{'fraction_non_alphanumeric': 0.05328731030820449, 'fraction_numerical': 0.01233634933927003, 'mean_word_length': 3.8737806091976905, 'pattern_counts': {'":': 0, '<': 22, '<?xml version=': 0, '>': 31, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Scattering resonances of large weakly open quantum graphs

In this paper, we consider a sequence of open quantum graphs, with uniformly bounded data, and we are interested in the asymptotic distribution of their scattering resonances. Supposing that the number of leads in our quantum graphs is small compared to the total number of edges, we show that most resonances are close to the real axis. More precisely, the asymptotic distribution of resonances of our open quantum graphs is the same as the asymptotic distribution of the square-root of the eigenvalues of the closed quantum graphs obtained by removing all the leads.


Weyl's laws for scattering resonances
When dealing with Schrödinger operators in R d , with a compactly supported potential, the spectrum of the operator consists of [0, +∞), along with finitely many negative eigenvalues. Thus, a lot of relevant spectral information about such operators consist, not in the spectrum and the eigenfunctions themselves, but rather in the scattering resonances and scattering states. These states are eigenfunctions which are not square-integrable, but instead are outgoing at infinity. The associated eigenvalues are the scattering resonances, which are complex numbers having the following interpretation: the real part is associated to the speed of oscillations, while the imaginary part corresponds to the escape rate of the scattering state towards infinity, when propagated by the wave equation.
For instance, in dimension 1, a number z ∈ C is a resonance of an operator −∆ + V if there exists a function f ∈ C ∞ (R) such that −∆f + V f = z 2 f f (x) = a±e ±izx when ±x >> 1.
We refer the reader to the recent monograph [16] both for a friendly introduction to scattering resonances, and for a comprehensive presentation of this field of research.
Scattering resonances can be seen as (square roots of) eigenvalues of a non-selfadjoint operator. It is notorious that the spectrum of non-selfadjoint operators is much more difficult to understand than that of self-adjoint ones; this is mainly due to pseudospectral effects, where a small perturbation in the operator can produce large perturbations of the spectrum.
Thus, it is complicated to determine the localization of the resonances. In dimension d = 1, it is possible to count the number of resonances in large disks. Namely, if we denote by N (R) the number of resonances in D(0, R) counted with multiplicity, we have where Vol(chsupp(V )) denotes the length of the convex hull of the support of V . Equation (1) was first proved for some special case of V in [35], and then generalized in [40] (see also [20] and [37]). It can be seen as an analogue of the classical Weyl's law for the eigenvalues of the Laplacian in a bounded domain.
In higher dimension, it is not known if the analogue of (1) holds. All that is known in general is the upper bound proven in [41], saying that N (R) ≤ CV R d .
Another interesting problem is to count the number of resonances not in disks, but in strips below the real axis. From a physical point of view, this is more relevant, as the resonances close to the real axis are related to resonant states which spend a long time in the system, before they escape to infinity. This question is often studied in the context of geometric scattering, or in the semiclassical framework. We refer to [42] for an account of the recent developments in this direction.

Scattering resonances on quantum graphs
As explained in the previous section, it is easier to count resonances of Schrödinger operators on R than on R d . A popular family of one-dimensional systems is that of quantum graphs. A quantum graph is a graph where each edge is endowed with a length, and each vertex, with a boundary condition. Eigenfunctions on the quantum graph are eigenfunctions of the Laplacian on each edge (and hence, are complex exponentials), satisfying the required boundary conditions at each vertex. The most natural boundary conditions are the Kirchhoff (or Neumann-Kirchhoff ) boundary conditions (see Definition 2); in all this paper, we will be working with these conditions.
Though very simple to describe, quantum graphs and their spectrum have been successfully used as a model of quantum chaos since the pioneering work of Kottos and Smilansky [28]. Here, we are interested in quantum graphs where, to some of the vertices, we attach semi-infinite edges (or leads). Scattering resonances on such open quantum graphs are complex numbers z such that there exists an eigenfunction satisfying the boundary conditions at each vertex, and equal to a constant times e izx on every lead. Resonances can also be seen as poles of the scattering matrix or as poles of the scattering determinant (see [19]).
Counting resonances in disks It is natural to wonder if (1) holds on open quantum graphs. A natural candidate to replace the constant Vol(chsupp(V )) would be the length of the (finite part of the) quantum graph, i.e., the sum of the lengths of all the edges that are not leads.
Actually, there are examples where N (R) behaves asymptotically like a constant times R, where the constant is not 2 π times the length of the graph. Such an open quantum graph is sometimes called a non-Weyl graph. In [14], it was shown that, on an open quantum graph with Kirchhoff boundary conditions at every vertex, we have N (R) R −→ c, where c is 2 π times the length of the graph if and only if the graph is unbalanced, i.e., at every vertex, the number of internal edges is different from the number of leads.
The simplest example of a non-Weyl (thus, not unbalanced) quantum graph is the following: if we have a vertex to which one internal edge and one lead are attached, the situation is equivalent to the one we obtain by removing the vertex and the lead, and replacing the edge by a lead; this would thus reduce the length of the quantum graph. Other examples of non-Weyl graphs were constructed in [14], [32], [18], [13]. The article [31] provides a nice presentation of all these results.
Counting resonances in strips Just as for Schrödinger operators on R d , it is interesting to count the number of resonances in strips below the real axis, instead of disks. In [14], it is shown that, for quantum graphs satisfying Kirchhoff boundary conditions, the resonances all lie in a strip of the form Im z ∈ [−K, 0]. To our knowledge, no explicit expression has ever been given for K; we will do so in Lemma 1 below in the case of unbalanced quantum graphs.
It is natural to wonder if we can also have a non-trivial upper bound on the imaginary part of resonances, that is to say, if they do all lie in a strip of the form Im z ∈ [−K, −K ] for some This cannot hold in general, as there can be resonances on the real axis. This is something typical of quantum graphs, related to the fact that, on a quantum graph, there can exist eigenfunctions which vanish on a large part of the graph. For instance, if a quantum graph contains a cycle whose edges have lengths that are all commensurate, it is possible to build eigenfunctions which will vanish on every vertex of the cycle, and on all the edges that are not part of the cycle (see Figure 1 in the appendix). This is in strict opposition to what happens for Schrödinger operators in bounded domains or on manifolds, where the unique continuity principle forbids an eigenfunction to vanish on an open set.
Actually, this situation is highly non-typical: it is shown in [17] and [30] that, if a quantum graph has a resonance on the real axis, then a small perturbation of the lengths will turn it into a resonance with negative imaginary part. Also, it is shown in [12,Theorem 4.2] that, for almost all choices of edge lengths, there are no resonances on the real axis. So far, the only known examples of quantum graphs having resonances on the real axis either have some loop, some vertices of degree 2, or have the lengths of the edges on a cycle all commensurate (see Figure 1 in Appendix A). It is thus natural to think that, when all edge lengths are rationally independent, all degrees are at least 3, and the graph contains no loops, there cannot be any resonance on the real axis. We will show in Appendix A that this is not the case: we show (in a non-constructive way) the existence of quantum graphs containing no loops, having minimal degree larger than 3, and with rationally independent edge lengths, but having some resonances on the real axis.
Coming back to the existence of a resonance gap, i.e., to the question of whether resonances lie in a strip of the form Im z ∈ [−K, −K ] for some K > 0, the analogy with semiclassical Schrödinger operators may lead us to look for criteria that are less related to arithmetic of the edge lengths, and more to some classical dynamics. In the physics literature, this was proposed in [4], where the authors introduce some classical dynamics on a quantum graph, introduce a topological pressure 1 , and claim that, if this pressure is negative, there should be a resonance gap.
This statement was disproved in [23] thanks to physical and numerical arguments, and in [12] by mathematical proofs. Indeed, in [12], the authors show that, for quantum graphs that are not trees, and have rationally independent edge lengths, there exist resonances arbitrarily close to the real axis (and with arbitrarily large real parts). The authors are even able to give an interesting lower bound on the number of resonances close to the real axis.
Finally, let us mention that, along with scattering resonances on an open quantum graph, many other non-selfadjoint problems have been considered on quantum graphs, like the damped wave equation, Schrödinger operators with complex potentials, or non-selfadjoint conditions at the vertices. We refer the reader to the introduction of [36] for a presentation of the advances on such topics.

Resonances of large quantum graphs
In this paper, we will not be interested in the resonances of a single quantum graph. We will rather consider a sequence of open quantum graphs QN , having a number of vertices and edges of the order of N , and will consider the asymptotic distribution of their resonances. Namely the kind of question we will be interested in is the following: Given a region Ω ⊂ C, what is the asymptotic number of resonances of QN in Ω as N −→ +∞ ?
Hence, we consider here the large graph limits instead of the large frequency limits.
To our knowledge, this is the first time in the mathematical literature that resonances of large open quantum graphs are considered. However, for closed quantum graphs (i.e., quantum graphs without leads, which is the situation the most frequently considered), the large graph limits has been a subject of growing interest in the past few years. An important motivation for that was the question of Quantum Ergodicity, which is a property of equidistribution of the eigenfunctions in systems enjoying a chaotic (or just ergodic) classical dynamics. Although quantum graphs are a popular model of quantum chaos, it was shown in [11] that, in general, Quantum Ergodicity does not hold on a given quantum graph. It thus became natural to consider eigenfunctions on a sequence of quantum graphs of increasing size. See for instance [6], [22], [25] [9], [24] and [3] for examples of families of quantum graphs for which quantum ergodicity holds, and [7] for an example where it doesn't.
Let us also mention [2], where the question (2) is addressed for closed quantum graphs (hence, the resonances are just the square roots of the usual eigenvalues of selfadjoint operators, which are thus on the real axis). Instead of counting exactly the number of resonances in a set Ω, it is more natural to consider the empirical spectral measures where L(QN ) is the sum of the lengths of the edges of QN , Res(QN ) is the set of resonances of QN . It is thus a locally finite measure, which is supported on R in the case of closed graphs.
Benjamini-Schramm convergence of quantum graphs In [2], the authors show that a natural criterion to obtain convergence of the empirical spectral measures is to suppose Benjamini-Schramm convergence (also dubbed local weak convergence) of the quantum graphs. For a sequence of discrete graphs (GN ), we say that it converges in the sense of Benjamini-Schramm ( [10,1]) if for any finite rooted graph (F, o) and k ∈ N, the fraction of points x ∈ GN such that BG N (x, k) ∼ = (F, o) has a limit, where BG N (x, k) is the ball of radius k around x in the graph GN . The limit is then a probability measure on the set of rooted graphs.
This definition of Benjamini-Schramm convergence for discrete graphs can be extended to the case quantum graphs; we will recall how to do so in section 3.3.1. We will also recall that, if a sequence of quantum graphs has uniformly bounded degrees and edge lengths, we may always extract a subsequence which converges in the sense of Benjamini-Schramm.
The main result of [2] can be summed up as follows: If a sequence of closed quantum graphs (QN ) has uniformly bounded degrees and edge lengths, and converges in the sense of Benjamini-Schramm to some probability measure P on the set of rooted quantum graphs, then µQ N converges vaguely to some measure depending only on P. More precisely, for any continuous compactly supported function χ ∈ Cc(R), we have 2 where HQ is the Schrödinger operator on the limiting random quantum graph Q, χ √ HQ (x0, x0) is the value of the Schwartz kernel of the operator χ √ HQ at the root x0 and Q * is a set of (equivalence classes of) rooted quantum graphs.
Presentation of our main result The main result of this paper is a generalization 3 of (3) to the case of a sequence of open quantum graphs converging, in the sense of Benjamini-Schramm, to a probability measure supported on the set of rooted closed quantum graphs: Let (QN ) be a sequence of unbalanced open quantum graphs with N vertices, having uniformly bounded degree and edge lengths. Suppose that (QN ) converges, in the sense of Benjamini-Schramm, to a measure probability P supported on the set of rooted closed quantum graphs. Then QN satisfies (30). In particular, µQ N converges vaguely to some measure on R depending only on P.
Recall that, up to extracting a subsequence, we may always suppose that (QN ) converges in the sense of Benjamini-Schramm. The assumption that the limit is on the set of closed quantum graphs is equivalent to supposing that the number of leads is a o(N ).
An informal, but illuminating, formulation of our result is the following. Let QN be the quantum graph obtained from QN by removing all the leads from it. Then, if the number of leads of QN is a o(N ), the resonances of QN and the square-root of the eigenvalues of QN are asymptotically distributed the in same way.
In particular, this shows that, if K is a compact set which does not intersect the real axis, then the number of resonances of QN in K is a o(N ).
Resonances mildly close to the real axis In Theorem 4 below, we will show more precise estimates, depending on the number of leads in QN . In particular, if the number of leads in QN is bounded independently of N , we show that the number of resonances QN We don't believe this bound is optimal; it would be very interesting (and probably hard) to describe the asymptotic number of resonances QN in [a, b] + i[−∞, −δN ], for instance when δN is a negative power of N . To this end, the numerical computations done in [4, Section 7] could be a good starting point, although their N isn't very large, and the asymptotic they consider is not very clear (they seem to consider the asymptotic δN >> 1, although there are no resonances away from a horizontal strip).
Relation to other works The large quantum graph limit we consider (with few leads) has analogues in the realm of Schrödinger operators. Namely, consider a potential V ∈ L ∞ (R), spatially extended (for instance, periodic with respect to Z d , or a random potential), and write 2 Note that, in [2], the authors define the empirical spectral measures on closed quantum graphs as sums of Dirac masses on the eigenvalues, which are squares of resonances. This is why the function we integrate in (3) is χ( √ z). 3 Note however, that we only consider case of Kirchhoff boundary conditions here, while more general boundary conditions are treated in [2]. Actually, the proofs of [2] are much simpler when only Kirchhoff boundary conditions are considered.
VR(x) := V (x)1 |x|≤R . It is natural to wonder if the resonances of −∆ + VR will somehow converge to the real axis, and approach the spectrum of the fully extended operator −∆ + V . The reason why such a phenomenon would happen is that, to escape from the support of the potential and go to infinity, the waves must pass through a spherical layer close to the boundary of the ball. As the ball becomes large, this layer has a volume much smaller than the total volume of the ball. Hence, this situation is very similar to having few leads on a large quantum graph.
To our knowledge, this problem of resonances for potentials with a wide support was first studied in [27] for both periodic and random potentials in Z, which are truncated outside of large intervals. The author obtains very precise estimates on the localization of the resonances for both situations he considers. We cannot hope to obtain such precise results here, as we are working in the framework of Benjamini-Schramm convergence, which encompasses situations as different as those of periodic and random potentials. We refer the reader to [15] for a review of all the results involving resonances of potentials with a wide support.
Finally, we would like to mention that the methods we use here are very much inspired from those developed by J. Sjöstrand to study scattering resonances or the spectrum of the damped wave equation, as in [38] or [39].
Organisation of the paper After this long introduction, we will start in section 2 by stating our results in a more general setting involving holomorphic families of large matrices. In section 3, we will recall the definition of quantum graphs and of their resonances, and explain how they enter the framework developed in the previous section. In section 4, we will recall the tools of complex analysis we will use. Section 5 shows that there are few resonances away from the real axis, while section 6 gives a proof of our main result. Finally, Appendix A, which is independent from the rest of the paper, gives new examples of quantum graphs having scattering resonances on the real axis.
Acknowledgements It is a real pleasure to thank Frédéric Klopp for telling me about the problem of resonances for potentials with a wide support, which was the starting point of the reflection which led to this article. I also benefited a lot from discussions with Guillaume Klein and Martin Vogel on related topics.
Last but not least, I would like the thank the anonymous referee for indicating several interesting references, and pointing out a few imprecisions in the manuscript.

A general framework for our results
We would like to state our results in a general framework of holomorphic families of operators acting on some Hilbert spaces. In most of the paper, we will restrict ourselves to the finitedimensional case, which is the most relevant for quantum graphs.
Let Y0 ∈ (0, +∞]. We will write C Y 0 := {z ∈ C; Im z < Y0}. For each N ∈ N, let HN be a Hilbert space, and let UN (z) : HN −→ HN be a sequence of trace-class operators depending in a holomorphic way on z ∈ C Y 0 . Recall that, if U is a trace class-operator with trace-norm U 1, the determinant of Id + U is well-defined, and we have Recall furthermore that, if U is bounded and V is trace-class, we have We will be interested in solutions of the equation A solution of (6) will be called a resonance of UN . If z is a solution of (6), ker(UN (z) − Id) is a vector space of positive finite dimension. This dimension will be called the multiplicity of the solution of (6).
If Ω ⊂ C Y 0 is an open set, we will write NΩ(UN ) := number of solutions of (6) in Ω, counted with multiplicity.
Our aim will be to estimate the asymptotic behaviour of NΩ(UN ) as N → +∞. To this end, we make the following assumption on UN .
2. If K ⊂ {z ∈ C; 0 < Im z < Y0} is a compact set, we may find a constant C0(K) > 0, independent of N , such that for all N ∈ N and all z ∈ K, we have 3. There exists Y1 > 0 such that, for any 4. Let K ⊂ C Y 0 be a compact set. There exists C1(K) > 0 such that for all N ∈ N and all z ∈ K, we have Note that, by analyticity of UN (z), equation (9) implies that there exists a constant C 1 (K) > 0 such that for all N ∈ N and all z ∈ K, we have Our main assumption is that UN is close, in trace-norm, to an operator having all its resonances on the real axis. In the case of quantum graphs, these new operators will be built from the quantum graphs where all the leads are removed.
Hypothesis 2. We suppose that, for each N ∈ N, there exists another family UN (z) : HN −→ HN of invertible operators depending in a holomorphic way on z ∈ C Y 0 , such that the following holds.
1. If K ⊂ C Y 0 is a compact set, there exists a constant c(K) such that the following holds for all N ∈ N and all z ∈ K In particular, for all compact set K ⊂ C Y 0 \ R, there exists C2(K) such that for all N ∈ N and all z ∈ K, we have 2. There exists a sequence (γN ) of positive numbers such that the following holds.
Note that (12) implies that the resonances of UN (z) are all on the real axis. In the next section, if we are working with a quantum graph having N vertices, γN will be the number of leads, which should be small compared to N .
Our aim will be to count resonances in boxes. If a < b, c < d, we will write If a < b, we will also write Our first theorem gives a bound on the number of resonances of UN away from the real axis.
Theorem 1. Let (UN ) and ( UN ) satisfy Hypotheses 1, 2. Let a1 < a2 ∈ R, and let a3 > 0. There exists a constant C depending on a1, a2, a3 and on the constants in Hypotheses 1 and 2, but not on N , such that the following holds. Let (δN ) be a bounded sequence of positive numbers, possibly going to zero. Then there exists N0 such that for all N ≥ N0, In particular, if γN = o(N ) and Ω ⊂ C is a bounded set such that Ω ∩ R = ∅, we have Equation (15) follows from (14) by taking a sequence (δN ) converging to zero slowly enough. Note that equation (14) is non-trivial only when δN is not chosen too small.

Remark 1.
We will see in Theorem 2 that, if the dimension of HN grows like N , then, under reasonable assumptions, Na 1 ,a 2 ,−a 3 ,0(UN ) is of the same order as N . Hence, equation (14) implies that most of the resonances in Ra 1 ,a 2 ,−a 3 ,0 are actually in R a 1 ,a 2 ,−δ N ,0 , provided we have δN γ N N 2/5 . The quantity γ N N 2/5 may thus be thought of as a lower bound for the imaginary part of typical resonances. We would like to insist that we don't believe the power 5 2 appearing in (14) to be optimal. It would be very interesting to carry out numerical simulations to see what is the best power we can expect here.
We want to show that, if BN 1 is small, MN has few eigenvalues away from the real axis. The matrix BN being negative Hermitian, if z is an eigenvalue of MN , we must have We thus have We have In particular, when z is not an eigenvalue of MN , we have (UN (z) − Id) −1 ≤ 1+|z|+2C 2 (AN + iBN − zId) −1 . Points 2 and 3 of Hypothesis 1 follow easily, taking Y1 = 3C.
We also define for all z ∈ C such that Im z < 1 Note that UN (z) can be diagonalized in an orthonormal basis. More precisely , if v ∈ R N and λ ∈ R are such that Av = λv, Equation (11) follows easily.
Finally, we have Therefore, our matrices satisfy all the assumptions from Hypotheses 1 and 2, with γN = BN 1. We deduce that, for any sequence (δN ) of positive numbers, the number of eigenvalues This result seems elementary, but we could not find a trace of it in the literature.
To state our next theorem, we will need to restrict ourselves to the finite-dimensional setting. The main reason for this is that we want the operators UN to be unitary on the real axis; this assumption is compatible to being trace-class only in finite dimension.
3. The operators UN (x) from Hypothesis 2 are unitary for all x ∈ R.

For any
Note that, thanks to (16), if U : HN −→ HN , we have Furthermore, the last assumption of Hypothesis 3 implies that the resonances of UN and UN are symmetric with respect to the imaginary axis.
Our next theorem says that, in boxes close to the real axis, UN and UN have asymptotically the same number of resonances, up to changing slightly the size of the boxes.  Proof that Theorem 2 implies Corollary 1.
Step 1: Let us fix c > 0. We define a sequence of measures νN,c on R by Now, recall that the measure µ is regular, so that the limit of the right-hand side as ε → 0 is µ (a, b) . All in all, we have, for all a < b We may thus apply Portmanteau theorem to deduce that, for any c > 0, νN,c converges vaguely to µ as N → +∞.
Step 2: , and equal to one on (−δ, δ). We define two functions f δ , f δ ∈ C ∞ (C) by Note that, when f takes only non-negative values, we have for any δ > 0 so that, by the first step, for any δ > 0, Decomposing f into its positive and negative parts shows that (19) actually holds for all f ∈ C ∞ c (C).
On the other hand, by continuity, f δ − f δ C 0 = O(ε). Therefore, using Theorem 1 and our assumption, we have lim sup Finally, for any δ > 0, the function f − f δ is supported away from the real axis, so, by Theorem 1, we have lim sup Combining equations (19), (20) and (21) gives us the result.

Open quantum graphs and their resonances
We will now recall the definition of resonances on open quantum graphs, and explain how they enter the framework developed in the previous section.

Definition of resonances on open quantum graphs
An (open) quantum graph is given by a finite graph, where each edge is given a length and where infinite edges (called leads) are attached to some of the vertices. More precisely The quantum graph will be called finite if G is a finite graph. We will say that Q is a closed quantum graph if n(v) = 0 for all v ∈ V .
All the quantum graphs we will consider will always be either finite or closed. The graph (V, E) should be thought of as the compact part of our graph, the map L gives the length of the edges in the compact part, while the map n gives the number of infinite edges attached to each vertex. If v ∈ V , we denote by d(v) the (internal) degree of v, i.e., the number of e ∈ E to which v belongs. We define the total length of the graph by L(Q) := e∈E L(e).
We let B = B(G) be the set of oriented internal edges (or bonds) associated to E. If b ∈ B, we shall denote byb the reverse bond. We write o(b) for the origin of b and t(b) for the terminus of b.
We define the external bonds by Bext := v∈V A C 2 function on the graph will be a collection of maps We will write C 2 (Q) for the set of such functions. We may now define the boundary conditions we put at the vertices.

Definition 2.
We say that f ∈ C 2 (Q) satisfies Kirchhoff boundary conditions if it satisfies Let us now move to the definition of scattering resonances of quantum graphs.
Definition 3. Let Q be a finite quantum graph. A number z ∈ C is called a scattering resonance of Q if there exists f ∈ C 2 (Q) such that 1. f satisfies the Kirchhoff boundary conditions.

For all
If z is a resonance, we define its multiplicity, written mult(z) as the number of linearly independent f ∈ C 2 (Q) satisfying 1., 2. and 3. We will write Res(Q) := {Resonances of Q} ⊂ C.
Remark 2. If the quantum graph Q is closed, then condition 3. above is empty. A scattering resonance is then just the square-root of an eigenvalue of the Laplacian on Q with suitable selfadjoint conditions. We refer the reader to [5, Chapter 1 and 2] for more details on this situation. In particular, all the scattering resonances of a closed quantum graph belong to the real axis.
Remark 3. When z = 0, condition 3. means that f b is constant on every lead. In particular, the multiplicity of z = 0 does not change when we add or remove leads, i.e., if we change the value of n.
Note that, if we did not impose the last condition (and if n is not identically zero), then any number z ∈ C would be a scattering resonance. On the other hand condition 3. imposes that scattering resonances must have negative imaginary part. Otherwise, the function f would be an eigenfunction of a selfadjoint operator, associated to a non-real eigenvalue. We will see another proof of this fact in the next subsection.

A characterisation of scattering resonances of an open quantum graph
If a function f ∈ C 2 (Q) satisfies 1. and 3. in Definition 3, then for every b ∈ Bext, f b is entirely determined by f b (0), and we have f b (0) = izf b (0), so that the current conservation condition rewrites ∀v ∈ V, Remark 4. In particular, we see that z is a resonance if and only if there existsf (22) and −f b = z 2 f b , and such that the continuity condition and (23) are satisfied. This caracterization has the advantage of only implying (functions on) the compact part of the graph, along with the map n. The Let us rewrite the generalized Kirchhoff conditions in terms of the coefficients a(b) and a (b). The continuity condition rewrites The condition about derivatives can be expressed as Remark 5. Note that if we denote by σ (v) the d(v) × d(v) matrix whose lines and columns are indexed by the bonds of B such that o b = v, with entries σ where J has all its entries equal to one. In particular, σ (v) is symmetric and has eigenvalue −1 with multiplicity d(v) − 1, and −1 + 2d(v) n(v)+d(v) with multiplicity one. In particular, σ (v) is always subunitary.
When n(v) = 0 (hence, when there are no leads attached to v), then σ (v) is an isometry. Otherwise, we see that σ (v) is invertible unless v is a balanced vertex, i.e., a vertex such that Finally, recall that, condition (22) The continuity and generalized Kirchhoff conditions may then be rewritten as Let us define the matrices D(z), S and U (z) whose lines and columns are indexed by the elements of B by An important feature of Kirchhoff boundary conditions, essential in our proof, and which disappears for more general conditions, is that the matrix S does not depend on z.
Writing a = (a(b)) b∈B , equation (24) then rewrites as We have thus shown that, for any z ∈ C\{0}, and the multiplicities coincide. This characterisation of resonances, which can be found for instance in [29], is reminiscent of the secular equation introduced by Kottos and Smilansky in [28], which is a powerful tool to study the spectrum of closed quantum graphs.

Families of open quantum graphs
From now on, we consider a sequence of finite open quantum graphs QN = (VN , EN , LN , nN ), and we will be interested in the localization of its resonances. We shall always make the following assumption on our graphs.

Hypothesis 4 (Bounds). There exists
We will always make the following assumption on our graphs.

Lemma 1. Suppose that (QN ) satisfies (Bounds) and (Unbalanced). Then
In particular, Proof. Under hypothesis (Unbalanced), we see that the matrices σ (v) are all invertible, and we have ( In particular, if we furthemore suppose that (Bound) is satisfied, we get We see from Remark 5 that S ≤ 1. Furthermore, if J is the B × B matrix such that J b,b = δ b , b , then the matrix SJ is a block matrix, so it can be inverted block by block. We deduce from this and from Remark 5 that S −1 ≤ n0 + D. In particular, we see that U (z) is invertible, and that for any z ∈ C − , we have U (z) −1 ≤ (n0 + D)e Im zL min .

Benjamini-Schramm convergence for open quantum graphs
The set of rooted open quantum graphs, quotiented by ∼, will be denoted by ROQ. If  (Q, b0) is a rooted open quantum graph, we will denote by [Q, b0] its equivalence class.
If v ∈ G is a vertex in a graph and r ∈ N, we write BG(v, r) for the set of vertices which are at a (combinatorial) distance at most r from v. We write E(BG(v, r)) for the set of edges in E connecting two vertices of BG(v, r).
We introduce a distance between rooted quantum graphs as follows Note that this definition is independent of the representatives we chose in the equivalence classes [Q1, b1], [Q2, b2], so it is well-defined on ROQ. Furthermore, one can show that (ROQ, d) is a Polish space, i.e., a separable complete metric space.
Let P(ROQ) be the set of Borel probability measures on ROQ.
Definition 5. Any finite quantum graph Q = (V, E, L, n) defines a probability measure νQ ∈ P(ROQ) obtained by choosing a root uniformly at random: If (QN ) is a sequence of quantum graphs, we say that P ∈ P(ROQ) is the local weak limit of (QN ), or that (QN ) converges in the sense of Benjamini-Schramm to P, if (νQ N ) converges weakly to P.
Note that, in the special case of equilateral graphs with no leads, that is, if, for all N , we have nN ≡ 0 and LN ≡ L0, then (QN ) converges in the sense of Benjamini-Schramm if and only if the underlying discrete graphs (VN , EN ) converge in the sense of Benjamini-Schramm (recall that Benjamini-Schramm convergence for discrete graphs was discribed in section 1.3). We refer the reader to [2, §8] for examples of sequences of closed quantum graphs that converge in the sense of Benjamini-Schramm.
Let D, n0 ∈ N, 0 < m ≤ M . We define ROQ D,n 0 ,L min ,Lmax ⊂ ROQ as the subset of equivalence classes [Q, b0] = [(V, E, L, n, b0)] such that Q satisfies (27). Let us now present the main result of [2], showing that Benjamini-Schramm convergence of closed quantum graphs implies the convergence of their empirical spectral measures.
If Q is a finite quantum graph (open or closed), we define its empirical spectral measure as where HQ is the Schrödinger operator on the limiting random quantum graph Q, χ √ HQ (x0, x0) is the value of the Schwartz kernel of the operator χ √ HQ at the root x0 and Q * is a set of (equivalence classes of ) rooted quantum graphs.
We refer the reader to [2] for the fact that the operator χ √ HQ does indeed admit a continuous Schwartz kernel, and for other equivalent definitions of the measure µ P .

Resonances of sequences of open quantum graphs
Just as in the previous section, if a < b, c < d, we will denote by N a,b,c,d (QN ) the number of resonances of QN with a ≤ Re z ≤ b and c ≤ Im z ≤ d, and, more generally, by NΩ(QN ) the number of resonances of QN in Ω.
Our main assumption is that the number of leads of QN is small compared to the number of vertices.
We may then state the analogue of Theorem 1. There exists a C0 depending on the constants appearing in (Bounds) such that the following holds. Let (δN ) be a bounded sequence of positive numbers, and let 0 < a1 < a2, 0 < a3. Then there exists N0 such that for all N ≥ N0, In particular, if Ω ⊂ C is a bounded set such that Ω ∩ R = ∅, we have Proof. We want to show how this enters the framework of Theorem 1.
Recall that the resonances of QN that are not zero are characterized, along with their multiplicity, by the equation det(Id − UN (z)) = 0, with UN (z) as in (25). We must thus check that UN satisfies Hypotheses 1 and 2. We may take Y0 = +∞, as UN is well-defined on all of C. When Im z > 0, we have UN (z) ≤ SN e izL N ≤ e − Im zLmax , and the first point of Hypothesis 1 follows. In particular, if Im z > 0, then UN (z) < 1, so that (Id − UN (z)) can be inverted by a Neumann series, with (Id − UN (z)) −1 ≤ The first point of Hypothesis 2, when Im z > 0, is proved in exactly the same way as the first point of Hypothesis 1. When Im z < 0, we note that, from Remark 5, SN is an isometry, so that UN (z) is invertible, and UN (z) −1 ≤ e −L min | Im z| . Hence, and the first point of Hypothesis 2 follows. For the second point of Hypothesis 2, note that Now, SN − SN has all its entries bounded uniformly, and has a number of non-zero entries which is a O(g(N )). Therefore, the second point of Hypothesis 2 follows, with γN = g(N ).
We may thus apply Theorem 1 to deduce the result.
Recall that, by Hypothesis (Bounds) and Lemma 2, we may always extract a subsequence from (QN ) which converges in the sense of Benjamini-Schramm. Furthermore, by Hypothesis (Weakly open), the Benjamini-Schramm limits of our graphs are always measures supported on RQ, the set of equivalence classes of rooted quantum graphs such that n ≡ 0 (in other words, the set of equivalence classes of closed quantum graphs). Therefore, the following hypothesis is always satisfied, up to extracting a subsequence; furthermore, it implies Hypothesis (Weakly Open).
Hypothesis 7 (BS). The sequence (QN ) converges in the sense of Benjamini-Schramm to a measure P ∈ P(RQ).
Our main result is then an analogue of Corollary 1. Proof. We want to apply Corollary 1, with the notations of the proof of Theorem 4. Note that the resonances of UN (resp UN ) only correspond to the resonances of QN (resp QN ) that are different from zero. This is not a problem since, by Remark 3, zero has the same multiplicity as a resonance for QN and QN .
We need to check that Hypothesis 3 holds, with the notations of the proof of Theorem 4. The first point always holds if we label our sequence of graphs such that |VN | = N , and the second point then follows from Hypothesis (Weakly open).
The third point follows from the fact that SN is an isometry. Concerning the fourth point, note that LminId is a positive selfadjoint operator, since SN is an isometry and (LN − Lmin)Id is selfadjoint and non-negative.
As for the last point, we have This is the transpose of the matrix UN (z), which is thus similar to UN (z). The same proof shows that UN (z) is similar to U * N (−z). Therefore, Hypothesis 3 is satisfied. Finally, note that, since (QN ) converges in the sense of Benjamini-Schramm to P, Hypothesis (Weakly Open) implies that ( QN ) does also converge to P. Along with Theorem 3, this shows that all the assumptions of Corollary 1 are fulfilled, which concludes the proof.

Remark 6.
In what precedes, Hypothesis (Unbalanced) is used only to obtain the bound (8) from Hypothesis 1. We may thus wonder if this hypothesis is really necessary.
Actually, Theorem 5 cannot hold if a hypothesis like (Unbalanced) is not made, as can be seen in the following simple example. Consider a graph with edges x0, ..., xN , with edges of length L joining xj to xj+1 for 0 ≤ j ≤ N −1, and with a lead attached to x0 and to xN . Having a scattering resonance on such a graph is equivalent to having a scattering resonance for the free Laplacian on R, which can happen only at z = 0. However, if the leads are removed, resonances correspond to square-roots of the eigenvalues of the Neumann Laplacian on [0, LN ], which are regularly spaced at a distance π LN , so that 1 N µ Q N converges to a non-trivial measure. Hence, the result of Theorem 5 does not hold in this case.

Tools from complex analysis
The aim of this section is to recall the facts from complex analysis which we will use in the proof of Theorems 1 and 2.

Jensen's formula
Let z0 ∈ C and let f be a holomorphic function such that f (z0) = 0. For every t > 0, denote by n(t) the number of zeroes of f (z) such that |z − z0| < t. Recall that Jensen's formula tells us that r 0 n(t) t dt + log |f (0)| = 1 2π 2π 0 log |f (e iθ r)|dθ.
Therefore, for any ε > 0, we have
Let K ⊂ C be a bounded set containing Rz 0 ,s,t. Let f be a holomorphic function on C. We denote the zeroes of f in K, repeated with multiplicity, by (zj)j=1,...,J , with J = J(f, K). Note that, if we take K = Rz 0 ,s,t, then, thanks to Jensen's formula, we have We set so that, for all z ∈ Sz 0 ,r ⊂ K Since B f has the same zeroes as f in K, we deduce that the map is holomorphic on C and does not vanish in K.
We recall the following lemma, which was proven in [38, Lemma 8.1].
Hence, the harmonic function m(z) := 2M (f ) − log |G f (z)| remains positive in Rz 0 ,αs,αt. We may thus apply Harnack's inequality to find a constant C depending only on the ratio s t such that sup We deduce that, for any z ∈ Rz 0 ,αs,αt, we have In other words, if we write Since log |G f | is harmonic, we deduce from standard elliptic gradient estimates (see for instance [21,Theorem 3.9]) that ∀z ∈ R z 0 ,α 2 s,α 2 t , |∂z log |G f (z)|| ≤ C rm(1 − α) M (f, z0, s, t, α, K).
Now, log |G f | = Re log G f , where log G f is only well-defined modulo 2iπ, but is holomorphic. Hence, ∂z log G f (z) is well-defined, and by the Cauchy-Riemann equations, we have We may sum up the results obtained in this section so far in the following lemma.
Lemma 4. Let z0 ∈ C, let s, t > 0, 0 < α < 1, and let K ⊂ C be a bounded set containing Rz 0 ,s,t := RRe z 0 −s,Re z 0 +s,Im z 0 −t,Im z 0 +t. There exists a constant C = C(s/t) such that, if f is a holomorphic function on C, we have for all z ∈ R z 0 ,α 2 s, with M (f, z0, s, t, α, K) as in (42), and where (zj)j=1,...,J are the zeroes of f in K.

Proof of Theorem 1
We now proceed with the proof of Theorem 1. In section 2, our operators UN (z) were defined for Im z < Y0. To lighten notations, we will always take Y0 = +∞, which is always the case when considering quantum graphs. The case Y0 finite doesn't add any essential modification to the proof, as our complex-analytic constructions always take place below the real axis, or slightly above it.
Proof of Theorem 1. Without loss of generality, we may suppose that a3 > 2Y1, with Y1 as in Hypothesis 1.

Proof of Theorem 2
Thanks to the last point of Hypothesis 3 we know that the resonances of UN and UN are symmetric with respect to the imaginary axis. Hence, the number of resonances in a rectangle can be estimated using the number of resonances in rectangles that are symmetric with respect to the imaginary axis. Therefore, Theorem 2 will follow from the following proposition. Let a > 0. For any ε > 0 and any N ∈ N, we may find a + ∈ [a + ε, a + 2ε] and a − ∈ [a − 2ε, a − ε] such that, for any c > 0, we have Let us now proceed with the proof of Proposition 1. Let a ∈ R, and let ε > 0. All the values of the constants in the sequel will implicitly depend on a and ε. Let (ξN )N be a sequence decreasing to zero. Its exact value will be chosen later on.
For each x ∈ R, the operator UN (x) is unitary, so it can be diagonalised in an orthonormal basis. Let us denote by e iθ k (x) 1≤k≤d N its eigenvalues. The θ k are parametrised so that for each k, x → θ k (x) is smooth; let u k (x) be an associated normalized eigenvector.
For each x ∈ R, we write Lemma 6. Let ε > 0. There exists a + ∈ [a + ε, a + 2ε] and a − ∈ [a − 2ε, a − ε] depending of N , and c 1 > 0 independent on N such that Proof. We will only explain the construction of a + , as the construction of a − is exactly the same. We apply the results of section 4.2 for z0 = a + 3ε 2 + i ε 4 , s = t = ε 2 and α = 1 √ 2 , with f (z) := det(Id − UN (z)).
Lemma 5 gives us the existence of a + ∈ [a + ε, a + 2ε] such that Now, thanks to (18) and (5), we have log sup z∈D(z 0 , On the other hand, we have so that, thanks to (7) and (18) In particular, Therefore, taking logarithms, we have Recalling that the resonances of UN are isolated, we may do an arbitrarily small perturbation of a ± from the previous lemma such that (52) still holds, and we have ∀y ∈ R, a ± + iy is not a resonance of UN , From now on, we fix a ± such that (52) and (53) hold. We write We define a unitary matrix Σ ± N of size dN by Hence, U ± N (z) is a holomorphic family of matrices, and for any z in a compact set, there exists C such that Furthermore, if we denote by θ ±,N k (z) the eigenvalues of U ± N (z), we have Recall that, if a1 < a2 ∈ R, we denote by Na 1 ,a 2 ( UN ) the number of resonances of UN in [a1, a2] counted with multiplicity. We will also denote by Na 1 ,a 2 ( U ± N ) the number of zeroes of det Id − U ± (x) for x ∈ (a1, a2), counted with multiplicities. Proposition 1 will follow from the next two lemmas.
Lemma 8. We may find c ± > 0 independent of N such that Indeed, we have The first term is smaller than N −a ± ,a ± ,−c,− c ± Proof of Lemma 7. For each 1 ≤ k ≤ dN , we denote by N k (a1, a2) (resp. N ± k (a1, a2)) the number of zeroes of e iθ k (x) − 1 (resp. e iθ ± k (x) − 1) for x ∈ (a1, a2). To lighten notations, in all the proof, we will write a instead of a ± . By assumption, UN (−a) and UN (a) * are similar, so, for each k, there exists n k ∈ Z such that θ k (a) = −θ k (−a) + 2n k π.
Without loss of generality, we will suppose that θ k (−a) ∈ [0, 2π[. Differentiating the relation u k , UN (x)u k (x) = e iθ k (x) , we obtain In other words, we have In particular, thanks to (17), we see that this quantity is positive. Since θ k (x) is an increasing function, we see that On the other hand, we have All in all, we have Recall that Noting that , the same argument as above tells us that (57) and (52), we see that The result follows.
We may now proceed with the proof of Lemma 8.
Proof of Lemma 8. We will only bound the quantity is exactly the same. We know from (56) that Let z ∈ C with |z| ≤ 1. We have (59) Note that by (10), there exists C 1 such that Hence, we deduce from (59) that All in all, we may find c 2 > 0 such that det U + N (z) − Id does not vanish in B ±a + + i 2 ξN , and that we have On the other hand, thanks to (13) and (55), we have for all z in a compact set, where we write Let us set so that z ∈ C\R is a resonance of UN ⇐⇒ D + N (z) = 0. Thanks to equations (4), (5) and (61), we have, for all z in a compact set In particular, to (11) and to the fact that Σ + N is an isometry. Therefore, we have • If z ∈ D ±a + + i On the other hand, if z is such that Id − UN (z) is invertible, then we have An application of Jensen's formula (32) along with (64), (65) and (7) tells us that We write Sz,r := RRe z−r,Re z+r,Im z−r,Im z+r Let c 3 be such that Taking Y1 as in Hypothesis 1, we define Thanks to Hypothesis 1, we have NŴ 0 (UN ) = 0. Equation (66) tells us that N W ± (UN ) ≤ CζN , while Theorem 1 implies that provided we take ξN going to zero slowly enough. We denote by (zj)j=1,..., .
We are now going to apply the results of section 4.2 to estimate | log G + N | in W1. We will thus take K = W , and apply Lemma 4 in various rectangles.
All in all, we obtain that, for all z ∈ W , we have ∂z log G + N (z) ≤

A Irrational quantum graphs with real resonances
The simplest examples of open quantum graphs having resonances on the real axis are those whose compact part is just a segment, contains a loop (or a part that is equivalent to a loop, see the left side of Figure 1), or those where the lengths on a cycle are all rationally dependent (see the right part of Figure 1).
Apart from variants of these simple situations, it is not easy to build examples of quantum graphs having resonances on the real axis. Our aim here is to show that such graphs exist, thanks to the following proposition. if λ(L e1 + L e2 + L e3 ) ∈ Nπ, then we may build an eigenfunction with eigenvalue λ 2 , vanishing at x 1 , and on the lead connected to it. On the right, we may build an eigenfunction vanishing at every vertex, and on all the leads. Proposition 2. There exists a quantum graph Q = (V, E, L, n) such that • There no loops, i.e., no edges joining a vertex to itself; • Each vertex has degree ≥ 3; • The family (Le)e∈E is rationally independent and Q has a resonance on the real axis.
Proof. The proposition will follow if we show that there exists a closed quantum graph (V, E, L), with no loops, minimal degree ≥ 3 and lengths (Le)e∈E forming a rationally independent family, and having an eigenfunction ψ which vanishes at some vertex v ∈ V . Indeed, in this case, we may connect a lead to v, and ψ (extended by zero in the lead) will be a resonant state associated to a resonance on the real axis.
Hence, we will suppose for contradiction, that, for any graph (V, E) with no loops and minimal degree ≥ 3, we have (The closed quantum graph (V, E, L) has an eigenfunction vanishing on a vertex) =⇒ ( The lenghts (Le)e∈E are rationally dependent) .
To reach a contradiction, we will argue in three steps.
Let f be an eigenfunction associated to λn 0 = λn 0 (Le 1 , ..., Le J ) ≥ 2, and let b0 be an oriented edge of (V, E) associated to an edge ej 0 ∈ E. We know that there exists a, φ ∈ R such that f b (x) = a cos(λn 0 x + φ). Since λn 0 ≥ 1 and Le j 0 > 2π, the function f b must vanish at some point x b ∈ (0, π) ⊂ (0, Le j 0 2 ). We may add a new vertex at x b , and glue a new graph with six edges at x b , just as in figure  2. This new graph contains no loop, and the degrees are all larger than 3. The lengths 1, ..., 6 are arbitrary. Extending the function ψ by zero on the six new edges, we obtain an eigenfunction in this new graph, satisfying the Kirchhoff boundary conditions. We may thus apply (69) to deduce that the lengths are rationally dependent. Since the length 1, ..., 6 are arbitrary and the lengths (Le j ) 1≤j≤J are rationally independent, we deduce that for each 1 ≤ j ≤ J, there exists aj ∈ Q such that