William Borrelli

Raffaele Carlone

Lorenzo Tentarelli

A NOTE ON THE DIRAC OPERATOR WITH KIRCHOFF-TYPE VERTEX CONDITIONS ON NONCOMPACT METRIC GRAPHS

In this note we present some properties of the Dirac operator on noncompact metric graphs with Kirchoff-type vertex conditions. In particular, we discuss the specific features of the spectrum of the operator and, finally, we give some further details on the associated quadratic form (and on the form domain).

Introduction

The investigation of evolution equations on metric graphs (see, e.g., Figure 2) has become very popular nowadays as they are assumed to represent effective models for the study of the dynamics of physical systems confined in branched spatial domains. A considerable attention has been devoted to the Schrödinger equation, as it is supposed to well approximate (for p = 4) the behavior of Bose-Einstein condensates in ramified traps (see, e.g., [START_REF] Gnutzmann | Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory[END_REF] and the references therein).

This, naturally, has lead to the study of the graph versions of the laplacian, given by suitable vertex conditions, and, especially to the standing waves of the associated NonLinear Schrödinger Equation (NLSE) (see, e.g., [START_REF] Adami | Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs[END_REF][START_REF] Adami | NLS ground states on graphs[END_REF][START_REF] Adami | Threshold phenomena and existence results for NLS ground states on metric graphs[END_REF][START_REF] Adami | Negative energy ground states for the L 2 -critical NLSE on metric graphs[END_REF][START_REF] Adami | Multiple positive bound states for the subcritical NLS equation on metric graphs[END_REF][START_REF] Cacciapuoti | Variational and stability properties of constant solutions to the NLS equation on compact metric graphs[END_REF][START_REF] Cacciapuoti | Ground state and orbital stability for the NLS equation on a general starlike graph with potentials[END_REF][START_REF] Dovetta | Existence of infinitely many stationary solutions of the L 2 -subcritical and critical NLSE on compact metric graphs[END_REF][START_REF] Dovetta | Mass-constrained ground states of the stationary NLSE on periodic metric graphs[END_REF][START_REF] Dovetta | Ground states of the L 2 -critical NLS equation with localized nonlinearity on a tadpole graph[END_REF][START_REF] Dovetta | L 2 -critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features[END_REF][START_REF] Serra | Bound states of the NLS equation on metric graphs with localized nonlinearities[END_REF][START_REF] Serra | On the lack of bound states for certain NLS equations on metric graphs[END_REF][START_REF] Tentarelli | NLS ground states on metric graphs with localized nonlinearities[END_REF]). In particular, the most investigated subcase has been that of the Kirchhoff vertex conditions, which impose at each vertex:

(i) continuity of the function: u e (v) = u f (v), ∀e, f v, ∀v ∈ K, (ii) "balance" of the derivatives: where K denotes the compact core of the graph (i.e., the subgraph of the bounded edges), e v means that the edge e is incident at the vertex v and due dxe (v) stands for u e (0) or -u e (e ) according to the parametrization of the edge (for more see Section 2). The above conditions correspond to the free case, namely, where there is no interaction at the vertices which are then mere junctions between edges.

Moreover, recently [START_REF] Sabirov | Dynamics of Dirac solitons in networks[END_REF] proposed (although if in the prototypical case of the infinite 3-star graph depicted in Figure 1) the study of the NonLinear Dirac Equation (NLDE) on networks, where the Dirac operator is given by

D := -ıc d dx ⊗ σ 1 + mc 2 ⊗ σ 3 . (1)
Here m > 0 and c > 0 are two parameters representing the mass of the generic particle of the system and the speed of light (respectively), and σ 1 and σ 3 are the so-called Pauli matrices, i.e.

σ 1 := 0 1 1 0 and

σ 3 := 1 0 0 -1 . (2)
Precisely, as for the majority of the works on the NLSE, [START_REF] Sabirov | Dynamics of Dirac solitons in networks[END_REF] suggests the study of the stationary solutions, that is those 2-spinors such that χ(t, x) = e -iωt ψ(x), with ω ∈ R and with ψ solving Dψ -|ψ| p-2 ψ = ωψ. Remark 1.1. We observe that, strictly speaking, the parameter c in (1) corresponds to the speed of light only in truly relativistic models, whereas in other contexts it should be rather considered as a phenomenological parameter depending on the model under study. Nevertheless, for the sake of simplicity, we will refer to it as "speed of light" throughout the paper.

The attention recently attracted by the linear and the nonlinear Dirac equations is due to their physical applications, as effective equations, both in solid state physics and in nonlinear optics (see [START_REF] Haddad | The nonlinear Dirac equation in Bose-Einstein condensates: foundation and symmetries[END_REF][START_REF] Haddad | The nonlinear Dirac equation in Bose-Einstein condensates: II. Relativistic soliton stability analysis[END_REF]). While initially the NLDE appeared as a field equation for relativistic interacting fermions (see [START_REF] Esteban | Variational methods in relativistic quantum mechanics[END_REF][START_REF] Lee | Exact localized solutions of two-dimensional field theories of massive fermions with Fermi interactions[END_REF]), thereafter it was used in particle physics (to simulate features of quark confinement), in acoustics and in the context of Bose-Einstein Condensates (see [START_REF] Haddad | The nonlinear Dirac equation in Bose-Einstein condensates: II. Relativistic soliton stability analysis[END_REF]). Recently, it also appeared that certain properties of some physical models, as thin carbon structures, are effectively described by the NLDE (see [START_REF] Arbunich | Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures[END_REF][START_REF] Borrelli | Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity[END_REF][START_REF] Borrelli | Multiple solutions for a self-consistent Dirac equation in two dimensions[END_REF][START_REF] Borrelli | William Weakly localized states for nonlinear Dirac equations[END_REF][START_REF] Cacciapuoti | The one-dimensional Dirac equation with concentrated nonlinearity[END_REF][START_REF] Fefferman | Honeycomb lattice potentials and Dirac points[END_REF][START_REF] Fefferman | Wave packets in honeycomb structures and two-dimensional Dirac equations[END_REF][START_REF] Pelinovsky | Survey on global existence in the nonlinear Dirac equations in one spatial dimension, Harmonic analysis and nonlinear partial differential equations[END_REF]).

On the other hand, in the context of metric graphs NLDE may describe the constrained dynamics of genuine relativist particles, or be regarded as an effective model for solid state/nonlinear optics systems (as already remarked). In particular, it applies in the analysis of effective models of condensed matter physics and field theory (see [START_REF] Sabirov | Dynamics of Dirac solitons in networks[END_REF]). Moreover, Dirac solitons in networks may be realized in optics, in atomic physics, etc. (see again [START_REF] Sabirov | Dynamics of Dirac solitons in networks[END_REF] and the references therein).

The study of the Dirac operator (and other kind of operators) on metric graphs has has been carried on in several works in the last years (see, e.g., [START_REF] Adamyan | Dirac-Krein systems on star graphs[END_REF][START_REF] Bolte | Spectral statistics for the Dirac operator on graphs[END_REF][START_REF] Bulla | The free Dirac operator on compact and noncompact graphs[END_REF][START_REF] Post | Equilateral quantum graphs and boundary triples[END_REF][START_REF] Kurasov | Quantum graphs: PT -symmetry and reflection symmetry of the spectrum[END_REF][START_REF] Matrasulov | PT-symmetric quantum graphs[END_REF]). On the contrary, concerning the existence of standing waves for the NLDE on metric graphs, to our knowledge, the first rigorous mathematical work on the subject is [START_REF] Borrelli | Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit[END_REF], where a nonlinearity concentrated on the compactcore of the graph is considered. We refer the reader to [START_REF] Borrelli | An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity[END_REF], for a survey on standing waves for NLDE and NLSE with concentrated nonlinearity.

Finally, it is worth stressing that (as for the NLSE), preliminarily to the study of the nonlinear case, it is necessary to find suitable vertex conditions for the operator D that make it self-adjoint. In this paper, we consider those conditions that converge to the Kirchhoff ones in the nonrelativistic limit, and that we call Kirchhoff-type, which represents (as well as Kirchhoff for Schrödinger) the free case. Roughly speaking, these conditions "split" the requirements of Kirchhoff conditions: the continuity condition is imposed only on the first component of the spinor, while the second component (in place of the derivative) has to satisfy the "balance" condition (see ( 4)&(5) below). We limit ourselves to the case of noncompact graphs with a finite number of edges since it is the most studied one in the nonlinear context.

The paper is organized as follows. In Section 2 we recall some fundamental notions on metric graphs and we give the definition of the Dirac operator with Kirchhoff-type vertex conditions. In Section 3, we give a justification of the self-adjointness of the operator and of its spectral properties. Then, a model case is presented in Section 4, in order to clarify the general arguments developed in Section 3. Finally, Section 5 accounts for the particular features of the quadratic form associated with D and its form domain.

Functional setting

Preliminarily, we recall some basic notions on metric graphs (more details can be found in [START_REF] Adami | NLS ground states on graphs[END_REF][START_REF] Berkolaiko | Introduction to quantum graphs[END_REF][START_REF] Kuchment | Quantum graphs. I. Some basic structures[END_REF] and the references therein).

Throughout, by metric graph G = (V, E) we mean a connected multigraph (i.e., with possibly multiple edges and self-loops) with a finite number of edges and vertices. Each edge is a finite or half-infinite segment of real line and the edges are glued together at their endpoints (the vertices of G) according to the topology of the graph (see, e.g., Figure 2).

Unbounded edges are identified with (copies of) R + = [0, +∞) and are called half-lines, while bounded edges are identified with closed and bounded intervals I e = [0, e ], e > 0. Each edge (bounded or unbounded) is endowed with a coordinate x e which posseses an arbitrary orientation when the interval is bounded and the natural orientation in case of a half-line. As a consequence, the graph G is a locally compact metric space, the metric given by the shortest distance along the edges. Clearly, since we assume a finite number of edges and vertices, G is compact if and only if it does not contain any half-line.

Consistently, a function u : G → C is actually a family of functions (u e ), where u e : I e → C is the restriction of u to the edge e. The usual L p spaces can be defined in the natural way, with norms

u p L p (G) := e∈E u e p L p (Ie) , if p ∈ [1, ∞), and
u L ∞ (G) := max e∈E u e L ∞ (Ie) ,
while H m (G) are the spaces of functions u = (u e ) such that u e ∈ H m (I e ) for every edge e ∈ E, with norm

u 2 H m (G) = m i=0 u (i) 2 L 2 (G) .
Accordingly, a spinor ψ = (ψ 1 , ψ Remark 2.1. Usually, graph Sobolev spaces are not defined as before. They also contain some further requirement on the behavior of the functions at the vertices of the graph (in the case m = 1, for instance, global continuity is often required). However, in a Dirac context it is convenient to keep integrability requirements and conditions at the vertices separated. Now, we can define the Kirchhoff-type realization of the Dirac operator on graphs.

Definition 2.2. Let G be a metric graph and let m, c > 0. We define the Dirac operator with Kirchhoff-type vertex conditions the operator D :

L 2 (G, C 2 ) → L 2 (G, C 2 ) with action D | Ie ψ = D e ψ e := -ıc σ 1 ψ e + mc 2 σ 3 ψ e , ∀e ∈ E, (3)
σ 1 , σ 3 being the matrices defined in (2), and domain dom(D) := ψ ∈ H 1 (G, C 2 ) : ψ satisfies (4) and ( 5)

,
where

ψ 1 e (v) = ψ 1 f (v), ∀e, f v, ∀v ∈ K, (4)
e v ψ 2 e (v) ± = 0, ∀v ∈ K, (5)
ψ 2 e (v) ± standing for ψ 2 e (0) or -ψ 2 e ( e ) according to whether x e is equal to 0 or e at v. An immediate, albeit informal, way to see why the previous one can be considered a Kirchhofftype realization of the Dirac operator is the following.

First, recall that the domain of the Kirchhoff Laplacian consists of the H 2 (G)-functions that also satisfy (i)&(ii) of Section 1. Therefore, since, roughly speaking, the Laplacian is (a component of) the square of the Dirac operator, one can square D and check if the resulting operator is in fact the Kirchhoff Laplacian. This is, indeed, the case, since D 2 clearly acts as (-∆) ⊗ I C 2 and since, considering spinors of the type ψ = (ψ 1 , 0) T , if one imposes that ψ ∈ dom(D 2 ) (namely, that ψ ∈ dom(D) and that Dψ ∈ dom(D)), then ψ 1 ∈ dom(-∆).

Self-adjointness and spectrum

In this section, we prove the self-adjointness of the operator D and present its main spectral features.

Preliminarily, we observe that the proof of the self-adjointness is not new (see, e.g. [START_REF] Albeverio | A remark on Krein's resolvent formula and boundary conditions[END_REF][START_REF] Bulla | The free Dirac operator on compact and noncompact graphs[END_REF][START_REF] Noja | Nonlinear Schrödinger equation on graphs: recent results and open problems[END_REF][START_REF] Posilicano | Self-adjoint extensions of restrictions[END_REF]). In particular, [START_REF] Bulla | The free Dirac operator on compact and noncompact graphs[END_REF] shows it for a wide class of vertex conditions (including the Kirchoff-type ones.

Here we give an alternative justification using the theory of boundary triplets.

The study the of D requires some introductory notions. Let A be a densely defined closed symmetric operator in a separable Hilbert space H with equal deficiency indices n ± (A) := dim N ±i ∞, where

N z := ker(A * -z)
is the defect subspace.

Definition 3.1. A triplet Π = {H, Γ 0 , Γ 1 }
A * f |g -f |A * g = Γ 1 f, Γ 0 g H -Γ 0 f, Γ 1 g H , f, g ∈ dom(A * ),
holds (with •, • H the scalar product in H) and the mapping

Γ := Γ 0 Γ 1 : dom(A * ) → H ⊕ H is surjective.
Definition 3.2. Let Π = {H, Γ 0 , Γ 1 } be a boundary triplet for the adjoint operator A * , define the operator

A 0 := A * | ker Γ 0
and denote by ρ(A 0 ) its resolvent set. Then, the γ-field and Weyl function associated with Π are, respectively, the operator valued functions γ(•) :

ρ(A 0 ) → L(H, H) and M (•) : ρ(A 0 ) → L(H) defined by γ(z) := (Γ 0 | Nz ) -1 and M (z) := Γ 1 • γ(z), z ∈ ρ(A 0 ). (6)
3.1. Self-adjointness. In order to apply the theory of boundary triplets to the definition of the Kirchhoff-type Dirac operator, one has to study the operator on the single components of the graph (segments and halflines) imposing suitable boundary conditions. Then, one describes the effect of connecting these one-dimensional components, according to the topology of the graph, through the vertex condtions (4)- [START_REF] Adami | Multiple positive bound states for the subcritical NLS equation on metric graphs[END_REF].

First, observe that the set E of the edges of a metric graph G can be decomposed in two subsets, namely, the set of the bounded edges E s and set of the half-lines E h . Fix, then, e ∈ E s and consider the corresponding minimal operator D e on H e = L 2 (I e , C 2 ), which has the same action of (3) but domain H 1 0 (I e , C 2 ). As a consequence, the adjoint operator possesses the same action and domain dom( D * e ) = H 1 (I e , C 2 ). On the other hand, a suitable choice of trace operators (introduced in [START_REF] Gernandt | Locally finite extensions and Gesztesy-Šeba realizations for the Dirac operator on a metric graph[END_REF]) is given by Γ e 0,1 :

H 1 (I e , C 2 ) → C 2 , with Γ e 0 ψ 1 e ψ 2 e = ψ 1 e (0) icψ 2 e ( e )
, Γ e One can check that {H, Γ 0 , Γ 1 }, with H = C M and M = 2|E s | + |E h |, is a boundary triplet for the operator D * (and hence one can compute gamma-field and Weyl function as before). On the other hand, note that boundary conditions (4)-( 5) are "local", in the sense that at each vertex they are expressed independently of the conditions on other vertices. As a consequence, they can be written using proper block diagonal matrices A, B ∈ C M ×M , with AB * = BA * , as

AΓ 0 ψ = BΓ 1 ψ
(see also the model case in Section 4). The sign convention of ( 5) is incorporated in the definition of the matrix B. Therefore the Dirac operator with Kirchoff-type conditions can be defined as 3.2. Essential spectrum. Now, we can focus on the essential spectrum of D. It can be studied adapting the strategy used for the Schrödinger case in [START_REF] Kostrykin | Kirchhoff's rule for quantum wires[END_REF].

D := D * , dom (
Preliminarily, note that the spectrum of D 0 is given by the superposition of the spectra of each addend, that is

σ(D 0 ) = e∈Es σ(D e ) ∪ e ∈E h σ(D e ).
Precisely, following [START_REF] Carlone | On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set[END_REF], each segment I e , e ∈ E s , contributes to the point spectrum of D 0 with eigenvalues given by

σ(D e ) = σ p (D e ) =    ± 2mc 2 π 2 2 e j + 1 2 2 + m 2 c 4 , j ∈ N    , ∀e ∈ E s ,
and each half-lines has a purely absolutely continuous spectrum

σ(D e ) = σ ac (D e ) = (-∞, -mc 2 ] ∪ [mc 2 , +∞), ∀e ∈ E h .
Now, one can check that a Krein-type formula for the resolvent operators holds, namely

(D -z) -1 -(D 0 -z) -1 = γ(z) (B M (z) -A) -1 Bγ * (z), ∀z ∈ ρ(D) ∩ ρ(D 0 ) (7)
(with γ(•) and M (•) the gamma-field and the Weyl function, respectively, associated with D -see [START_REF] Carlone | On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set[END_REF]). Hence, the resolvent of the operator D is as a perturbation of the resolvent of the operator D 0 . Since one can prove that the operator at the right-hand side of ( 7) is of finite rank, Weyl's Theorem [45, Thm XIII.14] gives Theorem 3.5. The essential spectrum of the operator D defined by Definition 2.2 is given by

σ ess (D) = σ ess (D 0 ) = (-∞, -mc 2 ] ∪ [mc 2 , +∞).
3.3. Absence of eigenvalues in the spectral gap. A natural question raised by Theorem 3.5 is: do there exist eigenvalues? If so, where do they are located? Unfortunately, there is no easy and general answer to these questions. In principle, eigenvalues can be located everywhere in the spectrum, suitably tuning the topology and the metric of the graph. Indeed, they can be both embedded in the essential spectrum, and at the thresholds. However, they cannot be in the spectral gap, as shown by the following computation.

Let λ ∈ σ(D) be an eigenvalue. As a consequence, there exists 0 = ψ ∈ dom(D) such that

Dψ = λψ,
or equivalently, such that

-ic dψ 2 dx = (λ -mc 2 )ψ 1 , (8)
-ic

dψ 1 dx = (λ + mc 2 )ψ 2 . (9)
If |λ| = m, then we can divide both sides of ( 9) by (λ + mc 2 ) and plug the value of ψ 2 into (8), obtaining

-c 2 d 2 ψ 1 dx 2 = (λ 2 -m 2 c 4 )ψ 1 . (10)
Furthermore, using ( 4)-( 5), we can prove that e v dψ 1 e dx (v) = 0, ψ 1 e i (v) = ψ 1 e j (v), ∀e i , e j v, so that ψ 1 is eigenfunction of the Kirchhoff Laplacian on G. Hence, multiplying (10) times ψ 1 and integrating, |λ| > mc 2 , namely Proposition 3.6. If λ ∈ R is an eigenvalue of the operator D (defined by Definition 2.2) then |λ| mc 2 .

3.4.

Graphs with eigenvalues at the tresholds. As already remarked, D may present eigenvalues at thresholds. This is the content of the following Proposition 3.7. Let G be a graph with two terminal edges incident at the same vertex v ∈ K. Then λ = ±mc 2 are eigenvalue of the operator D (defined by Definition 2.2). Remark 3.8. For simplicity we prove the result for the case depicted in Figure 3, which is the simplest one having the property stated above. The same proof applies to more general graphs satisfying such condition, simply considering spinors which vanish identically on all the edges except on the two terminal edges. Proof of Proposition 3.7. First, identity the bounded edges of the graph in Figure 3 with the compact intervals I j = [0, j ], j = 1, 2, and the common vertex with 0.

Let λ = mc 2 . Then, equations ( 8) and ( 9) read

dψ 2 dx = 0 , dψ 1 dx = 2imcψ 2 .
Now, let ψ f 1 ≡ 0. Integrating the above equations on f 1 , f 2 yields

ψ 1 f j (x) = 2imcA j x + B j , for x ∈ [0, j ], with j = 1, 2 and ψ 2 f j (x) ≡ A j , for x ∈ [0, j ], with j = 1, 2
where A j , B j ∈ C. Therefore, as ( 4) and ( 5) have to be satisfied at v 0, we find

B 2 = B 3 = 0, and A 3 = -A 2
and thus λ = mc 2 is an eigenvalue of D.

Let us, now, turn to λ = -mc 2 . In this case the eigenvalue equation becomes

dψ 2 dx = -2imcψ 1 , dψ 1 dx = 0 .
Set again ψ f 1 ≡ 0, we have

ψ 1 f j (x) ≡ E j , for x ∈ [0, j ], with j = 1, 2 and ψ 2 f j (x) = -2imcE j x + F j , for x ∈ [0, j ], with j = 1, 2
where E j , F j ∈ C, and again by ( 4) and ( 5)

E 2 = E 3 = 0, and
F 3 = -F 2 .
Then, also λ = -mc 2 is an eigenvalue of D and the proof is completed.

Remark 3.9. We guess that suitably choosing the topology and the metric of the graph, one could also construct eigenvalues embedded in the essential spectrum. However, a complete discussion of this point goes beyond the scopes of the present paper. We will deal with it in a forthcoming article.

A model case: the triple junction

The aim of the present section is to clarify the main ideas explained before by means of an example.

Consider a 3-star graph with one bounded edge and two half-lines, as depicted in Figure 4. In this case the finite edge is identified with the interval I = [0, e 3 ] and 0 corresponds to the common vertex of the segment and the half-lines. Here trace operators can be defined as

Γ 0 ψ =       ψ 1 e 1 (0) ψ 1 e 2 (0) ψ 2 e 3 ( e 3 ) icψ 1 e 3 ( e 3 )       , Γ 1 ψ =       icψ 2 e 1 (0) icψ 2 e 2 (0) icψ 2 e 3 (0) ψ 1 e 3 ( e 3 )      
, so that, again, Kirchoff-type conditions ( 4)-( 5) can be written as

AΓ 0 ψ = BΓ 1 ψ,
with AB * = BA * given by

A = 2 3       -2 1 1 0 1 -2 1 0 1 1 -2 0 0 0 0 a       , B = -ı 2 3       1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 b      
, where, properly choosing a, b ∈ C, one can fix the value of the spinor on the non-connected vertex. Remark 4.1. Looking at the example above, and recalling that conditions (4)-( 5) are defined independently on each vertex, one can easily see how to iterate the above construction for a more general graph structure, thus obtaining matrices A, B with a block structure, each block corresponding to a vertex.

In order to investigate the spectral properties of D on the graph depicted by Figure 4, we have to use Weyl's Theorem and computate the singularities of the resolvent (see [START_REF] Adamyan | Dirac-Krein systems on star graphs[END_REF]). Hence, it is necessary to compute (the gamma-field γ(z) and) the Weyl function M (z).

The structure of γ(z) and M (z) can be recovered using a block composition as in [START_REF] Carlone | On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set[END_REF][START_REF] Gernandt | Locally finite extensions and Gesztesy-Šeba realizations for the Dirac operator on a metric graph[END_REF]. In particular, the eigenvalues of the operator D are given by the zeroes of the determinant of (BM (z) -A), so that the computation of (BM (z) -A) -1 B, is needed. Remark 4.2. In order to simplify some notations in the following, we choose m = 1/2 in the definition of the operator (1), so that the thresholds of the spectrum become λ = ±c/2.

Let us define

k(z) := 1 c z 2 -(c 2 /2) 2 , z ∈ C , and
k 1 (z) := ck(z) z + c 2 /2 = z -c 2 /2 z + c 2 /2 , z ∈ C .
where the branch of the multifunction

√ • is selected such that k(x) > 0 for x > c 2 /2. It this way k(•) is holomorphic in C with two cuts along the half-lines (-∞, -c 2 /2] and [c 2 /2, ∞).
Then, the Weyl function reads

M (z) =      ick 1 (z) 0 0 0 0 ick 1 (z) 0 0 0 0 ck 1 (z) sin( e 3 k(z)) cos( e 3 k(z)) 1 cos( e 3 k(z)) 0 0 1 cos( e 3 k(z)) sin( e 3 k(z)) ck 1 (z)cos( e 3 k(z))      ,
Assume also, for the sake of simplicity, that a = 0, b = 1 and fix e 3 = c = 1. After some calculations, one sees that the zeroes of the determinant of BM (z) -A are given by the zeroes of the following function: Let us consider, initially, z ∈ (-1/2, 1/2). For such values √ 4z 2 -1 ∈ iR, and then [START_REF] Berkolaiko | Introduction to quantum graphs[END_REF] rewrites

f (z) = - 8 9 i sin 1 2 4z 2 -1 + sin 3 2 4z 2 -1 + 2i sec 4 1 2 4z 2 -1 . (11)
f (z) = 8 9 sinh 1 2 1 -4z 2 + sinh 3 2 1 -4z 2 + 2 sech 4 1 2 1 -4z 2 .
which is a positive real function. For z / ∈ (-1/2, 1/2), on the contrary, √ 4z 2 -1 ∈ R and so

(f (z)) = 16 9 sec 4 1 2 4z 2 -1 and (f (z)) = - 8 9 sin 2 4z 2 -1 cos 4z 2 -1 ,
so that the real part of (11) cannot vanishes, while the imaginary part is periodic and with alternate sign. We can thus conclude that there are is eigenvalue in the spectrum. This should be compared with the example presented in Section 3.4, where threshold eigenvalues do exist for a small change of the graph structure. Morever, we remark that the above analysis does not exclude the presence of resonances. They may appear, when one imposes Kirchoff-type conditions, due to the eigenvalues of the operator given by the direct sum of the Dirac operators on the edge (without any boundary condition) that are embedded in the continuous part of the spectrum.

However, although the explicit analysis of the spectrum for a Dirac graph can be complicated when the topological structure is complex, the spectral characterization and the analysis of resonances appear to be a challenging and interesting problems.

The form-domain

We conclude the paper with some remarks concerning the quadratic form associated with D. The reason for which this point deserves some particular attention can be easily explained.

In the standard cases (R d , with d = 1, 2, 3) the quadratic form associated with the Dirac operator can be easily defined using the Fourier transform (see, e.g., [START_REF] Esteban | Stationary states of the nonlinear Dirac equation: a variational approach[END_REF]). Unfortunately, in the framework of (noncompact) metric graphs this tool is not available. In addition, also classical duality arguments seems to be prevented as, generally speaking, H -1/2 (G) is not the topological dual of H 1/2 (G), due to the presence of the compact core of the graph.

Therefore, one has to resort to the spectral theorem, where the associated quadratic form Q D and its domain dom(Q D ) are defined as

dom(Q D ) := ψ ∈ L 2 (G, C 2 ) : σ(D) |ν| dµ D ψ (ν) , Q D (ψ) := σ(D) ν dµ D ψ (ν),
with µ D ψ the spectral measure associated with D and ψ. However, such a definition is very implicit and thus not useful in concrete cases (as, for instance, in [START_REF] Borrelli | Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit[END_REF]).

On the other hand, an useful characterization of the form domain can be obtained arguing as follows, using real interpolation theory (see, e.g., [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Ameur | Interpolation between Hilbert spaces[END_REF]). Define the space

Y := L 2 (G, C 2 ), dom(D) 1 2 , (12)
namely, the interpolated space of order 1/2 between L 2 and the domain of the Dirac operator. First, note that Y is a closed subspace of

H 1/2 (G, C 2 ) := e∈E H 1/2 (I e , C 2 ),
with respect to the norm induced by H 1/2 (G, C 2 ). This can be easily checked observing that dom(D) is clearly a closed subspace of H 1 (G, C 2 ) and that, arguing edge by edge, one can see that

H 1/2 (G, C 2 ) = L 2 (G, C 2 ), H 1 (G, C 2 ) 1 2 .
Hence the closedness of Y follows by the sole definition of interpolation spaces. Therefore, by Sobolev embeddings,

Y → L p (G, C 2 ), ∀p 2,
and that, in addition, the embegging in L p (K, C 2 ) is compact, due to the compactness of K.

Let us prove that indeed dom(

Q D ) = Y. (13)
This characterization turns out to be particularly useful, for instance, in the nonlinear case where one studies the existence of standing waves by variational methods [START_REF] Borrelli | Nonlinear Dirac equation on graphs with localized nonlinearities: bound states and nonrelativistic limit[END_REF]. In order to prove (13), we exploit again the spectral theorem, but in a different form (see Theorem 5.1 below). In particular, it states that, roughly speaking, every self-adjoint operator on a Hilbert space is isometric to a multiplication operator on a suitable L 2 -space. In this sense self-adjoint operators can be "diagonalized" in an abstract way.

(•)(U ψ)(•) ∈ L 2 (M, dµ), (
) if ϕ ∈ U (dom(H)), then U HU -1 ϕ (m) = f (m)ϕ(m), ∀m ∈ M . 2
The above theorem, in other words, states that H is isometric to the multiplication operator by f (still denoted by the same symbol) on the space L 2 (M, dµ), whose domain is given by dom

(f ) := ϕ ∈ L 2 (M, dµ) : f (•)ϕ(•) ∈ L 2 (M, dµ) , endowed with the norm ϕ 2 1 := M (1 + f (m) 2 )|ϕ(m)| 2 dµ(m)
The form domain of f has an obvious explicit definition, as f is a multiplication operator, that is

ϕ ∈ L 2 (M, dµ) : |f (•)| ϕ(•) ∈ L 2 (M, dµ)
and we will prove in the sequel that it satisfies (13) (we follow the presentation given in [START_REF] Adams | Sobolev spaces[END_REF][START_REF] Ameur | Interpolation between Hilbert spaces[END_REF]). Consider the Hilbert spaces H 0 := L 2 (M, dµ) with the norm x 0 := x L 2 (dµ) , and H 1 := dom(f ), so that H 1 ⊂ H 0 . The squared norm x 2 1 is a densely defined quadratic form on H 0 , represented by

x 2 1 = (1 + f 2 (•))x, x 0 , where •, • 0 is the scalar product of H 0 . Define, in addition, the following quadratic version of Peetre's K-functional

K(t, x) := inf x 0 2 0 + t x 1 2 1 : x = x 0 + x 1 , x 0 ∈ H 0 , x 1 ∈ H 1 .
By standard arguments (see e.g. [START_REF] Ameur | Interpolation between Hilbert spaces[END_REF] or [START_REF] Adams | Sobolev spaces[END_REF]Ch. 7] and references therein) the intermediate spaces

H θ := [H 0 , H 1 ] θ ⊂ H 0 , 0 < θ < 1,
are given by the elements x ∈ H 0 such that the following quantity is finite:

x 2 θ = ∞ 0 t -θ K(t, x) dt t < ∞.
Now we can prove the following equivalence.

Proposition 5.2. For every θ ∈ (0, 1), there results

x 2 θ = (1 + f 2 (•)) θ x, x 0 , for x ∈ H θ . (14)
Proof. Preliminarily, for ease of notation, set A := (1 + f 2 (•)). The operator A is positive and densely defined on H 0 and also its (positive) square root A 1/2 has a domain A dense in H 0 . Now, let us divide the proof in two step.

Step 1. There results

K(t, x) = tA 1 + tA x, x 0 , t > 0, x ∈ H 0 . (15)
First, observe that the bounded operator in [START_REF] Borrelli | William Weakly localized states for nonlinear Dirac equations[END_REF] is defined via the functional calculus for A. Then, take x ∈ A and fix t > 0. By a standard convexity argument one gets the existence of a unique decomposition x = x 0,t + x 1,t , such that K(t, x) = x 0,t 2

0 + t x 1,t 2 1
(16) (note also that x j,t ∈ A, j = 0, 1). Then, for all y ∈ A, using the minimality requirement in the definition of K, there results d ds x 0,t + sy 2 0 + t x 1,t -sy 2 1 |s=0 = 0, and then, recalling that x 2 1 = A 1/2 x 2 0 , we have A -1/2 x 0,t -tA 1/2 x 1,t , A 1/2 y 0 = 0.

Since the above inequality must be true for all y in the dense subset A ∈ H 0 we conclude that A -1/2 x 0,t = tA 1/2 x 1,t , so that we obtain

x 0,t = tA 1 + tA x, x 1,t = 1 1 + tA x (17)
Combining ( 16) and ( 17) we get the claim.

Remark 5.3. Observe that the bounded operators tA 1 + tA , 1 1 + tA , t 0 , are defined using the functional calculus for the self-adjoint operator A.

Step 2. Proof of [START_REF] Borrelli | Multiple solutions for a self-consistent Dirac equation in two dimensions[END_REF]. By Step 1 and exploiting the functional calculus for A, we get

x 2 θ = ∞ 0 t -θ K(t, x) dt t = ∞ 0 t -θ A 1 + tA x, x 0 dt = A ∞ 0 dt t θ (1 + tA) x, x 0 . (18)
Consider, then, the differentiable function

f (a) := ∞ 0 dt t θ (1 + ta) , a > 0.
Integrating by parts, one easily gets Note that we have set the integration constant equal to zero in order to get the correct formula as θ → 1 -. Combining the above observations, one sees that (18) reads , namely, is equal to the interpolated space of order 1/2 between L 2 and the operator domain.

Figure 1 .

 1 Figure 1. The infinite 3-star graph

Figure 2 .

 2 Figure 2. a general noncompact metric graph.

1 e

 1 ( e ) and hence, given the boundary triplet {H e , Γ e 0 , Γ e 1 }, with H e = C 2 , one can compute the gamma field and the Weyl function using (6) and prove that D * e has defect indices n ± ( D e ) = 2. Moreover, note that in this way we can define an operator D e with the same action of D * e (and D e ) and domain dom(D e ) = ker Γ e 0 , which is self-adjoint by construction. Analogously, fix now e ∈ E h and consider the minimal operator D e defined on H e = L 2 (R + , C 2 ), with the same action as before and domain H 1 0 (R + , C 2 ). The domain of the adjoint reads dom( D * e ) = H 1 (R + , C 2 ) and the trace operators Γ e 0,1 : H 1 (R + , C 2 ) → C can be properly given by Γ gamma field and the Weyl function are provided by (6) (with respect to the boundary triplet {H e , Γ e 0 , Γ e 1 }, with H e = C), while the defect indices are n ± ( D e ) = 1. In addition, one can once more define a self-adjoint (by construction) operator as D e := D * e , dom(D e ) := ker Γ e 0 . Then, we can describe the Dirac operator introduced in Definition 2.2 using Boundary Triplets. Let D 0 be the operator D 0 := e∈Es D e ⊕ e ∈E h D e ,defined on H = e∈Es H e ⊕ e ∈E h H e , with domain given by the direct sum of the domains of the addends. Consider, also, the operator natural definitions of the domains). Introduce, also, the trace operators Γ 0

  D) := ker(AΓ 0 -BΓ 1 ), and then, by construction, Theorem 3.3. The Dirac operator with Kirchhoff-type boundary conditions D, defined by Definition 2.2 is self-adjoint.Remark 3.4. The boundary triplets method provides an alternative way to prove the self-adjointness of D. More classical approaches à la Von Neumann can be found in[START_REF] Bulla | The free Dirac operator on compact and noncompact graphs[END_REF].

f 1 f 2 f 3 Figure 3 .

 233 Figure 3. A 3-star graph with two segments.

e 1 e 2 e 3 Figure 4 .

 234 Figure 4. A 3-star graph with a finite edge.

Figure 5 .

 5 Figure 5. The real and imaginary part of f (z) for z ∈ [-1/2, 1/2], in blue and yellow, respectively.

f

  (a) = a θ-1 .

x 2 θ 2 .

 22 = A θ x, x 0 , thus proving the claim. Finally, in view of the previous Proposition, if one sets for θ = 1 2 , then one recovers the form domain of the operator f and, hence, setting H = D and H = L 2 (G, C 2 ), one has that (12) is exactly the form domain of D, with Y = U -1 H 1 Consequently, (13) is satisfied and, summing up, we have shown the following Theorem 5.4. The form domain of D (defined by Definition 2.2) satisfies dom(Q D ) = L 2 (G, C 2 ), dom(D) 1 2

  is said a boundary triplet for the adjoint operator A * if and only if H is a Hilbert space and Γ 0 , Γ 1 : dom(A

* ) → H are linear mappings such that

  Theorem 5.1. ([44, thm. VIII.4]) Let H be a self-adjoint operator on a separable Hilbert space H with domain dom(H). There exists a measure space (M, µ), with µ a finite measure, a unitary operator U : H -→ L 2 (M, dµ) ,

	and a real valued function f on M , a.e. finite, such that
	(1) ψ ∈ dom(H) if and only if f


president Michel Cosnard

Alain Filloux

The creation of the unit was initiated by Ms Arlette DARFEUILLE-MICHAUD in 2008 in the form of a Young INRA team with the JE2526 label (USC-2018(USC- , since 2005)). Following the AERES evaluation in February 2011, the unit was transformed in 2012 into the UMR U1071 under the authority of INSERM and Université d'Auvergne. The unit name was then coined as M2iSH for "Microbe, Intestine, Inflammation and Host Susceptibility". The death of Ms DARFEUILLE-MICHAUD in 2014 resulted in the installation of the current director, Mr Nicolas BARNICH. In its current form, the unit mostly includes teaching staff from the Université d'Auvergne and physicians from the CHU Clermont-Ferrand, with only one full time researcher appointed by INSERM. The unit is made of one single team and is based on the campus of the Faculté de Médecine et de Pharmacie (Place Henri-Dunant, Clermont-Ferrand) where it occupies about 600 m 2 on a single floor.

Management team

The management team includes the director (Mr Nicolas BARNICH) and a deputy director (Mr Richard BONNET). The director can consult the laboratory council, which includes representatives of all personnel categories and meets every month.

HCERES nomenclature

SVE1_LS6, SVE1_LS4,

SVE1_LS7

Scientific domains

The unit has historically founded its activity and visibility by studying the impact of enteric bacteria, first on chronic inflammatory bowel diseases (Crohn's disease) and now on colorectal cancer. The expertise of the unit encompasses a variety of techniques from basic molecular microbiology, cellular biology and immunology up to in vivo models and physiopathology of the disease, with strong expertise in gastroenterology and oncology and involving the clinicians. , is interested in the evolution and the pathoadaptive development of bacteria within a host. In particular it studies pathogens involved in the etiology of acute and chronic inflammatory bowel diseases (IBD) and, more recently, of colorectal cancer. This unit gained an international recognition when it conducted a seminal work and first discovered the abnormal colonization of the ileal mucosa of Crohn's Disease (CD) patients with a particular pathotype of E. coli, called adherent invasive E. coli or AIEC. This work was published in 1998, and since then a continuous and impressive series of publications has unravelled many aspects of this very peculiar host-pathogen interaction.

Unit workforce

Unit workforce

During the past previous years, the unit capitalized on this expertise and developed three majors themes: i) the molecular characterization of intestinal disease-associated Escherichia coli; this was more specifically in AIEC in CD, but similar approaches have been extended to E. coli strains associated with the progression of tumours in colorectal cancer (CRC), with emphasis on colibactin-producing E. coli strains; ii) the host-bacteria cross-talk at the intestinal mucosa site; in this theme, the unit deciphers the mechanisms of bacterial translocation through the intestinal epithelium, the characterization of the bacterial and host factors associated with the process, and the overall impact of the colonization on the physiopathology; and iii) the development of innovative diagnostic tools and therapies, which makes use of very specific discoveries obtained in the unit such as the specific recognition of the CEACAM6 molecule by the bacterial adhesin FimH.

The future scientific strategy of the unit mostly pursues research work along these lines, which are promising and by far to be fully understood and exploited. This should lead to the development of improved diagnostic tools and therapeutic strategies. In brief, the research projects for the next five years will be focused on the interaction between pathogenic bacteria or pathobiont bacteria and the host, in the context of intestinal inflammation (Research Axis 1) or carcinogenesis (Research Axis 2) and the development of innovative diagnostic and/or therapeutic tools (Research Axis 3).

Global assessment of the unit

The evaluation committee recognized that this excellent to outstanding unit has acquired an international recognition in the study of the host-pathogen relationship associated with Crohn's disease. This covers a wide range of approaches that have unravelled detailed interactions between the E. coli strain AIEC and host receptors or other host susceptibility factors. What was initiated by the previous director has been not only continued with similar success but also with a continuous dynamic as suggested by the new project of the unit, although it is extremely ambitious. Several research lines have been built on previous preliminary data, and are ready to take off such as the importance of pathotypes in the progression of tumors in colorectal cancer, the role of exosomes in cell-cell communication and large omics approaches to identify new virulence determinants or host susceptibility factors. The committee considered that the integration between academics and clinicians is a tremendous success, which guarantees the project feasibility and the visibility of the unit. The niche which has been developed is quite unique and positioned the region of Clermont-Ferrand and the Université d'Auvergne at the forefront of international recognition in the domain of IBD and infection.

Strengths and opportunities in the context

By identifying a new pathotype of E. coli (AIEC) associated with Crohn's disease, the Unit has pioneered a field and is internationally recognized for it.

The unit comprises scientists and clinicians with multidisciplinary expertise, (microbiology, cell and molecular biology, immunology, gastroenterology, digestive surgery, internal medicine). The committee noticed that all the members interact extremely tightly and efficiently, demonstrating real synergy and willingness to work together.

The remarkable interface with local hospitals provides access to biological samples and allows the effective development of translational research projects. This is demonstrated by the ownership of 4 patents one of which was sold to a company with which the unit has a longstanding collaboration.

The unit is small and built around one single team but has an excellent publishing record.

The support/connection from/with the Université d'Auvergne is very strong and reflects the commitment of the unit members to teaching. This results in the ability of the unit to recruit excellent students Despite a geographic location that might not be so well known outside French borders, the unit is very attractive from an international point of view. Several foreign post-docs have joined the unit (although none have left at the moment) and the unit was able to recruit a formal post-doc back from the States, which is currently the only full time researcher.

Weaknesses and threats in the context

There was a need to maintain the leadership of the unit in the field of AIEC bacteria and Crohn's disease after the death of Arlette DARFEUILLE-MICHAUD (2014) who was internationally recognized. This was a threat but the committee was clearly reassured that the new director had the scientific legitimacy and was unanimously supported by all unit members, and hopefully this will continue.

The unit has only one technical and administrative staff and only one full-time researcher. This is critical since at the same time the unit lost the only INRA researcher they had in the past and who left for personal convenience. This very limited support, combined with the important teaching activities of the team members, clearly restricts the intrinsic potential of the unit in research.

The team has decided to take high scientific risks on several new topics: impact of physical activity on microbiota composition in chronic diseases, role of exosomes in cell-cell communication, study of bacterial glycosylation on virulence modulation, development of animal models for chronic infection, translational and clinical studies… The committee feels that, even though all these directions are potentially original and promising, the project may look way too ambitious and the visibility of the unit might be lost or diluted if priorities are not clearly identified.

The financial support coming form companies is very high (25%) and the director indicated a possible risk of losing some of these current important financial partnerships that were tightly dependent on relationships established by the previous director. Yet the evaluation committee felt that there is no obvious reason for this to happen. Instead, attention should be given that major ANR or European project applications be posted in a timely manner in order to secure funding for the unit on the long term.

The director mentioned the need of a researcher specialized in bioinformatics (for omics approaches and the microbiota studies) as well as in glycobiology. The evaluation committee felt that these expertises should better be implemented via collaboration or subcontracting rather than effective recruitment of staff in this area.

The arrival of two new academic researchers, more HDRs and consequently, more PhD students will require additional space, an issue that has been raised by most personnel categories. The evaluation committee understood that a new research building will be opened in the faculty of Medicine (hosting the GRED), which may free some space in the building currently occupied by the unit.

Recommendations

The evaluation committee feels that the transition after the death of the previous director has been successfully achieved. The new director is a very competent scientist with strong track record in publication and funding (including an ANR JC/JC). The committee was pleased to observe that the new director received support from all the personnel within the unit. Obviously this is an ideal situation which should not be considered as definitely granted, and the director is encouraged to maintain such cohesion through the managerial style he has demonstrated up to now. The new director and all members should also be encouraged to maintain a high level of participation to international conferences and, whenever possible, be involved in major editorial or advisory boards.

In light of this observation, the committee cannot really recommend what the best organisational option is for the unit. The mono-team strategy seems to work and the implementation of several teams might affect the synergy and cohesion between the team members. It will be at the appreciation of the director to decide on the evolution of the unit in the future, decision that might be considered for the next evaluation.

The team is not that small since it involves about 40 members although only one is a full time researcher. Nevertheless the committee feels that the new project is extremely ambitious and has branched on several research lines. The committee will recommend that clear priorities be identified to unambiguously establish the short and mid term objectives versus the longer terms goals. The diversification of the projects should also be clearly accompanied by the identification of the personnel involved in or leading each individual topic, so that the feasibility could be documented and the internal organisation of the unit identified. Some of the new topics are highly competitive such as microbiota (microbiota/cancer, microbiota/IBD) and a number of teams with state of the art bioinformatic facilities are specialized in microbiota studies. The committee feels that there is a risk here and encourages the director to define the niche of the unit more precisely. It is likely that if it remains E. coli centred, with studies on the molecular mechanisms, it will still be original and fully reliying on the expertise of the team. It is also important that the project on the role of the clb+ E. coli in tumours progression be clearly coordinated with other groups in France working on this very same topic.

In conclusion the committee feels that all the risks have been well mitigated, that the unit has a strong potential and the ability to train young generation of academics and clinicians at the interface of microbiology and gastroenterology.

  (Representative of the INSERM-CSS) Mr Nicolas LAPAQUE, INRA, MICALIS Scientific delegate representing the HCERES: Mr Kamel BENLAGHA Representatives of supervising institutions and bodies: Mr Philippe DULBBECO, Université d'Auvergne Mr Alain ESCHALIER, Université d'Auvergne Ms Emmanuelle MAGUIN, MICA, INRA Ms Stéphanie POMMIER, INSERM Mr André SALAGNAC, CHU Clermont-Ferrand Head of Doctoral School: Mr Jean-Marc LOBACCARO, ED n°65 « Santé Agronomie et Environnement » 1  Introduction History and geographical location of the unit

Number on 30/06/2015 Number on 01/01/2017 N1: Permanent

	professors and similar positions	13	15
	N2: Permanent researchers from Institutions and similar positions	1	1
	N3: Other permanent staff (technicians and administrative personnel)	6	8
	N4: Other professors (Emeritus Professor, on-contract Professor, etc.)
	N5: Other researchers from Institutions (Emeritus Research Director, Postdoctoral students, visitors, etc.)	2
	N6: Other contractual staff (technicians and administrative personnel)	5
	N7: PhD students	11

TOTAL N1 to N7 38 Qualified research supervisors (HDR) or similar positions 11 Unit record From 01/01/2010 to 30/06/2015

	Introduction
	The unit UMR U1071 INSERM/Université d'Auvergne, or M2iSH

signed by their chairman." (Article 8, paragraph 5) 2 The evaluation reports "are signed by the chairman of the expert committee". (Article 11, paragraph 2) PhD theses defended 13 Postdoctoral scientists having spent at least 12 months in the unit 11 Number of Research Supervisor Qualifications (HDR) obtained during the period


president Michel Cosnard

This report is the sole result of evaluation by the expert committee, the composition of which is specified below.

The assessments contained herein are the expression of an independent and collegial reviewing by the committee.

Unit name:

Nutrition, Inflammation and Gut-Brain axis dysfunction The current unit UMR 1073 is applying for renewal of both its university and INSERM labels. The research unit members propose for the next term is in close continuation with their on-going research theme. The unit is organized as an integrated single team unit, proposing a multidisciplinary approach on the physiology and pathophysiology of eating behaviour, gut barrier and visceral sensitivity, in the context of gut-brain axis dysfunction. The general project is structured by themes, and specific projects are run by principal investigators (PIs), often associated in tandems or trios to provide the best interdisciplinary or fundamental/clinical interactions.

Management team

The INSERM unit 1073 is headed by Mr Pierre DÉCHELOTTE, and Mr Moïse COËFFIER is associate director. An executive committee meets at least once a month and helps with all the decisions. It is composed of the director, the deputy director and representatives of scientists, technical staff and students.

HCERES nomenclature

Principal: SVE1_LS4 Physiologie, Physiopathologie, Endocrinologie Secondary: SVE1_LS7 Recherche clinique, Santé publique

Scientific domains

The unit uses multidisciplinary approaches to explore the physiology and pathophysiology of eating behaviour, gut barrier and visceral sensitivity, in the context of dysfunctions of the gut-brain axis. The unit has anchored its activities in translational approaches, aiming at evaluating gut metabolism and gut-brain axis dysfunctions associated with eating disorders, irritable bowel syndrome and inflammatory bowel diseases. They focus their approaches on trying to propose new therapeutic avenues to treat these diseases. 2  Overall assessment of the unit

Unit workforce

Introduction

The INSERM unit 1073 has been created in 2012 and is headed by Mr Pierre DÉCHELOTTE. The scientific scope of the unit for the next 5-years contract will be closely related to the research that has been conducted during the last term. Almost the same staff of principal investigators and directors will compose the research unit propose for the next term, with the reinforcement of two additional professors (M. David MALTÊTE and M. Jean-Nicolas CORNU, the latter planned to be nominated as professor in 2016) Members of the unit focus their experimental and clinical research projects on subjects that represent major concerns for public health, namely three groups of diseases: "Eating Disorders" (ED) (including anorexia nervosa, bulimia nervosa, hyperphagia and Obesity), "Irritable Bowel Syndrome" (IBS) and "Inflammatory Bowel Diseases" (IBD). All these pathologies are interconnected and overlap on different aspects. It is therefore logical to address them all, considering the expertise represented in this research unit.

  location of the unitThe INSERM unit 1073, headed by Mr Pierre DÉCHELOTTE, was created on the 01/01/2012. INSERM U 1073 is located on the Clinical Campus (Medicine/Pharmacy School and Rouen University Hospital) and is involved in the "Institute for Research and Innovation in Biomedicine" (IRIB). U 1073 focuses its research on nutritional and gastrointestinal physiology and pathophysiology.

Global assessment of the unit

The unit is composed of researchers with a long-standing history of very good scientific input, with excellent in house expertise covering most aspects of nutritional and gastrointestinal pathophysiology. An important strength of the unit is the expertise in epidemiological, clinical, biological and technical (neurostimulation devices) fields.

Research is made in an integrated and translational manner. It covers both epidemiological and clinical research, as well as physiology and cellular biology. Bench to bedside and bedside to bench approaches constitute the main core of the research activities of the unit. Over the years, the research performed within the unit has led to innovative therapeutic approaches, and to the validation of experimental models.

A combination of original research topics and a multidisciplinary approach uniquely positions the unit to perform very high quality research.

Strengths and opportunities in the context

The laboratory performs authentic translational research and benefits from strong university and hospital support.

The unit is very well integrated into a SFR "Structure Fédérative de Recherche", which offers a number of state-of-the-art core facilities to the scientific community in Rouen.

The unit has demonstrated very powerful technological transfer abilities, with 2 start-up companies. A licence is presently pending, and 5 patents are currently negotiated with "TargEDys", a start-up company. Another start-up company, "Stimulics", which has developed a very promising nerve stimulation device, is presently in the process of being created within the "Normandie incubator" (local SEINARI).

The unit is recognized at the national and international levels on the themes of gut metabolism, visceral sensitivity and regulation of food intake.

The INSERM label provided in 2012 had a very strong positive impact on the research activities of the unit, both with regard to the production, with highly visible and high impact factor publications, and from an operational point of view.

Weaknesses and threats in the context

The unit would benefit from having a full time researcher.

The expert committee acknowledged that the unit has nicely refocused its research interests as compared to the previous term; however, further prioritizing its scientific priorities would be beneficial.

Recommendations

Staff members of the unit are encouraged to apply (at the university or at INSERM) for a complete discharge of teaching and/or of clinical activities, in order for at least some of them to be present within the research unit on a full-time basis.

The recruitment of post-docs should be further developed.


Experts:

Ms (3) Endocrinology, Signalling and Cancer (two teams). The GReD researchers have expertise in plants, fungi, invertebrates such as Drosophila, and mammals including mouse and human. The unit is located at two separate sites ~3 km apart: the Faculty of Medicine of Clermont-Ferrand and the Cézeaux University campus. However, an important change will occur in 2017 when all research teams of the GReD will move to a new building, the Center for Clinical and Biological Research (CCBR), located on the medical campus.

Management team

The GReD director is Ms Chantal VAURY and the deputy-director Mr Jöel DREVET.

HCERES nomenclature

Principal: SVE1_LS3 Secondary: SVE1_LS4, SVE1_LS2 and SVE2_LS3

Scientific domains

Dynamics of genome -plasticity -normal and pathological development -reproduction -molecular endocrinology. 2  Overall assessment of the unit

Unit workforce

Unit workforce

Introduction

The GReD has focused research efforts on understanding basic biological mechanisms occurring at the genome and cell levels using diverse model organisms. The thematic areas represented by the three research axes (Genome Dynamics and Epigenetic Control, Reproduction and Development in Health and Disease, Endocrinology, Signaling and Cancer) will continue during the next five years . The GReD is placing a stronger emphasis on translational research by adding the team of Mr Vincent SAPIN, which is focused on epithelial injury and repair. Also, the addition of the new team of Mr David VOLLE focused on infertility, which spun out of the Mr Jean-Marc LOBACCARO team, is an effort to strengthen the translational aspect of the unit. Six of the GReD teams will be codirected by a group leader and a younger associate leader. The rationale is that co-led teams serve as a mechanism to mentor younger researchers who eventually could become team leaders.

Global assessment of the unit

The GReD research projects cover diverse scientific questions aimed at understanding genome integrity, regulation of gene expression, cell and tissue differentiation and organismal development. Research productivity has increased during the last four years as evidenced by a 30% increase in research expenditures due to success in obtaining both national and international grants. The interactions and technical cross training between the teams, as well as the development of cutting-edge platforms, provide an exciting environment for GReD researchers.

Examples of significant research findings during the last four years include (i) intronic DNA methylation in transcriptional control of the IBM1 gene, (ii) description of histone modifications at imprinting control regions, (iii) the flam piRNA cluster in Drosophila, (iv) the link between small RNAs and repair of chromosome breaks, (v) demonstrating essential roles of Nanog and Gata6 during early differentiation, (vi) finding genes that sensitize mice to a high cholesterol diet, (vii) abnormal activation of WNT signalling triggering adrenal tumorigenesis, (viii) understanding the role of bile acids in pathophysiology of the testis, (ix) histone chaperone complexes maintaining nucleosome occupancy at genes.

Strengths and opportunities in the context

The team leaders and members consist of INSERM and CNRS supported researchers as well as teacher/researchers supported by the universities. Because of their close connection with the universities, the GReD faculty play a major role in both master and Ph.D. training. Of note are the increasing interactions with the medical school and hospital for translational programs. GReD members are active participants in national review committees such as CNRS, INSERM and Institute de Recherche pour le Développement (IRD), as well as members of French scientific societies, and local university committees. The GReD has increased its international visibility by members organizing and attending meetings. In addition, the number of publications has increased to 35-45 articles per year, with almost half being in journals rated as middle to top tier. Recognition and reputation of GReD researchers are also reflected in the increasing number of awards (e.g. ERC starting grant, ATIP/Avenir, two CNRS bronze medals, and one Fondation pour la Recherche Médicale (FRM) team). The unit has an impressive record of recruiting eight young researchers during the last five years, which adds important vitality to the scientific environment. A significant strength of the GReD are the technology platforms established by the research teams that are available to all researchers of the unit as well as to researchers outside the unit. In addition, GReD researchers have access to additional technology provided by the university or other research units (bioinformatics, high throughput sequencing, cell sorting, integrated imaging, metabolomics, proteomics). The move of all research teams into the new CCBR building will have a significant positive effect on research interactions and the possibility of initiating more interdisciplinary projects.

Weaknesses and threats in the context

Despite success in increasing support, funding is a major limiting factor for the GReD achieving international pre-eminence as a research unit in their chosen scientific fields. The limited number of permanent positions, both at the faculty and technical level, is a significant problem for the GReD in future planning. For example, the unit will have space for additional teams in the new CCBR building but no support for funding and attracting new researchers. Also, many of the permanent technical staff (specifically engineers) are approaching retirement age and the GReD does not know whether these positions will be replaced. Post-doctoral training is also a significant issue for the GReD. Post-doctoral fellows are classified as temporary employees and theoretically can remain in their position for up to six years maximum. However, in practice it seems impossible for post-doctoral fellows to remain at the GReD longer than 3-4 years due to local, regional rules that are outside the control of GReD management. Given the complexity and demands on what is required to complete projects, 3-4 years is not enough time to produce competitive research papers. This enforced shortened post-doctoral training period is not good for the fellows and makes the GReD a much less attractive place to recruit the very best fellows. This limitation has and will continue to limit the GReD's international visibility if outstanding postdoctoral fellows are not allowed to complete their work and publish in premier journals. In addition to this problem, limited funding and the lack of sufficient permanent positions results in GReD research teams that are small compared to international research teams. This restricts their ability to compete at the highest level. Given the data intensive nature of research today, the GReD is also lacking sufficient internal bioinformatics support embedded within the science of the research teams. This slows progress while they wait for support from outside informatics groups or for support from shared informaticians. The merger of the region Auvergne, with the region Rhône Alpes and the merger of the University of Auvergne with the University Blaise Pascal represent unknowns regarding the impact on future support for the GReD, and this creates an environment of uncertainty.

Recommendations

The GReD needs to work closely with the newly merged regions and universities to (i) develop a reasonable budget plan for maintaining the unit, (ii) increase permanent funding to allow the unit to recruit three new team leaders once the building opens, and (iii) maintain, if not increase, the permanent support positions (technicians and engineers). The policy of establishing co-leaders for research teams (a senior and junior leader) is a good practice for mentoring junior researchers. However, the GReD leadership must not automatically assume that the junior leaders succeed senior leaders when they retire. This policy will result in a stagnant research environment and not allow for an assessment for the need for new research areas. It is recommended that, when a team leader retires, the GReD leadership confer with their Scientific Advisory Board to assess the research strengths of the unit and decide whether new thematic areas should replace existing focus areas. In addition, GReD leadership should be encouraged to conduct an open, competitive search for new team leaders. Bringing outside researchers provides new perspectives and possibilities for scientific evolution of the unit. The quality of publications has improved since the last evaluation; however, the output is still not high enough to elevate the GReD as one of the best research units -not only internationally -but also in France. Complex and rapidly evolving technology requires interdisciplinary teams to answer challenging questions. The competition is fierce and the time to publication is critical. Although interactions amongst GReD teams occur regarding the sharing of technology, there is little evidence of collaborations amongst research teams resulting in interdisciplinary publications. Given that GReD research teams are limited in size, and that post-doctoral fellows have a limited time in the lab, collaborations on focused questions could lead to more significant publications in a faster time frame and elevate the stature of the unit. GReD researchers need to continue to attend and speak at international meetings. Finally, more effort is needed to integrate newly added clinical team(s) into the rest of the unit. There is significant potential synergy that has not yet been achieved.

  Marie-Christine CHABOISSIER, Institut de Biologie Valrose, Nice (representative of the Inserm CSS) Mr Thierry DARRIBERE, Laboratoire de Biologie du Développement, Paris (representative of the CNU) Mr Marc DELPECH, Hôpital Cochin, Paris Ms Claire FRANCASTEL, Université Paris-Diderot, Paris Mr François KARCH, University of Geneva, Switzerland Mr Antonio MOSCHETTA, University of Bari, Italy Mr Holger PUCHTA, Karlsruhe Institute of Technology, Germany Mr Frédéric RELAIX, Faculté de Médecine Pitié-Salpétrière, Paris Mr Daniel VAIMAN, Institut Cochin, Paris (representative of the CoCNRS) Mr Lucas WALTZER, Centre de Biologie du Développement, Toulouse Scientific delegate representing the HCERES: location of the unit The Genetics Reproduction and Development (GReD) laboratory was founded in 2008 under the joint sponsorship of the Université Blaise Pascal and the Université d'Auvergne with support from the region Auvergne, the Centre National de la Recherche Scientifique (CNRS) and the Institut National de la Santé de la Recherche Médicale (INSERM). The GReD is a major biosciences research unit in Clermont-Ferrand with a focus on understanding the genetic and epigenetic programs underlying development, reproduction and abnormal, diseased states. When created, the laboratory consisted of 13 research teams of which 7 still exist. Since the last external review in 2011, three new research teams were added resulting in a total of 10 research teams for the unit. The research of the GReD teams are aligned along three axes: (1) Genome Dynamics and Epigenetic Control (four teams); (2) Reproduction and Development in Health and Disease (four teams);

Number on 30/06/2015 Number on 01/01/2017 N1: Permanent

  Other professors (Emeritus Professor, on-contract Professor, etc.)

	professors and similar positions	32	45
	N2: Permanent researchers from Institutions and similar positions	17	19
	N3: Other permanent staff (technicians and administrative personnel)	23	24
	N4: Unit record	From 01/01/2010
		to 30/06/2015
	PhD theses defended	32

N5: Other researchers from Institutions (Emeritus Research Director, Postdoctoral students, visitors, etc.) 23 N6: Other contractual staff (technicians and administrative personnel) 3 N7: PhD students 29 TOTAL N1 to N7 127 Qualified research supervisors (HDR) or similar positions 28 Postdoctoral scientists having spent at least 12 months in the unit 29 Number of Research Supervisor Qualifications (HDR) obtained during the period 13

Genetics Reproduction and Development, GReD, U Clermont 1, CNRS, INSERM, U Clermont 2, Ms Chantal VAURY