and satisfies Neumann boundary conditions on Σ - ext ∪ Σ - int and Dirichlet boundary conditions on the remaining parts of the boundary ∂Ω - ver . Since the spectrum of this operator is purely discrete, we are interested in the behaviour of its lowest eigenvalue. As α → -∞, the set Ω - ver converges in a sense to the open right triangle T determined by the vertices OAC. More specifically, |Ω - ver \ T | = O(|α| -1 ) as α → -∞. Using the convergence result [1, Thm. 29], it particularly follows that the lowest eigenvalue of -∆ Ω - ver DN converges to the lowest eigenvalue λ 1 (T ) of the Laplacian in T , subject to Neumann boundary conditions on the segment OA and Dirichlet boundary conditions on the other parts of the boundary. It remains to study the latter. By a trivial-extension argument, λ 1 (T ) is bounded from below by the lowest eigenvalue of the Laplacian in the rectangle (0, |OA|) × (0, |AC|), subject to Neumann boundary conditions on (0, |OA|) × {0} and Dirichlet boundary conditions on the other parts of the boundary. That is, λ 1 (T ) ≥ (1 + β -2 ) E 1 (β). Summing up, we have established the result inf σ(-∆ Ωver DN ) ----→ α→-∞ λ 1 (T ) ≥ (1 + β -2 ) E 1 (β) .

  Ω D is bounded from below by E 1 (β) for all sufficiently large α, in the following we prove less. Instead, we impose extra Neumann boundary conditions on Σ ext ∪Σ int (i.e. no boundary condition in the form setting) and show that the spectral threshold of the Laplacian with combined Dirichlet and Neumann boundary conditions in the decoupled subsets Ω ext , Ω ver , Ω int is bounded from below by E 1 (β). More specifically, by a standard Neumann bracketing argument (cf. [18, Sec. XIII.15]), we have inf

Acknowledgment D.K. was partially supported by the GACR grant No. 18-08835S and by FCT (Portugal) through project PTDC/MAT-CAL/4334/2014.


Tom Bordin
email: tom.bordin@inria.fr

Thomas Maugey
email: thomas.maugey@inria.fr

Semantic based generative compression of images for extremely low bitrates

We propose a framework for image compression in which the fidelity criterion is replaced by a semantic and quality preservation objective. Encoding the image thus becomes a simple extraction of semantic, enabling to reach drastic compression ratio. The decoding side is handled by a generative model relying on the diffusion process for the reconstruction of images. We first propose to describe the semantic using low resolution segmentation maps as guide. We further improve the generation, introducing colors map guidance without retraining the generative decoder. We show that it is possible to produce images of high visual quality with preserved semantic at extremely low bitrates when compared with classical codecs.

I. INTRODUCTION

An image is traditionally compressed with the aim of minimizing the error made in the reconstruction. The Mean Squared Error (MSE) naturally comes as a simple and efficient criterion to evaluate the fidelity of the output in terms of distortion [START_REF] Wang | Multiscale structural similarity for image quality assessment[END_REF]. This metric remains widely used to evaluate methods compressing images ranging from high to low bitrates. But what happens when compression is pushed to the extreme (∼0.02 bpp)? While excessive compression on sensitive data such as movies or vacations photos is not desirable, a drastic reduction of storage could be welcomed for the so-called "cold data". This massive amount of data that is stored but almost never accessed is estimated to represent 60% of today's storage while projected to become 80% by 2025 [START_REF] Goodwin | Tape and cloud: Solving storage problems in the zettabyte era o f data[END_REF]. In that case, compression at extremely low bitrates could be an alternative to deleting potentially useful data or keeping a huge amount of data that might not be used. However, when targeting such bitrates, the relevance of the MSE as an evaluation criterion drops. As showcased by Blau et al. [START_REF] Blau | The perception-distortion tradeoff[END_REF] there exists a tradeoff between the perceived quality of the output and the fidelity in terms of pixels. They call it the perception-distortion tradeoff. Moreover, Blau and Michaeli [START_REF] Blau | Rethinking lossy compression: The rate-distortion-perception tradeoff[END_REF] show that decreasing the bitrate exacerbates the opposing goals of the two metrics, see Fig. 1. Optimization of the MSE then leads to the apparition of numerous compression artifacts, which makes the use of such bitrates really unattractive. Recent neural methods, such as HiFiC [START_REF] Mentzer | High-fidelity generative image compression[END_REF] and its latest versions [START_REF] Hoogeboom | High-fidelity image compression with score-based generative models[END_REF] [START_REF] Theis | Lossy compression with gaussian diffusion[END_REF], propose to alleviate this issue by integrating perceptual metrics to their training. They favorably replace latest standards even at low bitrates (0.2 bpp), but this remains insufficient to cope with the explosion of data. Our focus is on drastically lower bitrates where MSE-driven generative methods would fail to maintain semantic fidelity. Discarding the MSE criterion, we instead choose to optimize the realism and semantic fidelity of the output. Following Theis et al. [START_REF] Agustsson | Generative adversarial networks for extreme learned image compression[END_REF], we encode the image by extracting a compact semantic description. Decoding from this representation then requires the use of a generative approach. This paradigm is referred to as semantic based generative compression (SGC).

We propose a new SGC scheme with semantic represented as information on labels and colors. While labels brings information on the content of the image, the colors guarantee a certain level of consistency in the generation, acting as an overview of the image. The quantity of semantic transmitted then depends on the precision of the description, i.e. the number and precision of the labels. The decoding side is handled using a generative model, which is able to generate images from abstract semantic concepts. For our decoder, we propose to rely on diffusion models. Indeed, they showed great results in guiding the generation using semantic descriptions. The recent models for text to image generation [START_REF] Ramesh | Hierarchical text-conditional image generation with clip latents[END_REF] are a good illustration of their performance, sometimes even able to cheat human eyes in art competitions. We choose to rely on the architecture and weights of Latent Diffusion Models (LDM) [START_REF] Rombach | High-resolution image synthesis with latent diffusion models[END_REF]. Their model is designed to generate images using segmentation map conditioning which allows semantic fidelity. As color guidance is not native in LDMs, we propose a way to guide the generative process with a color map without retraining their network. Our method yields better visual results at extremely low bitrates when compared to recent codecs such as VVC while conserving the semantic.

We present the formulation for the generative compression with semantic representation as well as the architecture of the model we rely on in Section II. In Section III we present results of our method and a comparison with recent codecs.

II. SEMANTIC BASED GENERATIVE COMPRESSION

A. Problem formulation

We define the semantic based generative compression framework and illustrate it in Fig. 2. The input image x is encoded into its semantic representation σ using a semantic encoder E. The decoder D, reconstructs the decoded output x using the described semantic requiring a generative approach. Unlike classical decoders Fig. 2(a), our goal is not to minimize d(x, x), a pixel error. Instead, we want to preserve the semantic information in decoding while maximizing the realism or perception of the output. In other words, we switch from a pixel fidelity to a semantic fidelity, Fig. 2(b). To measure the quality and fidelity to the input, we define a semantic reconstruction error. We still measure a distance, but we do it in the semantic space. Using a projection Φ from the pixel to the semantic domain, the error can be expressed as d(Φ(x), Φ(x)). Φ is a nonlinear function extracting the semantic information of an image. Using this metric, two images can be extremely close semantically while being far in terms of MSE. Ideally, such a metric is resistant to rotations, translations or any other operations which do not change the content of the image and only slightly alter the semantic.

We formulate our problem as a maximization of the visual quality under constraints of very low bitrates R < R low and semantic fidelity d

(Φ(x), Φ(x)) < d min . max E,D Ψ(D(E(x))) (1)
s.t R < R low and d(Φ(x), Φ(x)) < d min
where Ψ is an evaluation of the realism (higher is better). In addition to quality metrics (FID) presented by Rombach et al.

[10], we use several Image Quality Assessment (IQA) metric as Ψ to evaluate the generation. In the following, we present the choices made for the semantic encoding for E. We then introduce our generative decoder D, which uses the semantic representation as input.

B. Encoding the semantic: E

The choice of the semantic representation defines what should be important to keep inside the image. We propose to describe σ as a combination of a segmentation map and a color map.

1) The segmentation map σ s : gives information on objects and their position in the image. Using it as an embedding of our image, the encoding E(x) of an image then becomes the computation and compression of the segmentation map. In our case, it is estimated using the trained model DeepLabV3 [START_REF] Chen | Encoder-decoder with atrous separable convolution for semantic image segmentation[END_REF] on classes of the COCO-Stuff dataset. The segmentation map is then lossly compressed using downsampling operation. Using low resolution of segmentation map σ s as embedding still allows for a good reconstruction, losing only the small semantic labels in the process. At the decoding side, the segmentation map is upsampled and fed as input to the LDM model.

Images compressed using only segmentation maps present a lot of variability in samples. Indeed, labels only captures one kind of semantic and increasing the number of possible labels comes at the cost of an increased rate. Therefore, a photo during taken in the day will have the same labels as if it were taken during the night. This issue can be observed in the samples presented in Fig. 3(a). To address this, we decide to complete σ with a color map.

2) The color map σ c : describes roughly the color information of the image. The color map is computed using a series of downsampling and blurring operations. Using a low resolution color map σ c on its own is not enough to guide the generation. This reveals the same problems as an MSE optimization, i.e a blurred blank spot could then either be decoded as a sheep or as a dog. The two semantic descriptions that were chosen are thus complementary. In practice, we use 16 × 16 color map. This color map gives global information on the aspect of the image. Using a higher resolution map not necessarily bring more precision on the generation, as the information might be lost when the noise is added at the start of the diffusion process.

To decode an image from the semantic representation, we propose to use a generative decoder, in our case LDM. We modify this model to integrate color guidance without requiring training of the network.

C. Generative decoding with LDM: D

We choose to use Latent Diffusion Models (LDM) [START_REF] Rombach | High-resolution image synthesis with latent diffusion models[END_REF] as our generative decoder. Diffusion models [START_REF] Dhariwal | Diffusion models beat gans on image synthesis[END_REF] [13] are trained by maximizing the likelihood that the generated outputs follow the same distribution as the dataset. They repeatedly denoise a randomly initialized vector until convergence to the desired distribution, in our case a realistic image. Rather than doing the diffusion in the pixel domain, LDM's authors propose to do it in the latent space of a separately trained VAE(E ′ , D ′ ) [START_REF] Esser | Taming transformers for high-resolution image synthesis[END_REF]. The role of the diffusion model ϵ θ then becomes to generate embeddings of images in this latent space. They present and provide several models trained conditionally on various inputs such as text, segmentation maps or layout.

Following the standard process of diffusion, the generation of samples using an LDM with a conditioned input for image generation is illustrated in Fig. 3(a). A noise z T is first randomly drawn with gaussian distribution in the latent space and repeatedly guided towards an image embedding z 0 with z t-1 = ϵ θ (z t , t, σ s ) for T timesteps. The semantic description σ s serves as a guide for ϵ θ and conditions the generation of the semantic. The VAE decoder D ′ is then used to reconstruct x, more details on the generation are present in their article.

We illustrate the generation process on an image of our dataset in the following figures. With samples generated using only the segmentation map, there is a lot of variability in output. We thus added color map to the semantic representation.

In LDMs ϵ θ does not natively take a color map as input. Rather than retraining an already trained, powerful and large network, we propose an alternative to conditioning for guiding using colors. This method is illustrated in Fig. 3(b). Inspired by previous work [START_REF] Meng | Sdedit: Image synthesis and editing with stochastic differential equations[END_REF] we propose to use the color map at the start of the diffusion model. Instead of initializing the diffusion with a random gaussian noise as the first embedding z T , we use the color map encoded in the latent space using the VAE encoder E ′ (σ c ) it corresponds to z 0 in Fig. 3(b). Then adding noise, the diffusion process is intercepted at a timestep t < T as if the denoising already started. The rest of the diffusion proceeds normally for t timesteps. This method is possible as the VAE introduced in LDM was trained optimizing an MSE loss. The latent space thus contains information on the pixels of the color map, i.e. the colors.

Note that the choice of the timesteps t at which we intercept the diffusion has an impact on the generation. Indeed, as shown [START_REF] Balaji | ediffi: Text-to-image diffusion models with an ensemble of expert denoisers[END_REF] the smaller t is the more z t is close to a realistic sample and the harder it is to redirect the generation. On the other hand, if t is chosen too close to T then z t does not contain enough information on the colors for the output to respect the color map at generation as too much noise was added. We empirically find that using 70% of T is ideal in our case. Using the same image encoded, we show samples generated using this method in Fig. 3(b). The color adds consistency to generation by transmitting an overview of the original image.

III. EXPERIMENTS

In this section, we compare our method with standard codecs' compression targeting similar bitrates. Since our method is not deterministic, to avoid cherry-picking, we choose the first generated sample for each image presented. All the images present in figures are in 512×512. Reader can find further results on different images in supplementary material available at https://project.inria.fr/dare/publications/sgc.

A. Sampling parameters

We use the LDM previously described as our generative decoder for semantic based image generative compression. The architecture and weights of the model [START_REF] Rombach | High-resolution image synthesis with latent diffusion models[END_REF] which can be found on the LDM's GitHub 1 . We use the conditional semantic generation model trained on the Landscape [START_REF] Esser | Taming transformers for high-resolution image synthesis[END_REF] dataset, finetuned for the generation of 512 × 512 images. Samples are generated using 200 timesteps with a DDIM scheduler and a guidance scale of 2 for the diffusion process. The scale guiding the influence of the condition on the generation.

The segmentation maps were obtained using a pretrained DeepLabV3 [START_REF] Chen | Encoder-decoder with atrous separable convolution for semantic image segmentation[END_REF] model. This segmentation model is used in the encoding task of our framework. In order to further reduce the bitrate and redundancy, the segmentation maps are then downsampled. We show that downsampling the segmentation map up to 16 times does not affect the quality of the generation of images in 512 × 512, see Fig. 4. The network extrapolates outside the given boundaries to generate more realistic samples. The image is using compression on the downsampled 32 × 32 segmentation map. At decoder side, the segmentation map is upsampled to fit the input of the model, only smaller area of labels are lost in the process.

B. Discussion

Following the formulation in (1), we evaluate our methods on image quality and semantic similarity. Results of our method are showcased in Fig. 5 along with evaluation metrics in Table I(↑ higher is better, ↓ lower is better). Frechet Inception Distance (FID) evaluation of the chosen architecture are available along more details on LDMs in the original article [START_REF] Rombach | High-resolution image synthesis with latent diffusion models[END_REF]. We compare our results to VVC, for which we use the latest version of the intra coder(v1.6) with Vvenc implementation [START_REF] Wieckowski | Vvenc: An open and optimized vvc encoder implementation[END_REF]. Other learned codecs [START_REF] Mentzer | High-fidelity generative image compression[END_REF], [START_REF] Theis | Lossy compression with gaussian diffusion[END_REF], [START_REF] Yang | Lossy image compression with conditional diffusion models[END_REF] were 1 https://github.com/CompVis/latent-diffusion not trained for such bitrates. They would require retraining to target similar bitrates for comparison. Images showcased from our method show a good visual quality and preservation of the semantic. The visual results are backed by the corresponding metrics. The perceptual quality Ψ of the image is evaluated using several image quality assessment metrics. We measure the quality over the dataset Landscape, and we systematically achieve better quality than VVC and score closely to images of the dataset on two of those metrics. The semantic similarity Φ is evaluated on two criteria. Firstly, on the labels and their position, using a binary cross entropy between estimated segmentation maps on decoded images and input images. We see from this metric that the segmentation model is not able to predict correctly the classes for the images encoded using VVC. Our method has a better preservation of the semantic when measured on the 182 possible labels. For the second criteria, we use a CLIP alignment metric. CLIP is a model originally used to measure alignment between text and images, however it can also be used to compare two images. Projecting the input and decoded images into CLIP latent using the image encoder, the angle between the two normalized vectors gives an alignment score. We present the measures as the scalar product between the two vectors, with values ranging from 0 for no alignment to 1 for a perfect match. We use the ViT-L/14 version of CLIP. Note that this metric is different from CLIP-IQA which relies on text and image alignment to assess the quality. We notice that even though our method was not trained to optimize these criteria, we still score really close to the input and higher than the VVC encoded image. SGC thus propose an alternative to the distortion optimization, offering quality and semantic preservation at extremely low bitrates.

A significant improvement lies in the absence of compression artifacts when using our method. The reduction in bitrate comes at the cost of a loss of information rather than a loss in image visual quality. Metrics shows that the loss of semantic for classical coding is more important than our methods. With an important constraint on the rate, the precision of the semantic description limits the fidelity we can get. Indeed, using a higher number of labels dedicated to the description of landscapes would bring decoded images closer to originals. This is noticeable in the last two rows of images in Fig. 5, where semantic has been preserved, but resemblance is somewhat lacking, even though both semantic metrics still advantages SGC on those particular examples. Fig. 5. Illustration of our methods compared to VVC. jpeg2000 [START_REF] Taubman | JPEG2000 Image Compression Fundamentals, Standards and Practice: Image Compression Fundamentals, Standards and Practice[END_REF] does not manage to satisfyingly reach similar bitrates. Our methods yield better image quality while conserving the transmitted semantic.

IV. CONCLUSION

We proposed a new framework for image compression based on a semantic representation. We proposed to guide the generation using color maps on a model trained model using segmentation maps as input. The decoding relies on a conditional diffusion process to generate images faithful to the semantic. We showed that using a semantic description of the image is enough to produce samples close to the image with high visual quality. Comparing to recent codecs, decoded image are highly detailed through synthetic information. This method could further be applied by doing selective generation, synthesizing only unnecessary information.

An interesting improvement for future work would be to have a fine grain semantic representation to navigate into. This would allow a control on the distribution of the semantic over the image and a better control on the bitrate.

Fig. 1 .

 1 Fig.1. Illustration of the perception-distortion tradeoff. Pixel (VVC) and semantic fidelity (SGC) compression methods side-by-side at extremely low bitrates (0.022 bpp). VVC which optimizes the MSE loses the semantic, while our approach manages to show semantic and visual quality at the same time.

Fig. 2 .

 2 Fig. 2. a) Classical and b) SGC framework for image compression. With an encoder-decoder(E, D) structure, we propose to use a semantic representation σ. In contrast to classical frameworks, we choose to optimize the semantic fidelity formulated as d(Φ(x), Φ(x)) rather than the pixel fidelity d(x, x).

Fig. 3 .

 3 Fig. 3. a) Illustration of the LDM for generation of images conditioned on segmentation map σs as input, the 2 center images are results of our method. The generated samples may visually differ a lot from the source, with a lot of variability between samples. b) The proposed alternative for color map σc guidance without retraining the model. The samples are then more consistent with respect to the color of the source image. Size is indicated here in bytes(B).

Fig. 4 .

 4 Fig. 4. Using downsampled segmentation maps does not affect neither the quality of generation nor the fidelity to the input in terms of semantic. The network is able to extrapolate and generate correct labels, not strictly inside the given map.

TABLE I EVALUATION

 I OF SGC

			IQA metrics ↑ Ψ		Φ		R
		DBCNN MUSIQ HYPERIQA CLIP-IQA BCE ↓ CLIP ↑ bpp ↑
		[20]	[21]	[22]	[23]		[24]
	Input	62.2	67.5	0.58	0.56	-	-	-
	VVC(qp 55)	19.8	24.2	0.23	0.32	0.93	0.62 0.0229
	SGC(Ours)	49.8	60.6	0.54	0.47	0.40	0.80 0.0209