Ivan Beschastnyi
email: i.beschastnyi@gmail.com

Ugo Boscain
email: ugo.boscain@upmc.fr

Eugenio Pozzoli
email: euge-nio.pozzoli@inria.fr

Quantum confinement for the curvature Laplacian -∆ + cK on 2D-almost-Riemannian manifolds

Keywords: Grushin plane, quantum confinement, almost-Riemannian manifolds, coordinate-free quantization procedures, self-adjointness of the Laplacian, inverse square potential

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set Z, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator ∆) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement.

In this paper we study the self-adjointness of the curvature Laplacian, namely -∆ + cK, for c ∈ (0, 1/2) (here K is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.

Introduction

A 2-dimensional almost-Riemannian Structure (2-ARS for short) is a generalized Riemannian structure on a 2-dimensional manifold M, that can be defined locally by assigning a pair of smooth vector fields, which play the role of an orthonormal frame. It is assumed that the vector fields satisfy the Hörmander condition (see Section 2 for a more intrinsic definition).

2-ARSs were introduced in the context of hypoelliptic operators [START_REF]Une métrique associée à une classe d'opérateurs elliptiques dégénérés[END_REF][START_REF]A certain class of hypoelliptic operators[END_REF] and are particular case of rank-varying sub-Riemannian structures (see for instance [START_REF] Pozzoli | A comprehensive introduction to sub-Riemannian geometry[END_REF][START_REF]The tangent space in sub-Riemannian geometry[END_REF][START_REF]Uniform estimation of sub-Riemannian balls[END_REF][START_REF]A tour of subriemannian geometries, their geodesics and applications[END_REF][START_REF]Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities[END_REF]). The geometry of 2-ARSs was studied in [START_REF]A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF][START_REF]Two-dimensional almost-Riemannian structures with tangency points[END_REF][START_REF]Normal forms and invariants for 2-dimensional almost-Riemannian structures[END_REF][START_REF]Lipschitz classification of almost-Riemannian distances on compact oriented surfaces[END_REF] while several questions of geometric analysis on such structures were analyzed in [START_REF]Geodesics in the sub-Riemannian problem on the group SO(3)[END_REF][START_REF]The Laplace-Beltrami operator in almost-Riemannian geometry[END_REF][START_REF]Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds[END_REF][START_REF]On the essential self-adjointness of singular sub-Laplacians[END_REF][START_REF]On geometric quantum confinement in Grushintype manifolds[END_REF][START_REF]Quantum Confinement in α-Grushin planes[END_REF][START_REF] Michelangeli | Quantum confinement on non-complete Riemannian manifolds[END_REF]. For an easy introduction see [START_REF] Pozzoli | A comprehensive introduction to sub-Riemannian geometry[END_REF]Chapter 9]. 2-ARSs appear also in applications; for instance in [START_REF]Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy[END_REF][START_REF]Optimal control in laserinduced population transfer for two-and three-level quantum systems[END_REF] for problems of population transfer in quantum systems and in [START_REF]Metrics with equatorial singularities on the sphere[END_REF][START_REF]Conjugate and cut loci of a two-sphere of revolution with application to optimal control[END_REF] for orbital transfer in space mechanics.

Let us denote by D p the linear span of the two vector fields at a point p. Where D p is 2-dimensional, the corresponding metric is Riemannian. On the singular set Z, where D p is 1-dimensional, the corresponding Riemannian metric is not well defined, but thanks to the Hörmander condition one can still define the Carnot-Carathéodory distance between two points, which happens to be finite and continuous. The Hörmander condition prevents the existence of points where D p is zero dimensional. When the set Z is non-empty, we say that the 2-ARS is genuine.

In most part of the paper we make the hypothesis that the 2-ARS is 2-step i.e., that for every p ∈ M we have dim(D p + [D, D] p ) = 2. Such a hypothesis guarantees that the singular set Z is a (closed) 1-dimensional embedded submanifold and that for every p ∈ Z we have that D p is transversal to Z. One of the main features of 2-ARSs is the fact that geodesics can pass through the singular set, with no singularities even if all Riemannian quantities (as for instance the metric, the Riemannian area, the curvature) explode while approaching Z. This is easily illustrated with the example of the Grushin cylinder that is the 2-ARS on R × S 1 defined by the vector fields

X 1 (x, y) = ∂ ∂ x , X 2 (x, y) = x ∂ ∂ y
, here x ∈ R, y ∈ S 1 . For such a structure, the geodesics cross the singular set Z = {(x, y) ∈ R × S 1 | x = 0} without singularities (see Figure 1), while the Riemannian metric g, the Riemannian area ω and the Gaussian curvature K are deeply singular on Z:

g = 1 0 0 1/x 2 , ω = 1 |x| dx dy, K = - 2 x 2 . (1)
However even if geodesics cross the singular set, this is not possible for the Brownian motion or for a quantum particle when they are described by the Laplace-Beltrami operator ∆ associated to the 2-ARS. This is due to the explosion of the Riemannian area while approaching Z that makes appearing highly singular first order terms in ∆. For instance for the Grushin cylinder the Laplace-Beltrami operator is given by

∆ = div ω • grad g = ∂ 2 ∂ x 2 + x 2 ∂ 2 ∂ y 2 - 1 x ∂ ∂ x .
This phenomenon is described by the following Theorem.1

Theorem 1 ([15]). Let M be a compact oriented 2-dimensional manifold equipped with a genuine 2-step 2-ARS. Let Ω be a connected component of M \ Z, where Z is the singular set. Let g be the Riemannian metric induced by the 2-ARS on Ω and ω be the corresponding Riemannian area. The Laplace-Beltrami operator ∆ := div ω • grad g with domain C ∞ 0 (Ω), is essentially self-adjoint on L 2 (Ω, ω).

Notice that by construction each connected component of ∂Ω is diffeomorphic to S 1 , Ω is open and (Ω, g) is a non-complete Riemannian manifold. In Theorem 1 the compactness hypothesis is not necessary, but simplifies the statement. In particular the conclusion of the theorem holds for the Grushin cylinder. Versions of Theorem 1 in more general settings have been proved in [START_REF]On the essential self-adjointness of singular sub-Laplacians[END_REF][START_REF] Michelangeli | Quantum confinement on non-complete Riemannian manifolds[END_REF].

The main consequence of Theorem 1 is that the Cauchy problems for the heat and the Schrödinger equations (here ∆ is the Laplace-Beltrami operator, is the Planck constant, and m is the mass of the quantum particle)2

∂ t φ(t, p) = ∆ φ(t, p), φ(0, •) = φ 0 ∈ L 2 (Ω, ω), (2)
i ∂ t ψ(t, p) = - 2 2m ∆ ψ(t, p), ψ(0, •) = ψ 0 ∈ L 2 (Ω, ω). (3)
are well defined in L 2 (Ω, ω) and hence nothing can flow outside Ω, that is, e t∆ φ 0 (resp. e it 2m ∆ ψ 0 ) is supported in Ω, for all t ≥ 0 (resp. t ∈ R). This phenomenon is usually known as quantum confinement (see [START_REF]On the essential self-adjointness of singular sub-Laplacians[END_REF][START_REF] Michelangeli | Quantum confinement on non-complete Riemannian manifolds[END_REF] and, for similar problems, [START_REF]On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R n[END_REF]).

Given that the geodesics cross the singular set with no singularities, the impossibility for the heat or for a quantum particle to flow through Z implied by Theorem 1 is quite surprising. For what concerns the heat, a satisfactory interpretation of Theorem 1 in terms of Brownian motion/Bessel processes has been provided for the Grushin cylinder in [START_REF]Extensions of brownian motion to a family of grushin-type singularities[END_REF] and from [START_REF]Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling[END_REF] one can extract an interpretation of Theorem 1 in terms of random walks. Roughly speaking random particles are lost in the infinite area accumulated along Z that, as a consequence, acts as a barrier.

Although for the heat-equation the situation is relatively well-understood, this is not the case for the Schrödinger equation since semiclassical analysis (see for instance [START_REF]Semiclassical analysis[END_REF]) roughly says that for → 0 sufficiently concentrated solutions of the Schrödinger equation move approximately along classical geodesics. Clearly semiclassical analysis breaks down on the singularity Z.

It is then natural to come back on the quantization procedure that permits to pass from the description of a free classical particle moving on a Riemannian manifold to the corresponding Schrödinger equation. This is a complicated subject that has no unique answer. The resulting evolution equation for quantum particles depends indeed on the chosen quantization procedure.

Most of coordinate invariant quantization procedures modify the quantum Hamiltonian by a correction term depending on the scalar curvature R. In dimension two, the scalar curvature is twice the Gaussian curvature K and the modified Schrödinger equation is of the form

i ∂ t ψ(t, p) = 2 2m -∆ + cK(p) ψ(t, p),
where c ≥ 0 is a constant. Values given in the literature include:

• path integral quantization: c = 1/3 and c = 2/3 in [START_REF] D W -M | A stochastic scheme for constructing solutions of the Schrödinger equations[END_REF], c = 1/2 in [START_REF]Cambridge Monographs on Mathematical Physics[END_REF];

• covariant Weyl quantization: c ∈ [0, 2/3] including conventional Weyl quantization (c = 0) in [START_REF]Pseudodifferential operators, covariant quantization, the inescapable Van Vleck-Morette determinant, and the R/6 controversy[END_REF];

• geometric quantization for a real polarization: c = 1/3 in [START_REF]Geometric quantization, Oxford Mathematical Monographs[END_REF];

2For the heat equation all constant are normalized to 1. This has not been done for the Schrödinger equation since the role of the Planck constant is important for further discussions.

• finite dimensional approximations to Wiener Measures: c = 2/3 in [6]3.

We refer to [START_REF]Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds[END_REF][START_REF]Pseudodifferential operators, covariant quantization, the inescapable Van Vleck-Morette determinant, and the R/6 controversy[END_REF] for interesting discussions on the subject. [START_REF]Two-dimensional almost-Riemannian structures with tangency points[END_REF] Purpose of this paper is to study the self-adjointness of the curvature Laplacian -∆ + cK depending on c to understand if quantum confinement holds for the dynamics induced by this operator. Before stating our main result, let us remark that the curvature term cK interacts with the diverging first order term in ∆.

For instance for the Grushin cylinder a unitary transformation (see Section 4.1, ( 14)) permits to transform the operator

∆ = ∂ 2 ∂ x 2 + x 2 ∂ 2 ∂ y 2 - 1 x ∂ ∂ x on L 2 R × S 1 , 1 |x| dx dy in ∆ = ∂ 2 ∂ x 2 + x 2 ∂ 2 ∂ y 2 - 3 4 1 x 2 on L 2 (R × S 1 , dx dy)
and hence the adding of a term of the form -cK = -c -2 1 x 2 (that remains untouched by the unitary transformation) to ∆ changes the diverging behaviour around x = 0. In particular for c = 3/8 the diverging potential disappears and -∆ + cK is not essentially self-adjoint in L 2 (R + × S 1 , dx dy) while -∆ does. The same conclusion applies to

-∆ + cK in L 2 (R + × S 1 , 1 |x | dx dy).
The main result of the paper is that the perturbation term given by the curvature destroys the essential self-adjointness of the Laplace-Beltrami operator.

Theorem 2. Let M be a compact oriented 2-dimensional manifold equipped with a genuine 2-step 2-ARS.

Let Ω be a connected component of M \ Z, where Z is the singular set. Let g be the Riemannian metric induced by the 2-ARS on Ω, ω be the corresponding Riemannian area, K the corresponding Gaussian curvature and ∆ = div ω • grad g the Laplace-Beltrami operator. Let c ∈ [0, 1/2). The curvature Laplacian -∆ + cK with domain C ∞ 0 (Ω), is essentially self-adjoint on L 2 (Ω, ω) if and only if c=0. Moreover, if c > 0, the curvature Laplacian has infinite deficiency indices.

The non-self-adjointness of -∆ + cK implies that one can construct self-adjoint extensions of this operator that permit to the solution to the Schrödinger equation to flow out of the set Ω, in the same spirit of [START_REF]Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces[END_REF][START_REF]Geometric confinement and dynamical transmission of a quantum particle in Grushin cylinder[END_REF]. The study of these self-adjoint extension and how semiclassical analysis applies to them is a subject that deserves to be studied in detail.

Remark 3. We remark that in Theorem 2 the hypothesis c ∈ [0, 1/2) guarantees the non-negativity of the operator -∆ + cK modulo a Kato-small perturbation. For further details, see also Remark 7,Lemma 25,and Remark 26 Remark 4. Notice that in Theorem 2 one can also consider the case c < 0. In this case, one can prove that the curvature Laplacian is essentially self-adjoint (applying for example the criterion for the self-adjointes of operators of the form -∆ + V on non-complete Riemannian manifolds, found in [START_REF] Michelangeli | Quantum confinement on non-complete Riemannian manifolds[END_REF]). However, if this case admits a physical interpretation is not known to the authors.

3Notice that the Schrödinger equation that one finds in the references [START_REF]Finite-dimensional approximations to Wiener measure and path integral formulas on manifolds[END_REF][START_REF]Cambridge Monographs on Mathematical Physics[END_REF][START_REF] D W -M | A stochastic scheme for constructing solutions of the Schrödinger equations[END_REF][START_REF]Pseudodifferential operators, covariant quantization, the inescapable Van Vleck-Morette determinant, and the R/6 controversy[END_REF][START_REF]Geometric quantization, Oxford Mathematical Monographs[END_REF]

is i ∂ t ψ(t, p) = 2 m -1 2 ∆ + c R(p) ψ(t, p). Hence, c = 4c .
4There are also other approaches to the quantization process on Riemannian manifolds that provide correction terms depending on the curvature. For instance if one consider the Laplacian on a -tubular neighborhood of a surface in R 3 with Dirichlet boundary conditions, then for → 0 after a suitable renormalization, one gets an operator containing a correction term depending on the Gaussian curvature and the square of the mean curvature (see [START_REF]Spectrum of the laplacian in narrow tubular neighbourhoods of hypersurfaces with combined dirichlet and neumann boundary conditions[END_REF][START_REF]Effective Hamiltonians for thin Dirichlet tubes with varying cross-section[END_REF]).

As in Theorem 1 the compactness hypothesis is useful to simplify the statement of the theorem. A version without the compactness hypothesis is given here where also the orientability assumption of M is not needed. Theorem 5. Let M be a 2-dimensional manifold equipped with a genuine 2-step 2-ARS.

Assume that

• the singular set Z is compact;

• the 2-ARS is geodesically complete.

Let Ω be a connected component of M \ Z, and c ∈ [0, 1/2). With the same notations of Theorem 2, the curvature Laplacian -∆ + cK with domain C ∞ 0 (Ω), is essentially self-adjoint on L 2 (Ω, ω) if and only if c=0. Moreover, if c > 0, the curvature Laplacian has infinite deficiency indices.

For the sake of simplicity, we prove Theorem 2 only. Theorem 5 can be proved following the same ideas.

Theorem 5 applies in particular to the Grushin cylinder with curvature Laplacian -∆

+ cK = -(∂ 2 x + x 2 ∂ 2 y -1 x ∂ x ) + 2c x 2 .
For this case, the fact that the deficiency indices are infinite means that all Fourier components of -∆ + cK are not self-adjoint.

Notice that under the hypothesis of the theorem, each connected component of ∂Ω is diffeomorfic to S 1 . Of course if c > 0, the manifold does not need to be geodesically complete.

If one removes the 2-step hypothesis the situation is more complicated since tangency points [3, 4] may appear. In presence of tangency points even the essential self-adjointness of the standard Laplace-Beltrami operator (without the term -cK) is an open question [START_REF]The Laplace-Beltrami operator in almost-Riemannian geometry[END_REF]. Without the 2-step hypothesis results can indeed be very different. To illustrate this, we study the α-Grushin cylinder for which computations can be done explicitly for every value of c. Proposition 6. Fix α ∈ R. On R × S 1 consider the generalized Riemannian structure for which an orthonormal frame is given by

X 1 (x, y) = ∂ ∂ x , X (α) 2 (x, y) = x α ∂ ∂ y , here x ∈ R, y ∈ S 1 .
Let c ≥ 0. On R + × S 1 the structure is Riemannian with Riemannian area 1 |x | α dx dy. Let -∆ α + cK α be the curvature Laplacian with domain

C ∞ 0 (R + × S 1 ) acting on L 2 (R + × S 1 , 1 |x | α dx dy). Denote by α c,± = (-2c + 1) ± 2 (c -2 + √ 3)(c -2 - √ 3) 4c -1 . • If 0 ≤ c < 1/4, -∆ α + cK α is essentially self-adjoint if and only if α ≥ α c,+ or α ≤ α c,-; • if c = 1/4, -∆ α + cK α is essentially self-adjoint if and only if α ≥ 3; • if 1/4 < c ≤ 2 - √ 3, -∆ α + cK α is essentially self-adjoint if and only if α c,-≤ α ≤ α c,+ ; • if 2 - √ 3 < c < 2 + √ 3, -∆ α + cK α is not essentially self-adjoint ∀α ∈ R; • if c ≥ 2 + √ 3, -∆ α + cK α is essentially self-adjoint if and only if α c,-≤ α ≤ α c,+ .
The regions where -∆ α + cK α is essentially self-adjoint are plotted in Figure 2. Note that for some of the quantizations listed earlier, -∆ α +cK α is essentially self-adjoint for |α| sufficiently big. The α-Grushin cylinder is an interesting geometric structure studied in [START_REF]Extensions of brownian motion to a family of grushin-type singularities[END_REF][START_REF]Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces[END_REF][START_REF]Geometric confinement and dynamical transmission of a quantum particle in Grushin cylinder[END_REF]. For α = 0 it is a flat cylinder, for α positive integer is a (α + 1)-step 2-ARS; for α negative it describes a conic-like surface (in particular for α = -1 describes a flat two-dimensional cone).

0 -1 1 α c 1/4 2 + √ 3 2 - √ 3 3 + 2 √ 3 -3
Figure 2: Regions of the (α, c)-parameter space where the operator -∆ α + cK α is essentially self-adjoint.

Remark 7.

Notice that for α = 1 the α-Grushin cylinder is the standard Grushin cylinder, for which in Proposition 6 we obtain that -∆ 1 + cK 1 , c ≥ 0, is essentially self-adjoint if and only if c = 0. This suggests that the hypothesis c ∈ [0, 1/2) of Theorem 2 is technical and that the same result could be extended to the range of values c ≥ 0.

The proof of Proposition 6, which is instructive since it is simple and presents already some crucial ingredients necessary for the general theory, is given in Section 3.

Structure of the paper.

In Section 2 we give the key definition and results for 2-ARS. In Section 3 we introduce the basic concepts to study the self-adjointness of symmetric operators and give a proof of Proposition 6. The proof of Theorem 2 spans Sections 4 and 5. A local version around a singular region is studied in Section 4, where a description of the closure and adjoint curvature Laplacian operators is given. The main tools needed for our proof are the Hardy inequality, the Kato-Rellich perturbation theorem, the Fourier transform, and Sturm-Liouville theory applied here in the context of 2D operators. We then extend the results on the whole manifold in Section 5.

We conclude this introduction by remarking that while an operator of the form -∆ + cK(p) is useful to describe a quantum particle in a Riemannian manifold, it is not meaningful in the description of the evolution of the heat. Indeed a heat equation of the form ∂ t φ = (-∆ + cK(p))φ would describe the evolution of a random particle on a Riemannian manifold with a rate of killing proportional to the Gaussian curvature. 1. U is an Euclidean bundle over M of rank 2. We denote each fiber by U q , the scalar product on U q by (• | •) q and the norm of u ∈ U q as |u| = (u | u) q .

2D almost-Riemannian structures

2. f : U → T M is a smooth map that is a morphism of vector bundles i.e., f (U q ) ⊆ T q M and f is linear on fibers. [START_REF]A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds[END_REF]. the distribution D = { f (σ) | σ : M → U smooth section}, is a family of vector fields satisfying the Hörmander condition, i.e., defining

D 1 := D, D i+1 := D i + [D 1 , D i ], for i ≥ 1, there exists s ∈ N such that D s (q) = T q M.
A particular case of 2-ARSs is given by Riemannian surfaces. In this case U = T M and f is the identity.

Let us recall few key definitions and facts. We refer to [START_REF] Pozzoli | A comprehensive introduction to sub-Riemannian geometry[END_REF] for more details. • The (almost-Riemannian) norm of a vector v ∈ D p is

• Let D p = {X(p) | X ∈ D} = f (U p ) ⊆ T p M.
v := min{|u|, u ∈ U p s.t. v = f (p, u)}.
• An admissible curve is a Lipschitz curve γ : [0, T] → M such that there exists a measurable and essentially bounded function u :

[0, T] t → u(t) ∈ U γ(t) , called control function, such that γ(t) = f (γ(t), u(t)), for a.e. t ∈ [0, T]
. Notice there may be more than one control corresponding to the same admissible curve.

• If γ is admissible then t → γ(t) is measurable. The (almost-Riemannian) length of an admissible curve γ : [0, T] → M is (γ) := ∫ T 0 γ(t) dt.
• The (almost-Riemannian) distance between two points p 0 , p 1 ∈ M is

d(p 0 , p 1 ) = inf{ (γ) | γ : [0, T] → M admissible, γ(0) = p 0 , γ(T) = p 1 }.
Thanks to the bracket-generating condition, the Chow-Rashevskii theorem guarantees that (M, d) is a metric space and that the topology induced by (M, d) is equivalent to the manifold topology.

• Given a local trivialization Ω × R 2 of U, an orthonormal frame for the 2-ARS on Ω is the pair of vector fields {F 1 ,

F 2 } := { f • σ 1 , f • σ 2 } where {σ 1 , σ 2 } is an orthonormal frame for (• | •) q on Ω × R 2 of U.
On a local trivialization the map f can be written as

f (p, u) = u 1 F 1 (p) + u 2 F 2 (p).
When this can be done globally (i.e., when U is the trivial bundle) we say that the 2-ARS is free.

Notice that orthonormal frames in the sense above are orthonormal frames in the Riemannian sense out of the singular set.

• Locally, for a 2-ARS, it is always possible to find a system of coordinates and an orthonormal frame that in these coordinates has the form

F 1 (x, y) = 1 0 , F 2 (x, y) = 0 f(x, y) , (4)
where f : Ω → R is a smooth function. In these coordinates we have that

Z = {(x, y) ∈ Ω | f(x, y) = 0}.
Using this orthonormal frame one immediately gets:

Proposition 9. The 2-ARS is 2-step in Ω if and only if for every (x, y) ∈ Ω such that f(x, y) = 0, we have ∂ x f(x, y) 0.

Moreover, the implicit function theorem applied to the function f directly implies: Proposition 10. If the 2-ARS is genuine and 2-step then Z is a closed embedded one dimensional submanifold.

In particular if Z is compact, each connected component of Z is diffeomorphic to S 1 .

• Out of the singular set Z, the structure is Riemannian and the Riemannian metric, the Riemannian area, the Riemannian curvature, and the Laplace-Beltrami operator are easily expressed in the orthonormal frame given by (4):

g (x,y) = 1 0 0 1 f(x,y) 2 , ( 5
)
ω (x,y) = 1 |f(x, y)| dx dy, (6)
K(x, y) = f(x, y)∂ 2 x f(x, y) -2 (∂ x f(x, y)) 2 f(x, y) 2 , (7)
∆ = ∂ 2 x + f 2 ∂ 2 y - ∂ x f f ∂ x + f(∂ y f)∂ y . (8)
• To prove the main results of this paper, the following normal forms are going to be important. Proposition 11 ([3]). Consider a 2-step 2-ARS. For every p ∈ M there exist a neighborhood U of p, a system of coordinates in U, and an orthonormal frame {X 1 , X 2 } for the ARS on U, such that p = (0, 0) and {X 1 , X 2 } has one of the following forms:

F1. X 1 (x, y) = ∂ ∂ x , X 2 (x, y) = e φ(x,y) ∂ ∂ y , F2. X 1 (x, y) = ∂ ∂ x , X 2 (x, y) = xe φ(x,y) ∂ ∂ y ,
where φ is a smooth function such that φ(0, y) = 0.

A point p ∈ M is said to be a Riemannian point if D p is two-dimensional, and hence a local description around p is given by F1. A point p such that D p is one-dimensional, and thus D] p is two-dimensional, is called a Grushin point and a local description around p is given by F2.

D p + [D,
When M is compact orientable, each connected component of Z is diffeomorphic to S 1 and admits a tubular neighborhood diffeomorphic to R × S 1 . In this case the normal form F2 can be extended to the whole neighborhood. Proposition 12 ([15]). Consider a 2-step 2-ARS on a compact orientable manifold. Let W be a connected component of Z. Then there exist a tubular neighborhood U of W diffeomorphic to R × S 1 , a system of coordinates in U, and an orthonormal frame {X 1 , X 2 } of the 2-ARS on U such that W = {(0, y), y ∈ S 1 } and {X 1 , X 2 } has the form

X 1 (x, y) = ∂ ∂ x , X 2 (x, y) = xe φ(x,y) ∂ ∂ y . ( 9
)

Self-adjointness of operators

Let A be a linear operator on a separable Hilbert space H , (•, •) H . The linear subspace of H where the action of A is well-defined is called the domain of A, denoted by D(A). We shall always assume that D(A) is dense in H . Following [START_REF]Methods of modern mathematical physics. II. Fourier analysis, selfadjointness[END_REF], we recall several definitions and properties of linear operators:

• A is said to be symmetric if (Au, v) H = (u, Av) H for all u, v ∈ D(A).

• A is said to be closed if D(A) with the norm

• A := • H + A • H is complete.
• A linear operator B, D(B) such that D(A) ⊂ D (B) and Bu = Au for all u ∈ D(A) is called an extension of A. In this case we write A ⊂ B.

• If A is symmetric and densely defined, there exists a minimal closed symmetric extension A of A, which is said to be the closure of A. We describe this construction: take any sequence (u n ) n ⊂ D(A) which converges to a limit u ∈ H , and for which the sequence (Au n ) n converges to a limit w ∈ H . Then, by symmetry of A, we have that

(w, v) H = lim n→∞ (Au n , v) H = lim n→∞ (u n , Av) H = (u, Av) H , ∀v ∈ D(A).
Since D(A) is dense in H , w is uniquely determined by u. The closure of A is defined by setting Au = w, and the domain D(A) is the closure of D(A) with respect to the norm • A . One can easily see that A is closed, symmetric, and any closed extension of A is an extension of A as well.

• Given a densely defined linear operator A, the domain D(A * ) of the adjoint operator A * is the set of all v ∈ H such that there exists w ∈ H with (Au, v) H = (u, w) H for all u ∈ D(A). The adjoint of A is defined by setting

A * v = w. • A is said to be self-adjoint if A * = A, that is, A is symmetric and D(A * ) = D(A).
• A is said to be essentially self-adjoint if its closure is self-adjoint.

• If B is a closed symmetric extension of A, then A ⊂ A ⊂ B ⊂ A * .

• If a densely defined operator B is such that D(A) ⊂ D(B) and there exist a, b ≥ 0 such that

Bu ≤ a Au + b u , ∀u ∈ D(A), (10)
then B is said to be small with respect to A (or also, Kato-small w.r.t. A). The infimum of the set of a ≥ 0 such that (10) holds is called the A-bound of B. If a can be chose arbitrarily small, B is said to be infinitesimally small w.r.t A.

We will need the following classical result in perturbation theory:

Proposition 13 (Kato-Rellich's Theorem). Let A, B be two densely defined operators and assume that B is small with respect to A.

Then D(A) ⊂ D(A + B). If moreover a < 1 in (10), then D(A) = D(A + B).
In order to study the self-adjointness and more in general describe the extensions of a symmetric operator A, one may use the fundamental Von Neumann decomposition ([36, Chapter X])

D(A * ) = D(A) ⊕ A ker(A * + i) ⊕ A ker(A * -i), (11)
where the sum is orthogonal with respect to the scalar product

(•, •) A = (•, •) H + (A * •, A * •) H .
As a first direct consequence of [START_REF]Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy[END_REF], one has the fundamental spectral criterion for self-adjointness: Proposition 14. Let A, D(A), be a symmetric operator, densely defined on the Hilbert space H . The following are equivalent:

(a) A is essentially self-adjoint;

(b) Ran(A ± i) is dense in H ; (c) ker(A * ± i) = {0}.
• The dimensions of the vector spaces ker(A * + i) and ker(A *i) are called deficiency indices of A.

Always using [START_REF]Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy[END_REF] [START_REF]The Laplace-Beltrami operator in almost-Riemannian geometry[END_REF]). Then, since the eigenvalue equation

-u (x) + V(x)u(x) = ±i u(x)
has always two linearly independent solutions, the quotient D(A * )/D(A) has at most dimension four. Moreover, let us recall the limit point-limit circle Weyl's Theorem (see, e.g, [36, Appendix to Chapter X.1]) which says that the self-adjointness of a 1D Sturm-Liouville operator can be deduced by regarding the solutions to the ODE -u (x)

+ V(x)u(x) = 0. (12)
• If all solutions to (12) are square-integrable near 0 (respectively ∞), then V is said to be in the limit circle case at 0 (resp. ∞). If V is not in the limit circle case at 0 (resp. ∞), it is said to be in the limit point case at 0 (resp. ∞).

Proposition 15 (Weyl's Theorem). The operatord 2 dx 2 +V(x) with domain C ∞ 0 (R + ) has deficiency indices • (2, 2) if V is in the limit circle case at both 0 and ∞;

• (1, 1) if V is in the limit circle case at one end point and in the limit point at the other;

• (0, 0) if V is in the limit point case at both 0 and ∞.

In particular, -d 2 dx 2 + V(x) is essentially self-adjoint on L 2 (R + , dx) if and only if V is in the limit point case at both 0 and ∞. Some useful criteria to determine whether a potential V is in the limit point or limit circle case (it is also said to be quantum-mechanically complete or incomplete, respectively) at 0 and ∞ are the following:

Proposition 16. Let V ∈ C 1 (R + ) be real and bounded above by a constant E on [1, ∞). Suppose that ∫ ∞ 1 1 √ E-V (x) dx = ∞ and V /|V | 3/2 is bounded near ∞. Then V is in the limit point case at ∞. Proposition 17. Let V ∈ C 0 (R + ) be real and positive near 0. If V(x) ≥ 3
4x 2 near 0 then V is in the limit point case at 0. If for some > 0, V(x) ≤ ( 34 -) 1

x 2 near 0, then V is in the limit circle case at 0.

Proposition 18.

Let V ∈ C 0 (R + ) be real, and suppose that it decreases as x ↓ 0. Then V is in the limit circle case at 0.

Weyl's Theorem and these criteria are, respectively, [36, Theorem X.7, Corollary to Theorem X.8, Theorem X.10 and Problem X .7 ].

Here we give the proof of Proposition 6, which makes use of the limit point-limit circle argument.

Proof. of Proposition 6

The Laplace-Beltrami operator (with domain

C ∞ 0 (R + × S 1
)) and the curvature associated to the orthonormal frame X 1 , X (α) 2 are given by

∆ α = ∂ 2 ∂ x 2 + x 2α ∂ 2 ∂ y 2 - α x ∂ ∂ x , K α = - α(α + 1) x 2 .
We perform a unitary transformation

U α : L 2 R + × S 1 , 1 |x| α dx dy → L 2 (R + × S 1 , dx dy), ψ → |x| -α/2 ψ,
which gives the operator

L α,c := U α (-∆ α + cK α )U -1 α = - ∂ 2 ∂ x 2 -x 2α ∂ 2 ∂ y 2 + (1 -4c)α + (2 -4c)α 4x 2
.

Via Fourier transform with respect to the variable y ∈ S 1 , one obtains the direct sum operator

L α,c := k ∈Z (L α,c ) k , (L α,c ) k := - ∂ 2 ∂ x 2 + x 2α k 2 + (1 -4c)α 2 + (2 -4c)α 4x 2
, acting on the Hilbert space 2 (L 2 (R + )), with domain

D( L α,c ) = {( f k ) k ∈Z ∈ 2 (L 2 (R + )) | f k ∈ D((L α,c ) k ) ∀k ∈ Z, f k = 0 for almost every k}, where D (L α,c ) k = C ∞ 0 (R + ) for all k.
Moreover, as a general fact concerning direct sum operators, L α,c is essentially self-adjoint if and only if (L α,c ) k is so for all k ([17, Proposition 2.3]). Let us thus study the essential self-adjointness of (L α,c ) 0 : it is a Sturm-Liouville operator of the form -d 2 dx 2 + V α,c (x), where

V α,c (x) = (1 -4c)α 2 + (2 -4c)α 4x 2 =: k(α, c) x 2 . ( 13
)
The potential V α,c is quantum-mechanically complete at infinity, for all (α, c) ∈ R × [0, ∞) (as one can check by applying Proposition 16). So, applying Propositions 17 and 18, we can conclude that 2(L α,c ) 0 is essentially self-adjoint if and only if V α,c (x) ≥ 3 4x 2 near zero, since V α,c (x) = k(α, c)/x 2 and when k(α, c) < 0 then V α,c decreases for x ↓ 0. By using the explicit formula (13) for V α,c , we obtain k(α, c) ≥ 3/4 which yields the values of (α, c) given in the statement. For what we said before, when (L α,c ) 0 is not essentially self-adjoint, neither is so L α,c . Finally, when (L α,c ) 0 is essentially self-adjoint, then

x 2α k 2 + (1 -4c)α 2 + (2 -4c)α 4x 2 ≥ 3 4x 2 , so (L α,c
) k is essentially self-adjoint too, for all k, and hence L α,c is so.

Remark 19.

As a by-product of the proof of Proposition 6 we obtain that for the Grushin cylinder (α = 1) with c > 0 all Fourier components of -∆ 1 + cK 1 are not essentially self-adjoint, due to the inequality

x 2 k 2 + ( 3 4 -2c) 1
x 2 < 3 4x 2 which holds near zero for all k ∈ Z. Hence -∆ 1 + cK 1 has infinite deficiency indices, for any c > 0. When c ∈ (0, 1/2), Theorem 2 extends this result to any two-dimensional almost Riemannian manifold of step 2 (under some natural topological assumptions).

Closure operator

We shall work on the manifold Ω + , being the case Ω -analogous. Then the statement for Ω follows from the decomposition

L 2 (Ω, ω) = L 2 (Ω -, ω) ⊕ ⊥ L 2 (Ω + , ω).
For a metric g of the form diag(1, f(x, y) -2 ), plugging f(x, y) = xe φ(x,y) into ( 6), [START_REF]The tangent space in sub-Riemannian geometry[END_REF], and (8) one has the following:

ω (x,y) = 1 xe φ(x,y) dxdy, K(x, y) = - 2 x 2 - 2∂ x φ(x, y) x + ∂ 2 x φ(x, y) -(∂ x φ(x, y)) 2 , ∆ = ∂ 2 x + x 2 e 2φ(x,y) ∂ 2 y - 1 x ∂ x -∂ x φ(x, y)∂ x + ∂ y φ(x, y)x 2 e 2φ(x,y) ∂ y .
We perform a unitary transformation

U : L 2 (Ω + , ω) → L 2 (Ω + , dxdy), ψ → (xe φ ) -1/2 ψ, (14)
and the corresponding transformed Laplacian is given by

L = U∆U -1 = ∂ 2 x + x 2 e 2φ ∂ 2 y + 2x 2 e 2φ (∂ y φ)∂ y - 3 4x 2 - ∂ x φ 2x - 1 4 (∂ x φ) 2 + 1 2 ∂ 2 x φ + 3 4 x 2 (∂ y φ) 2 e 2φ + 1 2 x 2 (∂ 2 y φ)e 2φ .
We shall analyze the self-adjointness of the operator

-L + cK = U(-∆ + cK)U -1 = -∂ 2 x -x 2 e 2φ ∂ 2 y -2x 2 e 2φ (∂ y φ)∂ y + 3 4 -2c 1 x 2 + 1 -4c 2
∂ x φ x + 1 4 -c (∂ x φ) 2 + c - 1 2 ∂ 2 x φ - 3 4 x 2 (∂ y φ) 2 e 2φ - 1 2
x 2 (∂ 2 y φ)e 2φ = H c + η c , (15)
where we have defined the operator

H c = -∂ 2 x -x 2 e 2φ ∂ 2 y -2x 2 e 2φ (∂ y φ)∂ y + 3 4 -2c 1 x 2 + 1 -4c 2 ∂ x φ x ,
and the multiplicative operator

η c = 1 4 -c (∂ x φ) 2 + c - 1 2 ∂ 2 x φ - 3 4 x 2 (∂ y φ) 2 e 2φ - 1 2
x 2 (∂ 2 y φ)e 2φ ,
both with domain C ∞ 0 (Ω + ), acting on the Hilbert space L 2 (Ω + , dxdy) =: L 2 (Ω + ). For later convenience, we also define the operator

T c = -∂ 2 x -x 2 e 2φ ∂ 2 y -2x 2 e 2φ (∂ y φ)∂ y + 3 4 -2c 1 x 2 , D(T c ) = C ∞ 0 (Ω + ), acting on L 2 (Ω + ),
and the 1D operator, usually called inverse square potential or Bessel operator in the literature

s c = - d 2 dx 2 + 3 4 -2c 1 x 2 , D(s c ) = C ∞ 0 (R + ), acting on L 2 (R + ). (16)
The closure of s c is known:

Proposition 23. ([5, Theorem 5.1 (ii)]) Let c ∈ (0, 1/2]. Then, D(s c ) = H 2 0 (R + ). Remark 24.
The assumption c ∈ (0, 1/2] in Proposition 23 guarantees that s c is a non-negative operator. Anyway, in the recent paper [START_REF]On the domains of Bessel operators[END_REF] it is proved that

D(s c ) = H 2 0 (R + ) for all c > 0. Lemma 25. Let c ∈ [0, 1/2) and f ∈ L 2 (Ω + ) with supp f ⊂ (0, ) × S 1 , for some > 0. Then, f ∈ D -L + cK if and only if f ∈ D(T c ).
Proof. Since -L + cK = H c + η c , and η c is a bounded operator on (0, ) × S 1 , Proposition 13 implies that f ∈ D(-L + cK) if and only if f ∈ D(H c ), for all c > 0. Furthermore, we want to show that the singular term

g 1, c x := 1-4c 2 ∂ x φ
x is infinitesimally small w.r.t. T c , if c ∈ [0, 1/2). In order to do this we will use two main ingredients. The first one is Hardy inequality:

∫ ∞ 0 | f (x)| 2 x 2 dx ≤ 4 ∫ ∞ 0 | f (x)| 2 dx ∀ f ∈ C ∞ 0 (R + ).
The second one is perturbation theory, in particular Kato-Rellich's Theorem (Proposition 13).

For all functions f ∈ C ∞ 0 (R + × S 1 ) we have

g 1,c f x 2 L 2 (R + ×S 1 ) = ∫ R + ×S 1 |g 1,c (x, y)| 2 x 2 | f (x, y)| 2 dxdy = 1 - 3 -8c 4 -8c ∫ R + ×S 1 |g 1,c (x, y)| 2 | f (x, y)| 2 x 2 dxdy + 3 -8c 4 -8c ∫ R + ×S 1 |g 1,c (x, y)| 2 | f (x, y)| 2 x 2 dxdy ≤ |g 1,c | 2 L ∞ (R + ×S 1 ) 4 4 -8c ∫ R + ×S 1 |∂ x f (x, y)| 2 dxdy + 3 -8c 4 -8c ∫ R + ×S 1 | f (x, y)| 2 x 2 dxdy ≤ 4 |g 1,c | 2 L ∞ (R + ×S 1 ) 4 -8c ∫ R + ×S 1 |∂ x f (x, y)| 2 + x 2 e 2φ |∂ y f (x, y)| 2 + 3 4 -2c | f (x, y)| 2 x 2 dxdy = 4 |g 1,c | 2 L ∞ (R + ×S 1 ) 4 -8c (T c f , f ) L 2 (R + ×S 1 ) ≤ 2 |g 1,c | 2 L ∞ (R + ×S 1 ) 4 -8c δ T c f 2 L 2 (R + ×S 1 ) + 1 δ f 2 L 2 (R + ×S 1 ) ,
where we have used Fubini's Theorem and the Hardy inequality in the first inequality, we have integrated by parts in the third equality, and we have used the Young inequality in the last inequality that holds for every δ > 0. This proves that g 1, c

x is infinitesimally small w.r.t.

T c if c ∈ [0, 1/2). Proposition 13 then implies that D(H c ) = D(T c ) if c ∈ [0, 1/2).

Remark 26.

The assumption c ∈ (0, 1/2) is used in the proof of Lemma 25 to guarantee the non-negativity of T c .

For any function f ∈ L 2 (R + × S 1 ), we denote by f = k ∈Z f k (x)e iky its Fourier series.

(T c f ) k = s c f k - 1 2 x 2 m+m =k (-m 2 ) f m (e 2φ ) m -x 2 m+m =k (im) f m (∂ y φ e 2φ ) m ,
where s c is defined in ( 16), and we compute the norm using the triangular inequality

(T c f ) k L 2 (R + ) ≥ s c f k L 2 (R + ) - 1 2 x 2 m+m =k (-m 2 ) f m (e 2φ ) m L 2 (R + ) -x 2 m+m =k (im) f m (∂ y φ e 2φ ) m L 2 (R + ) .
We have,

m+m =k (-m 2 ) f m (e 2φ ) m L 2 (R + ) = m+m =k (-m 2 ) f m (e 2φ ) m -k 2 m+m =k f m (e 2φ ) m + 2k m+m =k m f m (e 2φ ) m L 2 (R + ) ≤ f L 2 (R + ×S 1 ) sup x ∈R + e 2φ H 2 (S 1 ) + k 2 sup x ∈R + e 2φ L 2 (S 1 ) + 2|k | sup x ∈R + e 2φ H 1 (S 1 ) ≤C φ,k f L 2 (R + ×S 1 )
thanks to the Cauchy-Schwartz inequality and the Plancherel formula. Similarly,

m+m =k (im) f m (∂ y φ e 2φ ) m L 2 (R + ) ≤ f L 2 (R + ×S 1 ) sup x ∈R + e 2φ ∂ y φ H 1 (S 1 ) + |k | sup x ∈R + e 2φ ∂ y φ L 2 (S 1 ) ≤ C φ,k f L 2 (R + ×S 1 )
Thus

s c f k L 2 (R + ) ≤ (T c f ) k L 2 (R + ) + 2 C φ,k f L 2 (R + ×S 1 ) ≤ T c f L 2 (R + ×S 1 ) + 2 C φ,k f L 2 (R + ×S 1 ) ∀ f ∈ C ∞ 0 ((0, ) × S 1 ). (17)
Now, if f ∈ D(T c ) and supp f ⊂ (0, ) × S 1 , we can consider (thanks to a localization argument) a sequence {ϕ n } n∈N ⊂ C ∞ 0 ((0, ) × S 1 ) that converges to f in the norm of T c , and hence [START_REF]Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces[END_REF] implies that the sequence of the k th -Fourier component { (ϕ n ) k } n∈N ⊂ C ∞ 0 (0, ) converges to f k in the norm of s c , for all k ∈ Z. Thus, f k ∈ D(s c ) ∀k ∈ Z. Then, the conclusion follows by applying Proposition 23, since every function f ∈ H

2 0 (R + ) satisfies f (x) = o(x 3
2 ) for x → 0.

Adjoint operator

We first consider the 1D Sturm-Liouville model operator given by

A = - d 2 dx 2 + g 2 x 2 + g 1 x , g 1 , g 2 ∈ R. (18)
Moreover, we introduce a C ∞ cut-off function 0 ≤ P ≤ 1,

P (x) = 1 if x ≤ /2, 0 if x ≥ . (19)
Lemma 28. Let g 1 ∈ R and g 2 ∈ (-1/4, 3/4). Consider the operator A acting on the Hilbert space

L 2 (R + ) with domain C ∞ 0 (R + ). Then, (a) for any f ∈ D(A), f (x) = o(x 3
2 ), as x → 0;

(b) D(A * ) = D(A) + span{ψ + P , ψ -P }, where P is the cut-off function defined in [START_REF]On the domains of Bessel operators[END_REF], and

ψ ± (x) = x α ± + a ± x α ± +1 , α ± = 1 2 ± 1 2 4g 2 + 1, a ± = g 1 (α ± + 1)α ± -g 2 ,
if g 2 0, and

ψ + (x) = x, ψ -(x) = 1 + g 1 x log(x), if g 2 = 0.
Remark 29. Lemma 28 is already known in the literature and holds also when g 2 < 3/4 (see, e.g., [START_REF]On radial Schrödinger operators with a Coulomb potential[END_REF]Proposition 3.1]), with the following modification of (b)

ψ + (x) = x 1 2 + g 1 x 3 2 , ψ -(x) = x 1 2 log(x) + g 1 x 3 2 log(x) + 2x 1 2 , if g 2 = -1/4.
Below we provide an alternative proof that uses perturbative arguments and does not need an introduction of the Bessel functions.

Proof. of Lemma 28

To show (a), we claim that g 1 x is infinitesimally-small w.r. x is infinitesimally-small w.r.t. T c if c < 1/2). Indeed, for g 2 > -1/4 and all f ∈ C ∞ 0 (R + ) we have

g 1 f x 2 L 2 (R + ) = ∫ R + g 2 1 x 2 | f (x)| 2 dx = 1 - 4g 2 1 + 4g 2 ∫ R + g 2 1 | f (x)| 2 x 2 dx + 4g 2 1 + 4g 2 ∫ R + g 2 1 | f (x)| 2 x 2 dx ≤ 4g 2 1 1 + 4g 2 ∫ R + | f (x)| 2 + g 2 x 2 | f (x)| 2 dx = 4g 2 1 1 + 4g 2 - d 2 dx 2 + g 2 x 2 f , f L 2 (R + ) ≤ 4g 2 1 1 + 4g 2 δ - d 2 dx 2 + g 2 x 2 f 2 L 2 (R + ) + 1 δ f 2 L 2 (R + ) ,
where we have used the Hardy inequality in the first inequality, we have integrated by parts in the third equality, and we have used the Young inequality in the last inequality that holds for every δ > 0. This proves the claim. As a consequence, for g 2 ∈ (-1/4, 3/4) we have

D - d 2 dx 2 + g 2 x 2 + g 1 x = D - d 2 dx 2 + g 2 x 2 = H 2 0 (R + ),
where we have used Kato-Rellich's Theorem (Propositon 13) in the first equality and Proposition 23 in the second equality.

To prove the second statement, we look for the solutions of

-u (x) + g 2 x 2 u(x) + g 1 x u(x) = 0. ( 20
)
These are two linearly independent functions which can be expressed via confluent hypergeometric functions, but since we are only interested in their behavior near x = 0, we can just use the Frobenius method (see, for instance, [START_REF]Ordinary Differential Equations and Dynamical Systems[END_REF]Chapter 4]) to understand their asymptotics. The first step is to write down the indicial polynomial, which is defined as

P(α) = (x -α+2 Ax α )| x=0 = α(α -1) -g 2 .
The construction depends whether or not the two roots of this polynomial are separated by an integer. The two roots are given by α ± = 1 2

± 1 2 4g 2 + 1.
Under the stated range of g 2 it follows that the only two cases where the two roots are separated by an integer are given by g 2 = 0. Assume that g 2 0. Then the Frobenius method states that there exist two independent solutions, which can be represented as converging series of the form

u ± (x) = x α ± ∞ i=0 a i x i . ( 21
)
We plug the ansatz ( 21) into ( 20) and obtain the following conditions for the dominating terms a 0 [α(α -1)g 2 ] = 0, a 1 (α + 1)αa 1 g 2 + a 0 g 1 = 0.

Setting a 0 = 1, we obtain that α ± are exactly the roots of the indicial polynomial, that a 1,± = g 1 (α ± + 1)α ±g 2 =: a ± , and that the solutions are u ± (x) = x α ± + a ± x α ± +1 + o(x α ± +1 ).

Assume now that g 2 = 0. Then the Frobenius method tells us that u + (x) is still a solution of (20) and the second solutions is given by u -(x) = Cu + (x) log(x) + x α - i=0 a i x i .

Plugging this series expression into [START_REF]On radial Schrödinger operators with a Coulomb potential[END_REF] allows us to recover ψ ± as the dominating terms of u ± . Moreover notice that, as a direct consequence of the Frobenius method, A(u ± -ψ ± ) is bounded near x = 0, and hence Aψ ± ∈ L 2 (0, 1).

So, let ψ ± as in the statement. Then (i) ψ ± ∈ L 2 (0, 1), (ii) Aψ ± ∈ L 2 (0, 1), (iii) ψ ± D(A),

where (i) and (ii) imply at once that ψ ± P ∈ D(A * ) and (iii) follows from part (a) and the asymptotics of ψ ± near x = 0. Since the functions ψ + P and ψ -P are linearly independent and the quotient D(A * )/D(A) has dimension at most 2 (as it follows from the fact that A is in the limit point case at ∞, by applying Proposition 16, which in turns implies that ker(A + ± i) have at most dimension 1, by applying Proposition 15), the thesis follows.

Now, we can use Lemma 28 to obtain informations on the adjoint of the 2D operator we are interested in, that is, -L + cK defined in [START_REF]The Laplace-Beltrami operator in almost-Riemannian geometry[END_REF], and complete the proof of Theorem 21. This concludes the proof of Theorem 2.

Figure 1 :

 1 Figure 1: Geodesics on the Grushin cylinder, staring from the point (-1/2, 0) with final time t f = 1.3, crossing smoothly the singular set Z (red circle).

Definition 8 .

 8 Let M be a 2D connected smooth manifold. A 2-dimensional almost-Riemannian Structure (2-ARS) on M is a pair (U, f ) as follows:

Lemma 27 . 3 2)

 273 Let c ∈ (0, 1/2), and let f ∈ D(-L + cK) be a function supported in (0, ) × S 1 , for some> 0. Then, f k (x) = o(x as x → 0 + , for every k ∈ Z. Proof. Let f ∈ C ∞ 0 ((0, ) × S 1 ). Lemma 25 shows that f ∈ D(-L + cK) if and only if f ∈ D(T c). Thus, we are left to study the behavior near x = 0 of a function f ∈ D(T c ). For any k ∈ Z, we have

Lemma 31 .

 31 φ n → h +, ,c and (-∆ + cK) φ n = (-∆ + cK) Ω φ n → G ,c | Ω , as n → ∞, in L 2 (Ω).

Remark 32 .

 32 Thus, we conclude by applying Lemma 31 which says that h +, ,c ∈ D((-∆ + cK) Ω ), which is impossible.Proof. of Lemma 31 Because of (i) and (ii), we have as n → ∞(i.1) φ nh +, ,c L 2 (U) → 0 , (i.2) φ n L 2 (M\U) → 0, (ii.1) (-∆ + cK)φ n -G ,c L 2 (U) → 0 , (ii.2) (-∆ + cK)φ n L 2 (M\U) → 0, since supp(h +,,c) and supp(G ,c ) are both contained in U. Then we have (as n → ∞)φ nh +, ,c L 2 (Ω) = φ nh +, ,c L 2 (U) + ξφ n L 2 (Ω\U) ≤ φ nh +, ,c L 2 (U) + φ n L 2 (M\U) → 0. Moreover, using that ∆(ξφ n ) = (∆ξ)φ n + 2∇ξ • ∇φ n + ξ(∆φ n ), we have (-∆ + cK) φ n -G ,c | Ω L 2 (Ω) ≤ (-∆ + cK)φ n -G ,c L 2 (U) + C (-∆ + cK)φ n + |∇φ n | + φ n L 2 (Ω\U) , where C is a constant such that C > ∆ξ L ∞ (Ω) , |∇ξ | L ∞ (Ω) , ξ L ∞ (Ω) . Since K is a bounded function on Ω \ U, we have K φ n L 2 (Ω\U) ≤ K L ∞ (Ω\U) • φ n L 2 (Ω\U) → 0, as n → ∞,and ∆φ n L 2 (Ω\U) ≤ (-∆ + cK)φ n L 2 (Ω\U) + cK φ n L 2 (Ω\U) → 0, as n → ∞.Finally, by Sobolev embedding, we have|∇φ n | L 2 (Ω\U) ≤ C( ∆φ n L 2 (Ω\U) + φ n L 2 (Ω\U) ) → 0, as n → ∞.To prove that the deficiency indices of -∆ + cK are infinite if c > 0, it suffices to consider the infinitedimensional vector space spanned by the family of functions {F k ,c } k ∈Z contained in D((-∆ + cK) * ) \ D(-∆ + cK) defined byF k ,c = e iky h +, ,c on Ω, 0 on M \ Ω.One can construct such family of functions close to any singular region of M, and each singular region has an infinite family of self-adjoint extensions; this gives room to self-adjoint extensions on the whole manifold, characterized by different boundary conditions to be imposed at each singular region.

  The set of points in M such that dim(D p ) < 2 is called singular set and it is denoted by Z. Since D satisfies the Hörmander condition, the subspace D p is nontrivial for every p and Z coincides with the set of points p where D is one-dimensional. The 2-ARS is said to be genuine if Z ∅. The 2-ARS is said to be 2-step if for every p ∈ M we have D p + [D, D] p = T p M.

  , one can deduce that A admits self-adjoint extensions if and only if its deficiency indices are equal ([36, Corollary to Theorem X.2]). Another immediate consequence of (11) is the following fact concerning 1D operators: let A be a 1D Sturm-Liouville operator -d 2 dx 2 + V(x), where V is a continuous real function on R + , acting on L 2 (R + , dx) with domain C ∞

0 (R + ) (for a general introduction to 1D Sturm-Liouville operators, see e.g.

[START_REF]Unbounded self-adjoint operators on Hilbert space[END_REF] Chapter

  t. -d 2 dx 2 + g 2x 2 if g 2 > -1/4 (exactly as in in Lemma 25 we showed that

g 1, c

1In[START_REF]The Laplace-Beltrami operator in almost-Riemannian geometry[END_REF] there is the additional hypothesis that Z is en embedded one-dimensional submanifold of M. However, as a direct consequence of the implicit function theorem, it is easy to see that such an hypothesis is implied by the fact that the structure is 2-step.

Acknowledgements: We thank M. Gallone and A. Michelangeli for very instructive discussions, and Stephen A. Fulling for sharing with us [25]. We are also deeply grateful to the anonymous reviewer for the crucial comments and corrections. The project leading to this publication has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 765267 (QuSCo). Also it was supported by the ANR projects SRGI ANR-15-CE40-0018 and Quaco ANR-17-CE40-0007-01.

However, from this proof we can also point out that this is not always the case for the α-Grushin cylinder, as for instance in the case of a flat cone (α = -1) with c ≥ 0 (note that this is not an ARS). In that case, the inequality (k 2 -1 4 ) 1 x 2 ≥ 3 4x 2 implies that the k-th's Fourier components of -∆ -1 + cK -1 are essentially self-adjoint for all k 0, even if (-∆ -1 + cK -1 ) 0 is not. This means that -∆ -1 + cK -1 has deficiency indices equal to 1, for all c ≥ 0.

We conclude this Section by considering a Riemannian manifold (M, g) without boundary, with associated Riemannian volume form ω, and Laplace-Beltrami operator ∆ = div ω • grad g acting on the Hilbert space L 2 (M, ω), with domain

Letting F be a real-valued continuous function locally L 2 (M, ω) seen as a multiplicative operator with domain C ∞ 0 (M), (i) and (ii) still hold true for the operator ∆ + F instead of ∆. Remark 20. Being ∆ + F a real operator (that is, it commutes with complex conjugation), its deficiency indices are equal ( [START_REF]Methods of modern mathematical physics. II. Fourier analysis, selfadjointness[END_REF]Theorem X.3]) and thus it always admits self-adjoint extensions.

Grushin zone

We focus now our attention around a Grushin point. We thus define the Riemannian manifold Ω = {(x, y) ∈ R × S 1 | x 0} with metric g = diag(1, x -2 e -2φ(x,y) ), where φ is a smooth function which is constant for large |x| . The smoothness of φ is guaranteed by Proposition 11, and even if it is only defined locally, we extend it constantly in the coordinates (x, y), since what matters is the analysis close to x = 0. Note that X 1 , X 2 given by Proposition 11 (F2) is an orthonormal frame for g. We then consider also the two connected components

We start by proving the following key result: Theorem 21. Consider the Riemannian manifold (Ω + , g), with associated Riemannian volume form ω, curvature K and Laplace-Beltrami operator ∆. Let c ∈ (0, 1/2). Consider the curvature Laplacian -∆ + cK, with domain C ∞ 0 (Ω + ), acting on L 2 (Ω + , ω). Then for every > 0 there exist functions

In particular, -∆ + cK is not essentially self-adjoint (here c 0). The same conclusions hold if we replace

Proof. of Theorem 21

We take the coefficient of 1

x evaluated at x = 0 (i.e., on the singularity) and treat the second variable y as a parameter. Indeed, setting

we obtain from Lemma 28 two functions ψ ±,c ∈ C ∞ (Ω) of both variables x, y. Then, we get the following:

Lemma 30. Let c ∈ (0, 1/2), and define h ±, ,c (x, y) = ψ ±,c (x, y)P (x) ∈ L 2 (Ω + ) ∩ C ∞ (Ω + ), where ψ ±,c have the same form as functions ψ ± from Lemma 28 with g 1 , g 2 given by [START_REF] D W -M | A stochastic scheme for constructing solutions of the Schrödinger equations[END_REF] and P is defined in [START_REF]On the domains of Bessel operators[END_REF].

Then,

Proof. Part (iii) is obvious, as supp(P ) ⊂ (0, ). To prove (i), we consider the operator R on the domain C ∞ 0 (Ω + ), whose action is defined by R := (-L + cK) -A, (where A is the operator whose action is defined in [START_REF]Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds[END_REF], but now is considered on the domain C ∞ 0 (Ω + ), and g 1 , g 2 are the functions defined in ( 22)), and we claim that R h ±, ,c ∈ L 2 (Ω + ), in the weak sense. Indeed, we have

(where η c and η c are bounded functions on (0, ) × S 1 ) and the claim follows from the C ∞ -regularity of h ±, ,c w.r.t. y. Then, by the very construction of h ±, ,c , we have that A h ±, ,c ∈ L 2 (Ω + ), which in turns implies that (-L + cK) h ±, ,c = (R + A) h ±, ,c ∈ L 2 (Ω + ) in the weak sense, and proves part (i).

To prove part (ii) we notice that the 0 th -Fourier component of h ±, ,c is given by

if c = 3/8 andwhere α ±,c = 1 2 ± √ 1 -2c and a ±,c is the function found in Lemma 28 (b) w.r.t. the function g 1 and the constant g 2 defined in [START_REF] D W -M | A stochastic scheme for constructing solutions of the Schrödinger equations[END_REF]. The conclusion follows by applying Lemma 27, since

2 ) for any c ∈ (0, 1/2).

To conclude the proof of Theorem 21, it suffices to consider h ±, ,c := U -1 h ±, ,c , where U is the unitary transformation defined in [START_REF]Lipschitz classification of almost-Riemannian distances on compact oriented surfaces[END_REF].

The proof of Theorem 22 is now an immediate consequence: Proof. of Theorem 22 It follows by considering the infinite-dimensional vector space spanned by the family of functions {h ±, ,c e iky } k ∈Z ⊂ D((-∆ + cK) * ) \ D(-∆ + cK).

Proof of Theorem 2

If c = 0, H 0 = -∆ is known to be essentially self-adjoint on L 2 (M, ω) ([15, Theorem 1.1]). Then, let c ∈ (0, 1/2).

Let Z = j ∈J W j be the disjoint union in connected components for the singular set, and M = ∪ i ∈I Ω i be an open cover such that, for every W j , there exist a unique Ω i j (Grushin zone) with W j ⊂ Ω i j and W j ∩Ω i = ∅ if i i j . Moreover, as previously remarked, we can assume that Ω i j is a tubular neighborhood of W j , i.e., Ω i j R × S We define the function

So we have

having integrated by parts (h +, ,c vanishes away from ∂Ω, and u vanishes away from W), which proves that

with 0 ≤ ξ ≤ 1, and define the sequence ( φ n = ξφ n ) n∈N ⊂ C ∞ 0 (Ω \ W). We have the following


Francesco Baldi
email: francesco.baldi@unibs.it

Silvia Agnelli

Luca Andena

Bamber R K Blackman

Leonardo Castellani

Patricia M Frontini

Jaroslav Kučera

Lucien Laiarinandrasana

Alessandro Pegoretti

Alicia Salazar

Laurent Warnet

Determination of the Fracture Resistance of Ductile Polymers: The ESIS TC4 Recent Experience

Keywords: J-integral, load separation criterion, fracture resistance, ductile polymers

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

To determine the low-rate fracture resistance of ductile polymers, for which the application of standard linear elastic fracture mechanics (LEFM) tests fails, the material crack growth resistance (J R ) curve (J vs crack extension, Δa), developed within the frame of elastic-plastic fracture mechanics (EPFM), is generally employed. This is usually constructed by means of a multispecimen approach (procedure developed by ESIS TC4, [START_REF] Hale | J-Fracture Toughness of Polymers at Slow Speed[END_REF] that is, the Technical Committee 4, "Polymers, Polymer Composites and Adhesives," of the European Structural Integrity Society, [START_REF]ESIS Home Page[END_REF] and ASTM D6068-10(2018), Standard Test Method for Determining J-R Curves of Plastic Materials 3 ). Specific ESIS TC4 round-robin (RR) tests showed that the uncertainties associated with the measurement of Δa often make this approach unreliable. Furthermore, in many cases, an initiation fracture resistance parameter, J Ic , cannot be obtained. With this in mind, the attention of ESIS TC4 has been recently paid to the analysis of a single-specimen testing scheme based on the load separation criterion (LSC), which does not require the measurement of Δa (see Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] ). This approach would (i) allow the determination of a material initiation fracture resistance parameter, J I,lim , and (ii) provide a rough measure of Δa produced during the fracture test in the plastic region. In order to assess the degree of reproducibility of the results obtainable with the application of this method, in view of its possible employment in a standardized procedure, a multilaboratory activity was begun in September 2011 under the direction of ESIS TC4, with Università degli Studi di Brescia (Italy) as the coordinating laboratory. Ten laboratories (indicated in the authors' affiliation list) participated in this activity, which was organized on three consecutive RR testing exercises (RR1 to RR3). RR1 consisted in a preliminary work aimed at setting the key points for the preparation of the reference draft testing protocol, [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] which was prepared in 2013 and then further reviewed in 2015. In RR2 and RR3, this drafted protocol (called RR protocol hereafter) was applied to the fracture characterization of polymeric materials that exhibit a ductile behavior but different degrees of stiffness and strength (from standard tensile tests). The outcomes were used to enhance the robustness of the method and to improve the protocol itself. The examined materials were as follows: an acrylonitrile-butadiene-styrene (ABS) and a high-impact polystyrene (HIPS) in RR2 (the RR2 results are presented in Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] ); and a rubber-toughened polybutylene terephthalate (RT-PBT) and a linear low-density polyethylene (LLDPE) in RR3. The present work describes the LSC-based testing procedure examined by ESIS TC4 and gives a summary of the results obtained during the RR activities.

The LSC-based Testing Method

The method described in the RR protocol, [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] founded on the LSC proposed by Ernst, Paris, and Landes, [START_REF] Ernst | Estimations on J-Integral and Tearing Modulus T from a Single Specimen Test Record[END_REF] was derived from Sharobeam and Landes' works published in the early 90s on metals. [START_REF] Sharobeam | The Load Separation Criterion and Methodology in Ductile Fracture Mechanics[END_REF][START_REF] Sharobeam | The Load Separation and η pl Development in Precracked Specimen Test Records[END_REF] The LSC assumes that, for a defined specimen geometry, material, and constraint, the load, P, recorded in a fracture test in the plastic region can be mathematically represented as the product of two independent functions, as equation (1) shows:

P = G b W • H u pl W (1)
where G(b/W) and H(u pl /W) are the geometry and the material deformation functions, respectively, W is the specimen width, b the specimen uncracked ligament length (that is W-a, with a that indicates the crack length), and u pl is the plastic displacement (that is the plastic component of the total displacement, u). The applicability of the LSC to polymeric materials, during both blunting and crack propagation phase, has been demonstrated (see Bernal, Cassanelli, and Frontini, 9,10 Bernal, Montemartini, and Frontini, [START_REF] Bernal | The Use of Load Separation Criterion and Normalization Method in Ductile Fracture Characterization of Thermoplastic Polymers[END_REF] Morhain and Velasco, 12, [START_REF] Morhain | J-R Curve Determination of Magnesium Hydroxide Filled Polypropylene Using the Normalization Method[END_REF] Salazar and Rodríguez, [START_REF] Salazar | The Use of the Load Separation Parameter S pb Method to Determine the J-R Curves of Polypropylenes[END_REF] and Baldi, Agnelli, and Riccò [START_REF] Baldi | On the Applicability of the Load Separation Criterion in Determining the Fracture Resistance (J Ic ) of Ductile Polymers at Low and High Loading Rates[END_REF] ). The testing method described in the RR protocol [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] is based on the construction of the "load separation parameter curve," i.e., the S sb curve, from tests in single edge-notched bending (SE(B)) configuration (see fig. 1). It requires the execution of two tests: on a sharp-notched (sN) specimen and on a blunt-notched (bN) specimen. A schematic representation of a sharp and of a blunt notch is shown in figure 1. In the sN specimen, fracture propagation occurs, whereas in the bN specimen, crack growth is hindered. From the load, P, versus displacement, u, curves obtained from quasi-static tests on sN and bN specimens, the separation parameter, S sb ,i s determined as follows:

S sb ðu pl Þ = P s P b u pl (2)
where P s and P b are load values read on P versus u pl curves of the sN and bN specimens, respectively, at a given value of plastic displacement, u pl , which, in the protocol examined here, is determined for each specimen as:

u pl = u -P • C 0 (3)
where C 0 is the initial elastic compliance of the specimen. With reference to u pl determined according to equation (3), that is from C 0 , it is worth pointing out that (i) for the sN specimen in the crack propagation phase, u pl data assume a nominal character in consideration of the fact that the true u pl data should be evaluated from the actual compliance, C, that increases during the crack growth phase but is not known (the determination of C would require test interruption and the measurement of Δa [see Appendix, where true u pl data determined from C have been used for the verification of the LSC validity for RT-PBT]); (ii) in consideration of the viscoelastic nature of polymeric materials, u pl represents the "non-elastic" component of the total displacement, u, which for these materials is in general the sum of three contributions: elastic, viscoelastic, and plastic. Preliminary investigations specifically carried out by the laboratory of Brescia showed that, to the aims of the testing method examined here (RR protocol [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] ), u pl data evaluated from C 0 can be successfully used. It is expected that the S sb curve, that is, S sb plotted against u pl , shows a "plateau" region, whereas S sb maintains an almost constant value (S sb,plateau ), followed by a decreasing S sb region. The former region corresponds to the crack blunting phase, whereas the latter region corresponds to the crack propagation phase in the fracture process of the sN specimen. The point between these two regions (called the limit point), at u pl = u pl,lim , corresponds to fracture initiation (or pseudo-initiation, especially considering that for ductile polymers fracture initiation can be a complex progressive process [START_REF] Baldi | On the Determination of the Point of Fracture Initiation by the Load Separation Criterion in J-Testing of Ductile Polymers[END_REF] ). J-integral value at u pl,lim , that is J I,lim , which can be taken as a material pseudoinitiation fracture resistance parameter in place of the more conventional J 0.2 computed by the J R curve (see the ESIS TC4 procedure 1 ), is evaluated as follows:

J I,lim = 2•U lim,c B • ðW -a 0 Þ (4)
where U lim,c is the energy up to the limit point, corrected for indentation (see the ESIS TC4 procedure 1 ), and B, W, and a 0 are the thickness, width, and initial crack length of the sN specimen, respectively.

Furthermore, the normalized separation parameter curve (R S curve), i.e., R S versus u pl , is traced (R S is obtained as the ratio of S sb over S sb,plateau ), and the parameter m S is determined in the region of fracture propagation as follows:

m S = - dR S du pl u pl >u pl,lim (5)
It has been shown 4 that parameter m S , which is a specimen characteristic (i.e., dependent on both specimen geometry and material), provides an indication of the crack advancement produced per unit of u pl and could be used, as a "ductility index," to classify the fracture propagation processes by the amount of crack growth occurring within the plastic region. The higher the value of m S , the higher the Δa per unit of u pl ;ifm S → 0, the process is governed by crack blunting.

Further details concerning this methodology can be found in Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] and in the RR protocol. [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] Experimental Materials ABS, HIPS, and LLDPE were provided by Versalis SpA (Mantova, Italy), whereas RT-PBT was provided by Radici Novacips SpA (Villa d'Ogna, Bergamo, Italy). Table 1 reports the supply form of the materials, their basic mechanical properties and the nominal dimensions of the SE(B) specimens used in the RR fracture tests. For ABS, HIPS, and RT-PBT the fracture resistance, J 0.2 data are also indicated. They were obtained from the J R curve constructed by the application of the ESIS TC4 multispecimen approach 1 on specimens having the same geometry and dimensions of the RR tests (for ABS and HIPS, J 0.2 data are from Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] ; for RT-PBT, the J R curve is reported in the Appendix). Each laboratory prepared and tested at least three sN and one bN specimens for each of the materials considered. The notching techniques were freely chosen by the laboratory, following the guidelines provided by the RR protocol. [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] For LLDPE, two sets of specimens (#a and #b in Table 1) that differed in size were examined. This was done in order to explore, for this very ductile polymer (which also showed the lowest degrees of stiffness and strength among the materials examined), how the results obtainable by the application of the RR protocol could be affected by the specimen size (geometrical constraint). All the experiments were performed by means of universal testing machines at ≈ 23°C and with a crosshead rate of 1 mm/min. The data were processed according to the RR protocol, [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] and the results (consisting of the S sb curve and data from J I,lim and m S , for each sN specimen tested) were sent to the laboratory of Brescia for the comparative analysis.

The applicability of the LSC to styrenic polymers and polyolefins has been examined in several literature works; in contrast, to the best of the authors' knowledge, the applicability of the LSC to RT-PBT has never been checked. Therefore, before starting the RR activity on the RT-PBT, the validity of the LSC has been experimentally verified (see Appendix).

Results and Discussion

Figure 2A and 2B report J I,lim and m S results, respectively, for the various materials examined. Each datum reported is the mean value obtained by averaging all the data from the various laboratories. The datum considered for each laboratory is the average of the data obtained from the various sN specimens tested at the laboratory.

The degree of repeatability of the results within the same laboratory was generally higher than that of reproducibility (represented by the data of standard deviation reported in fig. 2). Results with a very high degree of repeatability could be obtained, and this emerges clearly from figure 3A-E, in which the R S curves of different nominally identical sN specimens of a given material, obtained at a given (selected) laboratory, are compared (the curves are vertically shifted for clarity). Figure 3A refers to specimens of HIPS tested at Laboratory 10; figure 3B refers to specimens of ABS tested at Laboratory 11; figure 3C refers to RT-PBT tested at Laboratory 1 (the R S curves are reported both shifted and not); figure 3D and 3E refers to LLDPE specimens with size #a and #b, respectively, tested at Laboratory 6 (laboratory numbers are in accordance with the authors' affiliation list). In figure 3A and 3B, the R S curve indicated with an asterisk was built at Brescia based on the raw data provided by the laboratory that performed the tests. This was done, within the RR2 activities, just to check whether the data processing procedure described in the RR protocol [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] had been properly applied by the laboratory. Figure 3C FIG. 2 Mean values (± standard deviation) of (A) J I,lim and (B) m S for the various materials examined, which were obtained by averaging all the (mean) data from the various laboratories (refer to Table 1 for the dimensions of the specimens). Values in parentheses indicate the standard deviation expressed as a percentage of the corresponding mean value. For LLDPE, valid J I,lim data could not be obtained (see text). Data with an asterisk are from Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] shows the R S curves obtained at Brescia (Laboratory 1) from six sN specimens of RT-PBT, tested up to different levels of displacement (the rhomb indicates, for each sN specimen test record, the point at which the fracture test was interrupted, i.e., the final point). If not shifted vertically, these curves overlap well, and they practically draw one single curve in which the plateau region that extends up to the limit point is clearly visible. For the LLDPE, the degree of repeatability of the results was generally lower with respect to the other materials, as can be observed in figure 3E. This is related to the fact that LLDPE was largely the most difficult-to-characterize material. Even if relatively high displacement values were reached (especially with size #b; refer to fig. 3E), which is also necessary in consideration of the low Young's modulus of the material, reliable plateau regions could not be clearly identified in the S sb curves of the sN specimens tested. This indicates that, in the tests on the SE(B) specimens of the LLDPE examined herein, the crack blunting phase could not be distinguished from the crack propagation phase, and valid initiation fracture resistance data (J I,lim ) could not be obtained. Focusing attention on HIPS, ABS, and RT-PBT, for which data of both J I,lim and m S have been obtained, it can be observed that the degree of reproducibility for the latter data is higher than that for the former data (see fig. 2A and2B). This suggests that, with respect to the fracture propagation phase, which is what the m S parameter refers to, fracture initiation/early stages of crack growth, which is what J I,lim refers to, are less reproducible (at a macroscopic scale). Examination of the results collected from the various laboratories seems to indicate that the scattering observed for the J I,lim data is related to a combination of both testing and data analysis aspects.

(i) In relation to testing, the results suggest that the quality of the notch of the sN specimens played an important role. As underlined by Salazar et al. [START_REF] Salazar | Fracture Toughness of High Density Polyethylene: Fatigue Pre-cracking versus Femtolaser, Razor Sharpening and Broaching[END_REF] and Martínez et al., [START_REF] Martínez | Influence of the Notch-Sharpening Technique on Styrene-Acrylonitrile Fracture Behavior[END_REF] the influence of the sharp notch quality on the determination of the initiation fracture resistance of polymeric materials, in plane strain conditions via both LEFM and EPFM methods, is still not fully understood. Within ESIS TC4, an RR exercise aimed at identifying the most suitable notching technique for the preparation of the notched specimens for the execution of plane strain LEFM tests is currently underway. [START_REF]ESIS Home Page[END_REF] The outputs of this RR activity will be used, at a later stage, as the basis for an investigation into the effects of the sharp notch quality in J-testing of ductile polymers.

(ii) With regard to data analysis, the results clearly point out that a crucial role is played by the determination of the initial specimen compliance, C 0 . In addition, the procedure described in the RR protocol [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] for the identification of the limit point on the S sb curve can be further improved. Within ESIS TC4, specific activities aimed at carefully examining these aspects are in progress. More specifically, an RR exercise aimed at evaluating the reproducibility degree of the data of initial elastic compliance, C 0 , of SE(B) specimens of polymeric materials that exhibit a ductile behavior is currently underway.

Even if the degree of scattering observed for J I,lim data is higher than that of m S data, it can be acceptable within the field of fracture mechanics testing. Interestingly, not only for HIPS and ABS (as already pointed out by Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] ) but also for RT-PBT, the J I,lim value is lower than the value of the technological J 0.2 parameter read on the J R curve (see Table 1 and Appendix). This suggests that the LSC-based approach examined here is able to provide data of fracture resistance that is more conservative with respect to J 0.2 . It is worth noting, also, that for each material, the value of J I,lim verified the size criteria proposed in the ESIS TC4 protocol 1 for critical J-values, and this indicates that the J I,lim values refer to a plane strain state at the crack tip and that plasticity in the ligament is not excessive.

Figure 4 shows the R S curves constructed at the laboratory of Brescia in compliance with the RR protocol, for the SE(B) specimens of the different materials examined (one single curve for each material). For LLDPE, only the curve of size #b, which refers to specimens with a width-to-thickness ratio closer to those of the specimens of the other materials, is reported. The differences among the fracture processes of the various specimens emerge clearly. Interestingly, by comparing the R S curves of RT-PBT and HIPS (that is, at fixed specimen dimensions, see Table 1), it can be observed that the extension of the plateau region of the curve is larger for the former than for the latter. This indicates that the higher fracture resistance obtained for RT-PBT with respect to HIPS (J I,lim of RT-PBT is ≈ 3 times that of HIPS, see fig. 2A) can be related also to the fact that the blunting-to-fracture transition, in the fracture process of a sN specimen, occurs at a level of u pl that is higher for the former than for the latter.

The analysis of the fracture process based on the application of the RR protocol 5 brings to light, for the LLDPE, a behavior remarkably different from that of the other ductile polymeric materials examined. For the construction of the R S curve of LLDPE, in absence of a reliable S sb,plateau , following the protocol, [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] the value of S sb at u pl = 0.5 mm was used in place of S sb,plateau . The low values of m S obtained (see fig. 2B) clearly suggest that the process is governed by blunting: irrespective of the specimen size considered, either #a or #b, crack growth produced per unit of u pl is very small. The high degree of scattering obtained for m S data suggests that its fracture process is not easily reproducible, contrary to what has been observed for the other materials for which standard deviations of m S are quite small. This is one of the reason why a reliable J R curve could not be constructed for this material (see Table 1), for which, furthermore, valid data of Δa could not be obtained by the inspection of the fracture surface produced. Even if valid fracture resistance data (J I,lim ) have not been determined for LLDPE, the testing procedure examined here was able to highlight, through m S determination, that a testing scheme based on the propagation of a crack cannot be successfully applied to this material, working on SE(B) specimens with dimensions as in Table 1. It would be necessary to use different specimen geometry/ dimensions or to resort to another testing approach that does not require the use of precracked specimens, such as cutting tests (see Patel, Blackman, and Williams [START_REF] Patel | Determining Fracture Toughness from Cutting Tests on Polymers[END_REF] ).

The results obtained for RT-PBT and LLDPE, in RR3, strengthen the idea that was outlined by Agnelli et al. [START_REF] Agnelli | Application of the Load Separation Criterion in J-Testing of Ductile Polymers: A Round-Robin Testing Exercise[END_REF] : to attribute a key role to the m S parameter in the fracture characterization of ductile polymers. The uncertainties associated with the measurement of Δa, which impair the construction of the J R curve, can have various causes: instrumental, methodological, or related to the intrinsic fracture behavior of the material. The measurement of Δa is typically based on the optical analysis of the fracture surface generated during the test, and the greatest difficulty is associated with the correct evaluation of the crack front that defines the region on the fracture face characteristic of actual crack growth. More specifically, the obtainment of reliable Δa data can be particularly arduous if, depending on the material, the crack growth produced in the fracture test is very limited or it is overshadowed by blunting (that was the case of the LLDPE examined here), or it is complicated by the occurrence of specific phenomena, such as with multiple cracking. Results obtained in the RR exercise reported here show that the application of the RR protocol, [START_REF] Agnelli | A Testing Protocol for the Construction of the Load Separation Parameter Curve for Plastics[END_REF] which in principle requires only two tests, is effective at distinguishing between, using the m S parameter, fracture processes governed by blunting (m S → 0) and processes in which the crack growth actually occurs (with m S > 0). With this in mind, m S might be used as a key parameter in a criterion to check a priori if the application of the multispecimen approach for J R curve construction (founded on the measurement of Δa) to specimens with given geometry and dimensions is likely to fail. More specifically, with reference to SE(B) specimens (that is, the testing geometry considered in this work), a critical value of m S (m S,c ) might be fixed and used according to the following scheme:

• The LSC-based procedure is applied to a ductile polymer in the form of specimens with given dimensions, and the values of m S and J I,lim are determined (the possibility that the S sb plateau region is not clearly identified and, therefore, that J I,lim is not evaluated cannot be ruled out); • If m S > m S,c , then the value of J I,lim obtained is taken as the fracture resistance of the material (if the size criteria proposed for critical J-values in the ESIS TC4 protocol 1 are verified), and the application of the multispecimen approach for J R curve construction might be attempted; • If m S ≤ m S,c , then J I,lim (if determined) cannot be considered valid, and the multispecimen approach for J R curve construction cannot be applied; in this case, either the specimen dimensions are modified and new LSC-based tests performed or a different testing scheme that is not based on the use of precracked specimens (e.g., cutting tests [START_REF] Patel | Determining Fracture Toughness from Cutting Tests on Polymers[END_REF] ) is employed.

Of course, the analysis of the source of the errors in the measurement of Δa deserves a special consideration. To this aim, specific ESIS TC4 activities are currently being developed. More specifically, interlaboratory RR exercises focused on the instrumental and methodological aspects related to the optical analysis of the fracture surface for Δa evaluation will be organized. The possibility to resort to noncontact techniques (e.g., digital image correlation) for the indirect monitoring of crack advancement during the fracture test will also be considered.

Conclusions

The results obtained in the ESIS TC4 RR activity on the use of LSC in J-testing of ductile polymers are encouraging. The method under development has been successfully applied to ductile polymers that exhibit different degrees of stiffness and strength (ABS, HIPS, RT-PBT, and LLDPE). It allows determination of a material pseudoinitiation fracture resistance parameter (J I,lim ) as well as a crack propagation parameter (m S ). The results suggest that this latter parameter can play a key role in a criterion to check a priori if the multispecimen approach for J-testing (the ESIS TC4 procedure 1 or the ASTM D6068-10(2018) [START_REF]Standard Test Method for Determining J-R Curves of Plastic Materials, ASTM D6068-10[END_REF] ) applied to a ductile polymer (with given specimen geometry and dimensions) is likely to fail. The RR activity is still in progress, and special attention is being given to this latter outcome.

Cða=WÞ = 2•½f ða=WÞ 2 • Φða=WÞ E f ract • B (A.2)
where f(a/W)andΦ(a/W) are tabulated functions. [START_REF] Williams | Fracture Mechanics of Polymers[END_REF] The value of E fract , with a result of 1,400 ± 50 MPa, was previously determined by applying equation (A.2)-inverted-to the data of a 0 /W and C 0 (corrected by taking into account the specimen indentation compliance; see the ESIS TC4 procedure 1 ) of all the specimens used for the J R curve construction. A.1 J R curve of RT-PBT, constructed according to the ESIS TC4 procedure. [START_REF] Hale | J-Fracture Toughness of Polymers at Slow Speed[END_REF] The experimental (Δa; J) data points (rhomb), the power law best-fitting curve (solid line), the exclusion lines (vertical dash-dot lines), the blunting line (dashed line), and the points on the fitting curve corresponding to J bl and J 0.2 (asterisk)-see text-are indicated. The equation of the power law best-fitting curve and the correlation coefficient of the fitting (R 2 ) are also reported. indicates that u pl has no contribution on the value of S sb and, therefore, that the LSC is valid during crack propagation. For SE(B) configuration, the geometry function, G(b/W), can be expressed as (see Sharobeam and Landes 7 ) follows:

G b W = b W ηpl (A.3)
Therefore, a power law best-fitting curve-linear in the bilogarithmic plot of figure A.2-was forced on the fracture propagation data points. According to equation (A.3), the slope of the fitting line is η pl , which results in 1.8. The difference between this value and the theoretical value of 2 derived by Rice, Paris, and Merkle 22 for η pl , and widely adopted in literature, is very similar to those observed in other literature works in which values of η pl were determined experimentally (see Sharobeam and Landes, [START_REF] Sharobeam | The Load Separation and η pl Development in Precracked Specimen Test Records[END_REF] Bernal, Montemartini, and Frontini, [START_REF] Bernal | The Use of Load Separation Criterion and Normalization Method in Ductile Fracture Characterization of Thermoplastic Polymers[END_REF] Baldi, Agnelli, and Riccò, [START_REF] Baldi | On the Applicability of the Load Separation Criterion in Determining the Fracture Resistance (J Ic ) of Ductile Polymers at Low and High Loading Rates[END_REF] and Baldi and Riccò [START_REF] Baldi | High-Rate J-Testing of Toughened Polyamide 6/6: Applicability of the Load Separation Criterion and the Normalization Method[END_REF] ). Interestingly, the fitting curve passes very closely to the theoretical point at S sb = 1, which corresponds to a crack length in the sN specimen equal to the stationary crack length of the bN specimen.

The material deformation function, H(u pl /W), that characterizes the material during fracture propagation was then constructed-see equation (1). Following Sharobeam and Landes, [START_REF] Sharobeam | The Load Separation and η pl Development in Precracked Specimen Test Records[END_REF] it was built by referring to the normalized load, P N , that was evaluated as follows:

P N u pl W = Pð u pl W Þ B • W • Gð b W Þ (A.4)
where η pl = 1.8 was used in the expression of G(b/W). Figure A.3 shows the values of P N calculated at the final point of each sN specimen test record, plotted against the corresponding values of u pl /W. A power law best-fitting curve was forced to the experimental data points (this curve is known as "material key curve"; see Agnelli et al. [START_REF] Agnelli | Study of the Plastic Deformation Behaviour of Ductile Polymers: Use of the Material Key Curves[END_REF] ). In figure A.3, the P N versus u pl /W curve obtained from the P versus u curve of the bN specimen, which exhibits only blunting, is also reported. Interestingly, the material deformation function during the fracture propagation phase has the same form as during the blunting phase, and this is the experimental verification of the assumption on which the LSC is founded.

FIG. 1

 1 FIG. 1 Schematic representation of a SE (B) specimen. S is the span. Sharp (of a sharpnotched, sN, specimen) and blunt (of a bluntnotched, bN, specimen) notches are also represented; the blunt notch is a keyhole type.

FIG. 3 R

 3 FIG. 3 R S curves of different nominally identical sN specimens of (A) HIPS, from Laboratory 10; (B) ABS, from Laboratory 11; (C) RT-PBT, from Laboratory 1; (D) and (E) LLDPE, #a and #b, respectively, from Laboratory 6. The curves arevertically shifted by an h-factor. In (A), (B), and (C), the gray circles indicate the limit point. In (C), the curves are also represented as not shifted (in gray); rhomb indicates the final point (that is, the point at which the test was interrupted). See text for explanation of the asterisk in (A) and (B).

FIG. 4 R

 4 FIG. 4 R S curves obtained at Laboratory 1 from the fracture tests on the various materials examined (one curve for each material; refer to

Figure A. 2

 2 Figure A.2 shows the data points (b/W; S sb ) corresponding to the final points of the sN specimens test records considered. For S sb determination, the bN specimen test record of the RR experiments was used. These data points (b/W; S sb ), which are associated with different values of u pl , draw a single trend, and this

FIG. A. 2

 2 FIG. A.2 Separation parameter, S sb , evaluated at the final point of each sN specimen test record of RT-PBT examined (R S curves in fig. 3C), plotted against the corresponding value of the actual remaining ligament length divided by the specimen width, b/W (rhomb). The dashed line indicates the power law best-fitting curve forced to the experimental (b/W; S sb ) data at the final points. The theoretical point with S sb = 1 (asterisk)-see text-is indicated. The equation of the fitting curve and the correlation coefficient of the fitting (R 2 ) are also reported.

FIG. A. 3

 3 FIG. A.3Normalized load, P N , calculated at the final point of each sN specimen test record of RT-PBT examined (R S curves in fig.3C), plotted against the corresponding value of u pl /W (rhomb). The dashed line indicates the power law best-fitting curve forced to the experimental data (u pl /W; P N ), whose equation and correlation coefficient (R 2 ) are also reported. The solid line indicates the P N versus u pl /W curve obtained from the bN specimen of RT-PBT.

TABLE 1

 1 Supply form of the materials examined and their basic mechanical properties (Young's modulus, E, and tensile yield stress, σ y ); nominal dimensions of SE(B) specimens used in the fracture tests; and J 0.2 data

			Basic Mechanical Properties a	SE(B) Specimen Dimensions b
	Material (RR)	Supply Form	E (MPa)	σ y (MPa)	B (mm)	W (mm)	a 0 (W)	J 0.2 (kJ/m 2 )
	HIPS (RR2)	Injection-molded dumbbells c	1,760	18	4	10	0.6	2.84
	ABS (RR2)	Six-mm-thick compression molded plates	2,500	44	6	12	0.6	5.71
	RT-PBT (RR3) Injection-molded dumbbells c	1,450	31	4	10	0.6	6.58
	LLDPE d (RR3) Ten-mm-thick injection-molded plates	250	8	10	10 (#a)	0.6	-e
						20 (#b)

Note: a From quasi-static tests at room temperature. b B, thickness; W, width; a 0 , initial crack length (in bN specimen, notch tip radius of 1 mm); span used in fracture tests, S = 4W (ref. to fig. 1). c According to ISO 3167:2014, Plastics -Multipurpose Test Specimens; central narrow portion (with dimensions of 80 × 10 × 4mm 3 ) used for SE(B) specimen preparation. d Two sets of SE(B) specimens (#a and #b) used. e Not measured (J R curve not constructed).

Table 1

 1

	for the
	dimensions of the
	specimens). Gray circles
	indicate the limit point on
	the R S curves of HIPS,
	ABS, and RT-PBT. For
	each curve, m S is also
	represented.

ACKNOWLEDGMENTS

The authors are grateful to Versalis SpA (Mantova, Italy) and Radici Novacips SpA (Villa d'Ogna, Bergamo, Italy) for kindly supplying the materials tested in this study.

Appendix: Construction of J R Curve and Verification of the LSC Validity for RT-PBT

The J R curve was constructed according to the ESIS TC4 procedure. [START_REF] Hale | J-Fracture Toughness of Polymers at Slow Speed[END_REF] The same specimen geometry and dimensions (sN specimen, Table 1) and testing conditions (crosshead rate and temperature) of the RR tests were used. Figure A.1 shows the experimental (Δa; J) data points fitted by the power law best-fitting curve, the exclusion lines, and the blunting line, which was traced according to the following analytical expression:

where a value of m = 1 was used. The points on the J R fitting curve corresponding to J 0.2 (that is at Δa = 0.2 mm) and to J bl (that is, at the intersection with the blunting line) are also indicated. The value of J 0.2 is reported in As underlined by Sharobeam and Landes, [START_REF] Sharobeam | The Load Separation and η pl Development in Precracked Specimen Test Records[END_REF] if the load is separable during the fracture propagation phase, S sb data evaluated at different levels of u pl (when the crack is growing) should lie on one single curve if plotted against the actual values of b/W. For such verification, the sN specimens that have the highest geometrical similarity among the specimens tested for the construction of the J R c u r v e( w h i c ha ret h o s ew h o s eR S curves are reported in fig. 3C) were selected, and their P versus u curves and Δa data at the final point were considered. In this analysis, differently from what was done for the application of the RR protocol, the actual values of u pl at the final points-where the crack length, a,i sa 0 + Δa-were used. These u pl data were evaluated from the actual specimen elastic compliance, C(a/W), which was indirectly calculated by using the following expression 20 :