Absolutely Continuous Spectrum for Quantum Trees

We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates.


Introduction
Our aim in this paper is to establish the existence of bands of purely absolutely continuous (AC) spectrum for a large family of quantum trees. One of our motivations is to provide a collection of examples relevant for the Quantum Ergodicity result proven in [7].
For discrete trees, the problem is quite well understood when the tree is somehow homogeneous. The adjacency matrix of the (q + 1)-regular tree T q has pure AC spectrum [−2 √ q, 2 √ q] as is well-known [23]. If we fix a root o ∈ T q and regard the tree as descending from o, then the subtree descending from any offspring is the same (each is a q-ary tree), except for the subtree at the origin (which has (q + 1) children). We say that T q has two "cone types". It was shown in [21] that if T is a general tree with finitely many cone types, such that each vertex has a child of its own type, and all types arise in each progeny subtree, then the spectrum consists of bands of pure AC spectrum. This problem was revisited in [6], where these assumptions were relaxed to allow T to be any universal cover of a finite graph of minimal degree at least 2. In this case however, besides the bands of AC spectrum, a finite number of eigenvalues may appear. A natural question is whether AC spectrum survives if we add a potential. This is motivated by the famous Anderson model [9] where random independent, identicallydistributed potentials are attached at lattice sites. It remains a major open problem to prove such stability for the Anderson model on the euclidean lattice Z d , d ≥ 3 [32]. The first mathematical proof showing the stability of pure AC spectrum was obtained in [24] in the case of regular trees (Bethe lattice) under weak random perturbations, thus providing the first example of spectral delocalization for an Anderson model. More general trees were subsequently treated in [22,16], always in the setting of discrete Schrödinger operators. The stability of AC spectrum under perturbation by a non-random radial potential was proved in [21] in case of non-regular trees of finite cone type.
In this article we consider quantum trees, i.e. each edge is endowed some length L e and we study differential operators acting on the edges with appropriate boundary conditions at the vertices specified by certain coupling constants. The presence of AC spectrum for quantum trees appears to have been studied less systematically than in the case of discrete Schrödinger operators. In case of regular trees T q , it was shown in [12] that the quantum tree obtained by endowing each edge with the same length L, the same symmetric potential W on the edges and the same coupling constant α at the vertices, has a spectrum consisting of bands of pure AC spectrum, along with eigenvalues between the bands. The setting was a bit generalized quite recently in [13], where each vertex in a 2q-regular tree is surrounded by the same set of lengths (L 1 , . . . , L q ), each length repeated twice, similarly the same set of symmetric potentials (W 1 , . . . , W q ), and the boundary conditions are taken to be Kirchhoff. The nature of the spectrum is partly addressed, but the possibility that it consists of a discrete set of points is not excluded. Finally, it was shown in [1] that the AC spectrum of the equilateral quantum tree [12] remains stable under weak random perturbation of the edge lengths. The theorem however does not yield purity of the AC spectrum in some interval; one can only infer that the Lebesgue measure of the perturbed AC spectrum is close to the unperturbed one. We also mention the papers [17,30] which consider radial quantum trees, for which a reduction to a half-line model can be performed.
Our aim here is twofold. First, go beyond regular graphs. We are mainly interested in the case where the tree is the universal cover of some compact quantum graph. This implies the set of different lengths, potentials and coupling constants is finite, but the situation can be much more general than the special Cayley graph setting considered in [13]. We show in this framework that the spectrum will consist of (nontrivial) bands of pure AC spectrum, plus some discrete set of eigenvalues. Next, we consider random perturbations of these trees. We can perturb both the edge lengths and coupling constants. This setting is more general than [1], where the tree was regular and the coupling constants were zero. But our main motivation here is especially to derive the purity of the perturbed AC spectrum, along with a strong control on the resolvent, which is an important ingredient to prove quantum ergodicity for large quantum graphs. We do this in a companion paper [7].
1.1.1. Quantum graphs. Let G = (V, E) be a graph with vertex set V and edge set E. We will assume that there are no self-loops and that there is at most one edge between any two vertices, so that we can see E as a subset of V × V . For each vertex v ∈ V , we denote by d(v) the degree of v. We let B = B(G) be the set of oriented edges (or bonds), so that |B| = 2|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 map e : B −→ E by e((v, v ′ )) = {v, v ′ }. An orientation of G is a map or : E −→ B such that e • or = Id E .
A length graph (V, E, L) is a connected combinatorial graph (V, E) endowed with a map L : E → (0, ∞). If b ∈ B, we denote L b := L(e(b)).
A quantum graph Q = (V, E, L, W, α) is the data of: • A length graph (V, E, L), The underlying metric graph is the quotient . A function on the graph will be a map f : G −→ R. It can also be identified with a collection of maps (f b ) b∈B such that f b (L b − ·) = fb(·). We say that f is supported on e for some e ∈ E if f b ≡ 0 unless e(b) = e.
If each f b is positive and measurable, we define G f (x)dx := 1 When G = (V, E) is a tree, i.e., contains no cycles (which will be the case in most of the paper), we say that Q is a quantum tree, and we denote it by the letter T rather than Q, while the set G is called a metric tree and is denoted by T .

1.1.2.
Orienting quantum trees. Let T be a combinatorial tree, that is, a graph containing no cycles. We denote its vertex set by V (T) or just V , its edge set by E(T), and its set of oriented edges by B(T). In all the paper, we will often write v ∈ T instead of v ∈ V (T) to lighten the notations.
In this paragraph, we explain how we can present the tree T in a coherent view, that is to say, fix an oriented edge b o ∈ B(T), and give an orientation to all the other edges of T, by asking that they "point in the same direction as b o ".
More precisely, let us fix once and for all an oriented edge b o ∈ B(T), corresponding to an edge e o ∈ E(T). If we remove the edge e o from T, we obtain two connected components which are still combinatorial trees. We will write T + bo for the connected component containing t bo , and T − bo for the component containing o bo . Let v ∈ T + bo be at a distance n from t bo . Amongst the neighbours of v, one of them is at distance n − 1 from t bo : we denote it by v − , and say that v − is the parent of v. The other neighbours of v are at a distance n + 1 from t bo , and are called the children of v. The set of children of v is denoted by N + v . On the contrary, if v ∈ T − bo is at distance n from o bo , its unique neighbour at a distance n − 1 from o bo is called the child of v, and denoted by v + , and its other neighbours are its parents, whose set we denote by N − v . These definitions are natural if we see the tree at the left of Figure 1 as a genealogical tree. Let . One easily sees that e • b : V * −→ E(T) is a bijection, so that b • (e • b) −1 is an orientation of T. The map b serves to index all oriented edges: those in T + bo by their terminus, those in T − bo by their origin, and b o by its "midpoint" o. The latter makes sense once we turn T into a quantum tree T. We denote L v := L b(v) and W v := W b(v) . The metric tree T can be identified with the set A function on T will then be the data of ψ = (ψ v ) v∈V * , where each ψ v is a function of the variable On a quantum tree, we consider the Schrödinger operator , the set of functions (ψ v ) ∈ v∈V * W 2,2 (0, L v ) satisfying the so-called δ-conditions. Namely, for all v ∈ T + bo , . In a common convention we will refer to the α v = 0 case as the Kirchhoff-Neumann condition. Remark 1.1. The above conventions mean that we see T as a doubly infinite genealogical tree. This is what we called the coherent view; it can also be pictured by saying that we imagine an electric flow moving from T − bo to T + bo . There is another way of orienting the graph which we call the twisted view. This is done by turning b o into a V-shape and viewing V (T) as offspring of o. See Figure 1 for an illustration; here one should think that o is a source from which the electric flows moves outwards. When necessary to highlight this genealogical structure, we will write T + o for the set of offsprings of o. Each vertex v has a single parent v − and several children; all the edges take the form {v, v − } for a unique v.
The link between the two views is immediate: functions on T + bo in both views coincide, while on T − bo , one replaces b byb and derivatives take a sign. Hereb = (t b , o b ) is the edge reversal of b. Hence, in the twisted view 1 , all functions in the domain of H satisfy (1.3).
Given v ∈ V * , z ∈ C, let C z (x) and S z (x) be a basis of solutions of the problem Then any solution ψ v of the problem satisfies .
If W v ≡ 0 then the basis of solutions is It is a standard fact that S z (x), C z (x) are analytic functions of z ∈ C (see for instance [28,Chapter 1]).
1 These views of the tree have the advantage of avoiding the assumption that T has a special "root" vertex of degree one [1,30]. Such assumption simplifies the orientation a bit, but is not satisfied in many natural situations. As will be clear later, we will only need to study functions supported in T ± bo , which is why we did not specify what happens on bo. But one could specify that ψo from the coherent view becomes (ψ (1) o , ψ (2) o ) in the twisted view, with ψ (1) o (x) = ψo(x + Lo 2 ) and ψ

1.2.
Trees of finite cone type. We define a cone in T to be a subtree of the form T + b or T − b , for some b ∈ B(T). Each cone T + b has an origin t b , and each cone T − b has an end o b .
We say that two quantum cones T + b and T + b ′ are isomorphic if there is an isomorphism of combinatorial graphs ϕ : are defined the same way. We say that T is a tree of finite cone type if there exists b o ∈ B(T) such that: (i) There are finitely many non-isomorphic quantum cones T + (v − ,v) as v ∈ T + bo . (ii) There are finitely many non-isomorphic quantum cones T − (w,w + ) as w ∈ T − bo . Here (t bo ) − = o bo and (o bo ) + = t bo . Note that in a regular tree, all cones T ± b are isomorphic, but a necessary condition for it to be a quantum tree of finite cone type, is that its edges and vertices be endowed with finitely many lengths, potentials and coupling constants. 2 If T is a tree of finite cone type, with b o ∈ B(T) fixed, we may introduce a type function ℓ : since the corresponding isomorphism respects this information. Hence, any coherent quantum tree T of finite cone type comes with the following structure: encoding the lengths, potentials and coupling constants, respectively. More precisely, b o is endowed a special length L o and potential If we take the twisted view instead, we only need one alphabet A = A + ∪ A − and one corresponding matrix M = (M i,j ) i,j∈A .
A trivial example is the equilateral, (q + 1)-regular quantum tree, with identical potentials W on each edge and identical coupling constant α on each vertex [12]. In this case, all vertices in T ± bo have the same type, and we get two 1 × 1 matrices M = N = q . An important class of examples comes from universal covers of finite undirected graphs. More precisely, if G is a finite undirected graph and T is its universal cover, then T is a combinatorial tree satisfying condition (i). If we endow G with a quantum structure G and lift it to T in the natural way, then the corresponding T will be a quantum tree of finite cone type.
Quantum trees of finite cone type satisfying (a)-(c) will be our basic, "unperturbed" trees. We denote the Schrödinger operator (1.2) acting in this setting as H 0 . Later on, we shall study random perturbations of these trees, and denote the corresponding operator by H ω ǫ , where ǫ is the strength of the disorder. We make the following assumption on T: 2 Also note that it is not required that there are finitely many non-isomorphic quantum trees T − as v ∈ T + bo . To illustrate this point, consider the binary tree (so each vertex has 3 neighbors except for the special root ⋆ with 2 neighbors), let bo = (⋆, v), with v either neighbor. Then all cones T + b ⊂ T + bo look the same; they are binary trees. However, the backward cones T − b are distinct in each generation, because they "see" the special root at distinct distances. Despite this, T has finite cone type. See Remark 1.3 below for a discussion of this condition. We may now state a first theorem, which describes the structure of the spectrum of H 0 = H T on a tree T of finite cone type. We denote by G z 0 (x, y) = (H 0 − z) −1 (x, y) the Green's function of H 0 . Theorem 1.2. Let M, N satisfy (C1*). Then the spectrum of H 0 consists of a disjoint union of closed intervals and of isolated points: σ(H 0 ) = ( r I r ) ∪ P, where the I r are closed intervals, and P is a discrete set. The spectrum is purely absolutely continuous in the interior of each bandI r . For λ ∈I r , and for any v ∈ T, the limit G λ+i0 0 (v, v) exists and satisfies Im G λ+i0 z,0 be the Weyl-Titchmarsh functions of H 0 as defined in [1], see (2.2). Let R ± λ,0 = R ± λ+i0,0 when the limit exists. Theorem 1.2 implies that Im R + λ,0 (v)+Im R − λ,0 (v) > 0 inI r . We will need the stronger property that Im R + λ,0 (v) > 0 for all v. For this, we introduce the following strengthening of (C1*).
(C1) The quantum tree T is the universal cover of a finite quantum graph G of minimal degree ≥ 2 which is not a cycle.
Remark 1.3. Condition (C1*) means that on T + bo , any cone type l ∈ A + appears as offspring of any k ∈ A + after a finite number of generations, and similarly for T − bo . It is not required that cone types in A − appear in T + bo -we only need the matrices M and N to be separately irreducible. We can also allow for "rooted" trees where the root o has degree one. In this case the situation is a bit simpler actually; we only have to deal with one matrix M . Condition (C1*) applies in particular to trees with a "radial periodic" data, i.e. data that are periodic functions of the distance to the origin (such as some examples appearing in [30]).
Assumption (C1) implies (C1*) (see Remark 3.7), and is in fact more restrictive. In particular, T is "unimodular", that is, all data is somehow homogeneous as we move along the tree. This excludes for example the binary tree and more generally radial periodic trees, where the root plays a special role. However, such unimodular trees are still very general, they are actually the most interesting for us, and many techniques (such as a reduction to a half-line model) fail to tackle them. Even in the very simple case where the base graph G is regular but the edge lengths are not equal, the lifted structure in general will be neither radial periodic, nor identical around each vertex (in contrast to [13]).
Note that the case where G is a cycle is already known when the couplings are zero. In this case H T is just a periodic Schrödinger operator on R (of period ≤ |G|), it is well-known that the spectrum is purely AC in this case [29,Section XIII.16].
Theorem 1.4. If T satisfies (C1), then the spectrum of H 0 consists of a disjoint union of closed intervals and of isolated points: σ(H 0 ) = ( r I r ) ∪ P, where the I r are closed intervals, and P is a discrete set. The spectrum is purely absolutely continuous in the interior of each bandI r . For λ ∈I r , the limit R + λ+i0,0 (v) exists for any v ∈ V and satisfies Im R + λ,0 (v) > 0. Remark 1.5. In Theorems 1.2 and 1.4, it is not excluded in principle that r I r = ∅, i.e. the spectrum consists of isolated points. We think this never happens for infinite quantum trees of finite cone type with the δ-conditions we consider, i.e. we believe these should always have some continuous spectrum. We did not find such a result in the literature however. This is why we dedicate Section 4 to prove the following: if T satisfies either: (1) assumption (C1) and has a single data (L, α, W ) (all edges carry the same length, coupling and symmetric potential), (2) or has a general data (L e , α oe , W e ) e∈E(G) , but the finite graph G is moreover Hamiltonian, then H T always has some continuous spectrum, i.e. rI r = ∅. Recall that a finite graph is Hamiltonian if it has a cycle that visits each vertex exactly once. Note that as a discrete tree, T may cover many different graphs. We only need one of these finite graphs to be Hamiltonian. For example, we can consider any regular tree, despite the fact that some regular graphs (like the Petersen graph) are not Hamiltonian. In particular, the Cayley tree considered in [13] can be realized as the universal cover of the complete bipartite graph K 2q,2q , which is Hamiltonian. For this, use the fact that K 2q,2q has a proper 2q-edge-colouring and put the same length/potential on edges of the same colour. The lift of this is then a tree which has the same data around each vertex, and we may take L q+j = L j , W q+j = W j to be in the setting [13]. Then our theorems imply this tree has nontrivial bands of pure AC spectrum, thus enriching the results of [13]. Again, this is just one very special application of our framework.
1.3. Random perturbations of trees of finite cone type. Fix a quantum tree T satisfying (C1). As explained in Remark 3.7, any such tree is a tree of finite cone type. We fix an edge e ∈ E(T), and see our quantum tree in the twisted view (in which all vertices are descendent of a vertex o), so as to deal with a single alphabet A and a corresponding matrix M . We denote the lengths and coupling constants of the unperturbed tree T by (L 0 v ) v∈V * and (α 0 v ) v∈V . These can also be denoted (L 0 i ) i∈A∪{o} and (α 0 i ) i∈A . We assume there are no potentials on the edges and the couplings are nonnegative: We now want to analyze random perturbations of T. For this purpose, we introduce a probability space (Ω, F , P), a family of random variables ω ∈ Ω → (L ω v ) v∈V * representing random lengths, and a family of random variables ω ∈ Ω → (α ω v ) v∈V representing random coupling constants. In principle we could also consider random potentials ω ∈ Ω → (W ω v ) v∈V * , however here we assume there are no potentials on the edges even after perturbation. We also assume the perturbed couplings are nonnegative: W ω v ≡ 0 and α ω v ≥ 0. We make the following assumptions on the random perturbation (see Remark 1.1 for the notation T + o ): (P0) The operator H ω ǫ is the Laplacian on the edges acting on W 2,2 (0, L ω v ), satisfying δ-conditions with coupling constants (α ω v ) v∈T , which are assumed to satisfy o that share the same label, the restrictions of the random variables (α ω , L ω ) to the isomorphic forward trees of v and w are identically distributed. Remark 1.6. Assumptions (P1) and (P2) hold, in particular, for independent identically distributed random variables (which is the main case we have in mind).
We shall consider intervals I lying in the interior of the unperturbed AC spectrum: where I r are given in Theorem 1.4. We will also need to ensure that the various sin √ λL v do not vanish. More precisely, by (P0), the perturbed lengths all lie in j∈A∪{0} [L j,min (ǫ), L j,max (ǫ)], where L j,min (ǫ) = L 0 j − ǫ and L j,max (ǫ) = L 0 j + ǫ. We then assume (1.8) where the set D = D ǫ is a "thickening" of the Dirichlet spectrum, given by .
This ensures that sin √ λL ω v , sin √ λL 0 v = 0 for any λ ∈ I, v ∈ T and ω. Recall that the Weyl-Titchmarsh functions R + z (v) will be introduced in (2.2). Introduce the following condition: (Green-s) There is a non-empty open set I 1 and some s > 0 such that for all b ∈ T, sup λ∈I 1 ,η∈(0,1) Condition (Green-s) implies in particular that the spectrum in I 1 is purely AC, as long as it stays away from the Dirichlet spectrum, see Appendix A.2. Here (Green-s) refers to "Green's function" and the moment value s. In fact, such inverse bounds on the WT function imply moments bounds on the Green's function; see Corollary 2.5. Introduce the following assumptions: (C0) The minimal degree of T is at least 3.
(C2) For each k ∈ A, there is k ′ with M k,k ′ ≥ 1 such that for any l ∈ A: M k,l ≥ 1 implies M k ′ ,l ≥ 1. The second assumption ensures that each vertex v ∈ T has at least one child v ′ such that each label found in N + v can also be found in N + v ′ . See [6] for examples of such trees. Theorem 1.7. Let T satisfy (C0), (C1), (C2) and (α, L) satisfy (P0), (P1) and (P2), and be without edge potentials. Let I ⊂ Σ be compact with I ∩ D = ∅. Then for any s > 1, we may find ǫ 0 (I, s) such that (Green-s) holds on I for any ǫ ≤ ǫ 0 . In particular, σ(H ω ǫ ) has purely absolutely continuous spectrum almost-surely in I.
The "in particular" part is due to Theorem A.6. In the above theorem, the disorder window ǫ 0 (I, s) depends on the value of the moment s. We can actually obtain a disorder window valid uniformly for all s, but at the price of assuming some regularity on the δ-potential : (P3) For any v ∈ T, ℓ(v) = j, j ∈ A, the distribution ν j of α ω v is Hölder continuous : there exist C ν > 0 and β ∈ (0, 1] such that for any bounded I ⊂ R, This holds e.g. if the ν j are absolutely continuous with a bounded density (then β = 1). Theorem 1.8. Suppose in addition to the assumptions of Theorem 1.7 that (P3) is satisfied. Then there exists ǫ 0 (I) such that for any ǫ ≤ ǫ 0 and any s ≥ 1, (Green-s) holds on I.

Green's function on quantum trees
The aim of this section is to derive quantum analogs for the well-known recursive formulas of the Green's function on combinatorial trees. These identities will play a key role in the spectral analysis of the quantum tree, and may be of independent interest. In fact, we shall also need them when studying quantum ergodicity in [7]. Some of these identities appeared before in [1].
In all this section, we fix a quantum tree T, and denote by W 2,2 max (T) the set of ψ = (ψ v ) such that ψ v ∈ W 2,2 (0, , α vx = 0, and the lengths, potentials and coupling constants be the same as those of T + b on the rest of the edges. In a similar fashion, we define T − the set of ψ ∈ W 2,2 max (T ± u ) satisfying δ-conditions on inner vertices of T ± u , then for any z ∈ C + := H := {z ∈ C : Im z > 0}, there are unique z-eigenfunctions V + z;u ∈ D(H max is not self-adjoint, as there are no domain conditions at u.

Versions of this lemma previously appeared in [1, Lemma A.2] and [17, Lemma D.15].
We give the proof in Appendix A for completeness.
Since for each z ∈ C + , G z satisfies the δ-boundary conditions in each of its arguments, we deduce that, whenever ). These quantities will therefore be denoted by G z (v, ·) and G z (·, v) respectively.
As in [1], we define the Weyl-Titchmarch (WT) functions for x ∈ T by .
Note that we take here the coherent point of view, which is why there is a negative sign in the definition of R − z (x). Given an oriented edge i.e. the denominator does not vanish, as follows from (2.1) and the proof of [1, Theorem 2.1(ii)]. The proof also shows that z → ζ z (b) is holomorphic on C \ (a, ∞) and real-valued on (−∞, a].
In the case of combinatorial trees, [4]. In fact, by the multiplicative property of the Green function, we have . In the case that T is the (q + 1)-regular tree with equilateral edges, with identical coupling constants and potentials, then ζ z (b) is the quantity µ − (z) in [12], and is independent of b. Moreover, the limit µ − (λ) = lim η↓0 µ − (λ + iη) exists in this case, provided that λ is not in the Dirichlet spectrum, i.e., that sin(λL) = 0.
Finally, for the quantum Cayley graphs of [13], the ζ z (b) coincide with the multipliers µ m (z). Hence, there are finitely many distinct ζ z (b). Moreover, in this setting, ζ z (b) = ζ z (b), and ζ z (gb) = ζ z (b), where g is an element of the group acting on the graph.
Given an oriented edge b, we will denote by N + b the set of outgoing edges from b, i.e. the set of b ′ with o b ′ = t b and b ′ =b. Lemma 2.3. Let z ∈ C + . We have the following relations between ζ z and the WT functions R ± z : Moreover, , Finally, for any path b 1 , . . . , b k , we have . .
The first part of (2.5) follows by the Wronskian identity .
For the second part, we showed that . It follows from (2.13) and (2.14) that .
As previously observed, the Wronskian is constant on each , proving the other equality. Finally, by the first part of (2 For the following lemma, fix o, v ∈ V and consider the WT functions (2.2). Assume that We will need the notion of Herglotz functions [14] throughout the paper. A Herglotz function (a.k.a. Nevanlinna function or Pick function) is an analytic function from C + to C + . Herglotz functions form a positive cone: if f 1 , f 2 are Herglotz and a 1 , a 2 are positive constants, then a 1 f 1 + a 2 f 2 is Herglotz. Composition of two Herglotz functions is again a Herglotz function. The functions z → √ z and z → −1/z for example are Herglotz.
Every Herglotz function f has a canonical representation [14, Theorem II.I] of the form where A and B are constants and m is a Borel measure satisfying R (1 + t 2 ) −1 dm < ∞.
Then we may express arbitrary. Hence,

AC spectrum for the unperturbed tree
The aim of this section is to prove Theorems 1.2 and 1.4. Let T be a quantum tree of finite cone type, with the structure described in § 1.2. Given This notation is simply analogous to the one introduced in Section 1.1.2, and does not mean that ζ z is a function of the terminus alone. It simply means that each discrete edge in T + bo can be specified by indicating the terminus alone. We also let ζ The matrix elements M j,k were defined in §1.2(b). The system (3.1) is reminiscent of the finite system of equations that appears in the combinatorial case [21,6] for ζ z j = ζ z v − (v). In order to put it in a nicer form, we denote h j = S z (L j )ζ z j . Then we get the following system of polynomial equations: Sz(L j ) . An analogous system of equations involving the matrix N = (N i,j ) arises when considering cones in T − bo . We restrict ourselves to the above system; the other one is analyzed similarly.
We mention that a similar system of equations in a more special framework appeared recently in [13, eq. (4.8)]. In this case, one has M j,j = 1 for each j and M j,k = 2 for k = j.
Our aim in the following is to control the values of ζ λ+iη j as η ↓ 0. For the models [12,13], the ζ z j are uniformly bounded. The following simple criterion gives a sufficient condition for this to happen. Note the condition M j,j > 0 below implies that each vertex of label j has at least one offspring of its own type. Later we will relax that restriction.
The lemma implies in particular that |ζ λ+i0 j | ≤ 1 M j,j for any λ ∈ R. There are many models of interest for which the condition of Lemma 3.1 is not satistfied, so we next consider the general case. Now the limit ζ λ+i0 j may no longer exist, but we aim to show this problem can only occur on a discrete subset of R.
Proof. We follow the strategy in [6, §4]. The aim is essentially to decouple the system (3.2) and show that each h j satisfies an algebraic equation Q j (h j ) = 0. For this, we will use an algebraic tool from [25].
Let λ 0 ∈ R, and let We will show that z → J z is not the zero element of K ′ . For this, we first study the asymptotics of J z as z → −∞. Take z = −r 2 with r > 0 large. We remark that This follows from classical estimates [28,Chapter 1]. In fact, On the other hand, since h j is Herglotz (see Remark A.3), it has a representation of the form (2.15). If t 0 = inf σ(H T ), we also know from Remark 2.2 that h j (λ) is well-defined and real-valued for λ < t 0 . By [34,Theorem 3.23], the measure m is thus supported on [t 0 , ∞). Hence, for large r (say r 2 > −t 0 + 1), where we used that h j (−r 2 ) = lim η↓0 h j (−r 2 + iη) and dominated convergence (recall that Using dominated convergence again, we see that Therefore, recalling that the C j were defined in (3.4), we find that as r → ∞, , it follows that it cannot vanish identically on any neighbourhood It follows by [25,Proposition VIII.5.3] that each S z ζ z j is algebraic over K. By the Newton-Puiseux theorem (see e.g. [31, Theorem 3.5.2]), each h j thus has an expansion of the form Here m ∈ Z, d ∈ N, and the entire series n≥0 a n z n has a positive radius of convergence. In particular, z → S z ζ z j is analytic near any λ ∈ N λ 0 \ {λ 0 }. The set D corresponds to those λ 0 for which m < 0 in the Newton-Puiseux expansion at λ 0 , and the set D ′ corresponds to those λ 0 for which d > 1.
Our next aim is to show that all WT functions have a positive imaginary part on most of the spectrum.
Let σ D be the union of the Dirichlet spectra: We would like to index the WT functions by vertices, but the notation So we take the convention that This keeps with the convention of § 1.1.2 of indexing functions ψ(b) by their terminus 4 on T + bo and their origin on T − bo . As there are finitely many types of The notation is probably a bit awkward since . We stress however that R ± z is not a function of the vertex o b alone but depends on the whole directed edge b, so it should really be read as a function ψ(b), which we index by the terminus. By some abuse of notation, we assume the discrete sets D, D ′ of Proposition 3.2 are the same for the system analogous to (3.2) which involves the matrix (N i,j ).
. Then the limits R + λ (j) exist for λ ∈ R \ (D ∪ σ D ) and j ∈ A + . This follows from Proposition 3.2 and (2.4), which implies that for some j o ∈ A + , which exists by Proposition 3.2, so R + λ (o bo ) exists by (2.4). Similarly the result for the (N ij ) system implies the existence of ζ λ (b o ) and R − λ (t bo ). Finally if t bo has type j o ∈ A + , then R + λ (t bo ) = m k=1 M jo,k R + λ (k) − α t bo by (2.12), which exists by the previous paragraph.
. In particular, their zeroes do not accumulate. Hence, there is a discrete set D ′′ such that We may actually generalize the result of Remark 3.3 as follows.
Proof. By symmetry it suffices to prove (a) for is finite. Note that we already know is finite by (2.5). So consider any oriented edge b = (t bo , v). Applying (2.6) to b instead of b, we may express in terms of some C z , S z functions, plus ζ z (b ′ ) for b ′ ∈ N + b . One of them is ζ z ( b o ), whose limit on the real axis exists from Remark 3.3. The rest are precisely those with b ′ ∈ N + bo \ {b}, which also exist by Proposition 3.2. Thus, exists for any b = (t bo , v). By induction we get existence for any (v − , v) ∈ T + bo . It follows from (2.5) that Using (2.13), this implies G λ (v, v) exists. We now observe that In fact, by (2.7), as claimed. Using Proposition 3.2, we thus deduce the existence of G λ (w, w) for all w ∈ T + bo . Similarly the existence of w ∈ T − bo follows from the analog of Proposition 3.2 with the (N i,j ) system. Finally using (2.13) we see that (a) implies G λ (v, v) = 0.
The same conclusion holds if Im G λ+i0 (w, w) = 0 for some w ∈ T. (iii) For any v ∈ T, we have Proof. We first note that ζ λ j = 0, due to the relation λ) and the fact that the h k are finite.
Denote R λ k = R + λ+i0 (k). Suppose that Im R λ k > 0 for some k ∈ A + and let l ∈ A + . Then by (C1*), (M n ) l,k ≥ 1, so if v has label l and w has label k, there is a path (u 0 , . . . , u r ) with u 1 = v and u r = w. Denote b j = (u j−1 , u j ). Then applying (2.17) repeatedly, where the sum runs over all (r − 1)-paths (e 2 ; e r ) outgoing from b 1 , and the last inequality holds because Im R λ (o br ) = Im R λ k > 0 and all ζ λ j = 0. So under (C1*), if Im R λ j > 0 for some j ∈ A + , then Im R λ k > 0 for all k ∈ A + . So if Im R λ j = 0 for some j ∈ A + , then it must be zero for all j ∈ A + . The proof for R − λ+i0 is the same.
for all x ∈ e, contradicting the fact that C λ and S λ are linearly independent. We thus showed that {λ ∈ R \ D 0 : Lemma 3.6. If T satisfies (C1*), then: has a discrete set of zeroes. The same holds for σ(H) (ii) σ(H) is a union of closed intervals and isolated points, r I r ∪ P. The limits G λ+i0 (v, v) exist in the interiorI r and satisfy Im G λ+i0 (v, v) > 0, for any v ∈ T. (iii) The spectrum of H T is purely absolutely continuous in any compact subset K ⊂I r .
This proves the first part of (i). For the second part, suppose Im G λ (w, w) = 0 for some w ∈ T. By Lemma 3.5, this implies Im R + λ (j) + Im R − λ (k) = 0. Hence, λ must lie in the preceding discrete set of zeroes.
Finally, if K is a compact subset ofI r , we know that G λ (v, v) is uniformly bounded, and the same holds for R ± λ (v). In particular, if v = o e and ψ is supported in e, we get using respresentation (A.2) along with (2.13) that sup λ∈K | ψ, G λ ψ | < ∞. The claim follows by the density of the linear span of such ψ.
This completes the proof of Theorem 1.2. We next move to Theorem 1.4. Remark 3.7. Condition (C1) implies (C1*). In fact, as remarked in [6], all cone types are indexed by the directed edges of the finite graph G. If we consider the universal cover T rooted at the midpoint o of some b o ∈ B(G) (here o is not viewed as an added vertex, just a reference point), this means that the type of each vertex v ∈ T is determined by a directed edge, so there are at most |B(G)| types. By [27, Lemma 3.1], we know the non-backtracking matrix of B(G) is irreducible. This implies that if T is considered in the twisted view, and if M is the single matrix over some alphabet A encoding all cone types, then M satisfies: for any k, l ∈ A, there is n(k, l) such that (M n ) k,l ≥ 1. In particular, (C1*) holds if we take the matricesM , N encoding the types in T + bo and T − bo , respectively.
Proof of Theorem 1.4. Since (C1) implies (C1*), we already know that σ(H 0 ) has the structure given in Theorem 1.2. Let λ ∈I r be in the interior of an AC band. Within the twisted view, all vertices are offspring of o and we deal with the single, combined alphabet A. Under the stronger assumption (C1), we know the larger matrix M is irreducible. Consequently, if we suppose that Im R + λ+i0 (j) = 0 for some j ∈ A, then the statement in Lemma 3.5 (i) now implies that Im R + λ+i0 (j) = 0 for all j ∈ A. Now let v ∈ T. We know Im G λ (v, v) > 0, so by (2.13), Im R + λ (v) + Im R − λ (v) > 0, so either Im R + λ (v) > 0 or Im R − λ (v) > 0 by the Herglotz property. In the former case we are done. Suppose that Im R + λ (v) = 0. Say v = t b for some b ∈ B(T) and ℓ(v) = j.
. As mentioned in Remark 3.7, under (C1), all cone types are indexed by the directed edges of G, in particular T + b is one of the finitely many nonisomorphic cones 5 . In other words, Im R + λ (o b ) = Im R + λ (r) for some r ∈ A. Hence, Im R − λ (t b ) = 0. We thus get

Examples of nontrivial spectrum
For Theorem 1.4 to be interesting, we'll need to know that σ(H 0 ) is not reduced to the isolated points P. Our aim in this section is to give some examples in which this can be proved. We believe the phenomenon to be true for a wider class of examples.

Equilateral trees.
Let G be a discrete graph of minimal degree ≥ 2 and T = G its universal cover. We know from [10, Section 1.6] that the spectrum of the adjacency matrix σ(A T ) has a continuous part. Actually their argument remains valid for the normalized adjacency matrix P f (x) = 1 d(x) (Af )(x) (and also if we add potentials). Consequently, using [11,Theorem 3.18], the induced quantum tree with equilateral edge length, identical symmetric potentials, and identical coupling constants, will also have some continuous spectrum. Using Theorem 1.4, we can now conclude: If G is a graph of degree ≥ 2, if T = G is its universal cover, and we endow each edge of T with the same length L and potential W , and each vertex with the same coupling constant α, then σ(H T ) consists of non-empty bands of purely absolutely continuous spectrum, and possibly some isolated eigenvalues.
This generalizes the case of regular trees previously considered in [12,33]. We may easily extend this to graphs with several lengths which are rationally dependent. More precisely, if in G, we have L j = n j L for some n j ∈ N * , add n j vertices of degree 2 to the edge e j , with Kirchhoff-Neumann conditions. Then using the previous claim, we see that T also has nontrivial AC spectrum in this case.

4.2.
An argument of Bordenave-Sen-Virág. We now consider the non-equilateral case. For this, we start by adapting an argument from [10] to quantum graphs.
We begin with some definitions, which appear in a more general framework in [10]. Let G be a discrete graph and T = G its universal cover. A labeling (or colouring) of the vertices of T is a map η : V (T) → Z. With respect to a given labeling, we call a vertex v: (a) prodigy if it has a neighbour w with η(w) < η(v) and such that all other neighbours of w also have label less than η(v), (b) level if it is not a prodigy and if all of its neighbours have the same or lower labels, (c) bad if it is neither prodigy nor level.
The tree T = G is equipped with a natural unimodular 6 measure on the space of rooted graphs, namely P = 1 |G| x∈G δ [ G,x] . We say the labeling η on T is invariant if there exists a unimodular probability measure on the set of coloured rooted graphs, which is concentrated on {[T, v, η]} v∈T . See e.g. [4, Appendix A] for some background on coloured rooted graphs.
Let S ⊂ ℓ 2 ( G) be a subspace and let P S be the orthogonal projection onto S. We say that S is invariant if P S (gv, gw) = P S (v, w) for any g ∈ Γ, where Γ is the group of covering transformations with G/Γ ≡ G.
Given an invariant subspace S ⊂ ℓ 2 ( G), we define its von-Neumann dimension by A line ensemble in T is a disjoint union of bi-infinite lines (l i ). More precisely, L : Abusing notation, we then let L = {e : L(e) = 1}, which gives a subgraph consisting of disjoint lines.
We say a line ensemble L is invariant if there exists a unimodular probability measure on the space of weighted rooted graphs, which is concentrated on We say that T is Hamiltonian if there exists an invariant line ensemble L that contains the root with probability 1. In particular, the (q + 1)-regular tree T q is Hamiltonian, since it covers the complete bipartite (q + 1)-regular graph on 2(q + 1) vertices, which is Hamiltonian.
We may now state our adaptation of [10, Theorem 1.5] to quantum trees. Here, if G is a finite graph, we denote by G = G(α, L, W) the quantum graph obtained by endowing each edge with a length L e , a potential W e and each vertex a coupling constant α v , so the Schrödinger operator H = −∆ + W acts with δ-conditions. We say that T = G if T = G is endowed with the lifted structure α v = α πv , L (u,v) = L (πu,πv) and W (u,v) = W (πu,πv) .
Suppose on the contrary that S λ (L b ) = 0 for all b. Letφ = ϕ| V . We claim that and uv := (u, v). In fact, Let E λ ⊂ ℓ 2 (T) be the set of functions satisfying this eigenvalue equation, i.e. ψ ∈ E λ if and only if .
Note that E λ is invariant. In fact, let M λ = A λ − W λ . Then M λ is self-adjoint. This is because all weights are real-valued and symmetric. Moreover, Let C be a Hamiltonian cycle in G, so its lift L is a line ensemble as in Remark 4.1. Using the line ensemble, we may use the construction of [10, Theorem 1.5] to define for any k ∈ N * an invariant labeling η k : V (T) → Z k of the vertices of T by integers which satisfies: • b := P(o is bad ) ≤ 1/k, • vertices in L with η k (v) = 0 are prodigy. Vertices in L with η k (v) = 0 are bad, • vertices outside L are level.
In our case, all vertices are in L, so there are no level vertices. We now argue as in [10,Theorem 2.3]. Here the situation is simpler as there are no level vertices. Let B be the space of vectors which vanish on the set of bad vertices. Then We show E ′ is the trivial subspace by induction on the label j, showing that from low to high, any f ∈ E ′ vanishes on vertices with label j. Remember vertices v can only be prodigy or bad.
Recall that we have finitely many labels. Let j 0 be the smallest label and let v be of label j 0 . If v is a bad vertex, then f (v) = 0 since f ∈ B. Note that v cannot be a prodigy. Hence f (v) = 0 on vertices of smallest label. Now assume f ∈ E ′ vanishes on all vertices with label strictly below j. Since f ∈ B, we know f vanishes on bad vertices. If v is a prodigy vertex of label j, then v has a neighbour w such that f vanishes on w and all neighbours of w, except perhaps v. But (4.1) gives It follows that there is no ℓ 2 function on T such that A λ ψ = W λ ψ. By [15], it follows that there is no L 2 function on T such that H T ϕ = λϕ. In other words, λ is not an eigenvalue of H T (contradiction).

4.3.
More examples. Let T = G. We now show the spectral bottom a 0 = inf σ(H T ) is strictly below the smallest Dirichlet value. By virtue of Proposition 4.2, if G is Hamiltonian, this implies a 0 is not an eigenvalue. In particular, a 0 is not an isolated spectral value, so in view of Theorem 1.4, there is some pure AC spectrum near a 0 .
Recall that if Q j are the quadratic forms associated to operators H j , then In T there are finitely many different kinds of edges (lengths and potentials). To each oriented edge b, we associate the smallest Dirichlet: the smallest E such that S E (L b ) = 0. Denote the least of those values by E D and let (v, w) be the edge on which it is attained (choose one of them if there is more than one edge with the same lowest Dirichlet value).
Consider the quantum star graph around v, with the usual δ-condition at v, and Dirichlet conditions at the extremities w ′ ∼ v. Denote this (compact) graph by ⋆ and let E 0 be its smallest eigenvalue. We claim that For the first inequality, let For the second inequality, note that if f ∈ D(H ⋆ ) and H ⋆ f = Ef , then on the edge b, Due to the Dirichlet conditions at extremities of the star, f Evaluating at x b = 0, the centre of the star, Denote the left hand side of (4.2) as a function of E as Z(E). A solution Z(E) = α v will be an eigenvalue of the star graph. Let us consider the behaviour of Z(E) as E increases to E D . We first show that as E approaches E D from below, S E (L vw ) → 0 from above. For this, note that: • by [28, Theorem 6(a)], S E D (x) has exactly two zeros on [0, L vw ]. These are thus {0, L vw }. If E < E D , since S E (0) = 0 and S ′ E (0) = 1, we know that S E is positive near 0. If we show that its first zero on (0, ∞) occurs after L vw , this will imply that S E (L vw ) > 0, which is what we seek.
• We thus check that if E < E D , then the first zero of S E on (0, ∞) occurs after the first zero of S E D . For this, let , which is negative until the first zero of S E or S E D . Suppose for contradiction that the first zero of S E (call it L E ) is before the first zero of S E D . Then we get that f (0) = 0, f (L E ) > 0, but f ′ (x) < 0 on (0, L E ), which is absurd. This proves the claim.
On the other hand, from the proof of Proposition 3.2, we know that Together we have Since Z is continuous, there is a solution to Z(E) = α v strictly below E D , as claimed.

AC spectrum under perturbations
We now aim to prove Theorem 1.7. For this, we adapt the approach of [22], see also [16] for some earlier ideas.
A very sketchy outline of the argument is as follows. Our results in the previous sections tell us that we have a good control over the unperturbed operator: it has pure AC spectrum in Σ, and all relevant spectral quantities such as the Green's functions and WT functions have a limit on Σ, which has a strictly positive imaginary part. Let H z v be such a spectral quantity to be chosen later, where z ∈ C + and v ∈ V (T). The aim is now to prove an L p -continuity estimate in mean with respect to the disorder ǫ. More precisely, in some semi-metric γ on H, see (5.1), we aim to show that lim ǫ↓0 sup z∈I+i(0,1) E(γ(h z v , H z v ) p ) = 0, where h z v is the analogous spectral quantity for the perturbed operator (equal to H z v when ǫ = 0). Such a uniform stability result directly implies almost-sure pure AC spectrum in the combinatorial case, by classical results. In our case we will have to work further (Section 5.3 and Appendix A.2).
The important question now is which Herglotz function to choose for H z v . In the combinatorial case it is natural to take ζ z (b), with b = (v − , v). We considered something close in Proposition 3.2, namely S z (L b )ζ z (b). For the present continuity considerations, seems to behave better. Still, the function √ zS z (L b )ζ z (b) will also play an important role in fixing the disorder window later on (Appendix B). Of course it can be argued that all such quantities are related in Section 2, but one needs to be careful because the aim is roughly to prove strict contraction estimates on , so adding/multiplying terms to h z w is not a very good operation, although there are partial answers (Lemma 5.1 and Lemma 5.2).
We have not discussed how the L p -continuity actually proceeds; we outline the proof in § 5.2 after giving some important expansion estimates in § 5.1.
We will use the Möbius transformation C(z) = z−i z+i that sends the upper half plane model H isometrically to the disk model. Its inverse is C −1 (u) = i 1+u 1−u . Note that if, for g, h ∈ H, we set In fact, The following is a more adequate replacement of [22, Lemma 1] to our framework.
Lemma 5.1. Let K be a compact subset of the hyperbolic disc D. Then there exists a continuous function C K : R + −→ R + , such that C K (0) = 0, and for all z ∈ K and for all z ′ ∈ D, for all λ i ∈ C such that |λ i | ≤ 1.
More explicitly, if r K = max z∈K |z| < 1, we can take C K (t) = Proof. First assume λ 1 = 0. Suppose z = z ′ . We have where, to obtain the first inequality, we used that |λ i | ≤ 1.
We now consider T in the twisted view. If T is a tree with parameters The notation may seem a bit confusing since the WT function is evaluated at v − instead of v. However, this is in accordance with the notations of § 1.1, where the oriented edges are indexed by their endpoint, and where we write Then the δ-conditions (1.3) applied to V + z;o (x) give We shall assume the coupling constants α v ≥ 0 and potentials W ≥ 0. In this case we can ensure that h z v and H z v are Herglotz functions (Lemma 2.4), so their Cayley transform lies in D. In this case, g z v and Γ z v are also Herglotz by (5.5).

Remark 5.3.
Assume there is no potential on the edges: W v ≡ 0. Then the functions h z v and g z v are also connected by the following relations: √ zLv . From this, we find as previously observed in [1]. We also remark that we can invert the Möbius identity We finally define Lemma 5.4. Let K be a compact subset of the hyperbolic disc, and assume that z varies in a compact subset such that H z v ∈ H and C(Γ z v ) ∈ K for all v. Then (see equations (5.6), (5.7) and (5.8) below for the definition of the quantities q, α and Q) We will apply this when K = v {C(Γ z v ) : z ∈ I +i[0, 1]}, where I is a compact interval on which Γ λ+i0 v exists and Im Γ λ+i0 v > 0. Note that the union here is finite as the unperturbed model is of finite cone type. For the same reason, the supremum in c H is a maximum.
The quantities q, Q, α are defined by formulas similar to those in [22]: for x, y ∈ N + v , .
assuming h x = H z x and h y = H z y , otherwise we let Q x,y (h) = cos α x,y (h) = 0.

Dividing by (Im
Remark 5.5. Note that v + q v + = 1. On the other hand, Q v ′ ,w is a quotient of a geometric and arithmetic mean, so 0 ≤ Q v ′ ,w ≤ 1. Since −1 ≤ cos α v ′ ,w ≤ 1, this shows that −1 ≤ w∈N + v q w Q v ′ ,w cos α v ′ ,w ≤ 1. We now deduce a "two-step expansion". We will assume our tree satisfies (C2). In other words, for each vertex v, there is a vertex v ′ ∈ N + v such that every label found in N + v can also be found in N + v ′ . We then say that v ′ is chosen w.r.t. (C2). From now on, we denote S v = N + v . If * = t bo and * ′ is the vertex chosen w.r.t. (C2) corresponding to * , we let and for x ∈ S * ′ , let It follows from Remark 5.5 that c x (h) ∈ [−1, 1]. We next define for x ∈ S * , and for x ∈ S * ′ , (5.14) Then x∈S * , * ′ p x = 1. Note that c x (h) is a quantity that depends on the random parameters of the perturbed graph, whereas p x is non-random.
Proposition 5.6. Let I ⊂ Σ be a compact interval. There exists a continuous function Proof. Let K = e {C(Γ z e ) : z ∈ I + i[0, 1]}. We apply Lemma 5.4 to v = * , then to v = * ′ . If c I,H (ǫ, ǫ ′ ) = 2C K (ǫ ′ ) + c H (ǫ) + C K (ǫ ′ )c H (ǫ), the statement follows by taking To use this result under assumption (P0), note that Given the key results of § 5.1, much of the proof of Theorem 5.7 goes as in the combinatorial case [22], so we only outline the main ideas. The latter part of this proof becomes nevertheless more technical in the quantum setting, due to the more complicated relations among the Green's functions (see (5.23)), so we give the necessary modifications in Appendix B.
Recall that T = T + o is defined by a cone matrix M on a set of labels A satisfying (C0), (C1) and (C2). We previously denoted * = t bo . More generally, given The set S * j , * ′ j is then constructed analogously, and all results of § 5.1 apply without change (this works in particular for * j = o bo ).
Let us now discuss the proof of Theorem 5.7 in several steps: Step 1: The Euclidean bound follows from the hyperbolic one. In fact, if the γ-bound is proved, then using the Cauchy-Schwarz inequality, To bound the moment E(|h z v | p ), one uses the simple inequality (5.17) |ξ| ≤ 4γ(ξ, ζ) Im ζ + 2|ζ|, ξ, ζ ∈ H, applied to ξ = h z v and ζ = H z v .
Step 2: To prove the γ-bound, it suffices to show that for each j ∈ A, as long as P = (P j,k ) forms a nonnegative irreducible matrix (and c 1 (ǫ), C(ǫ) → 0 as ǫ → 0). Indeed, the Perron-Frobenius theorem then provides a positive eigenvector u ∈ R A such that P ⊺ u = u. If we consider the vector E p γ : δ , and the γ-bound easily follows.
Step 3: To prove (5.18), we apply the two-step expansion (Proposition 5.6) to get (5.19) The idea now is that for p ≥ 1, as follows from Jensen's inequality and the facts p x = 1 and |c x (h)| ≤ 1. However the first inequality is generally strict for p > 1, and this is what provides the (1 − δ 0 ) in (5.18) if we choose P = P (z) to be the matrix for j, k ∈ A, which satisfies the requirements of Step 2.
The RHS of (5.25) now becomes If all γ (π) x (h) ≈ 0 then we are done, so the nontrivial work is when h / ∈ B r (H), where This case is controlled by Proposition 5.8 below. To state it, recall that we assume I is away from the Dirichlet spectrum in (1.8). This implies, for the L in κ (p) * , (5.28) min Proposition 5.8 is remarkable as it gives a uniform contraction estimate on the random variable. Using it in (5.27), we get the (1 − δ 0 ) we were seeking in (5.18) (take δ 0 = min j∈A δ * j ), thus completing the proof of Theorem 5.7.
The proof of Proposition 5.8 is given in Appendix B.
7 Note that c (π) x (g) = cπx(g) in general. For example, if x ∈ S * and πx ∈ S * ′ , then c (π) x (g) is defined by a single sum while cπx(g) is a product of two sums, as in (5.11)-(5.12).

5.3.
Proof of pure AC spectrum. We may finally conclude with the proof of Theorem 1.7, which we recall: Now, assuming c 1 ≤ |S z (L b )| ≤ c 2 and |C z (L b )| ≤ c 1 4Qc 2 y ≤ 1 4y we have by (2.6) Here we used that − S ′ z (L) Sz(L) is Herglotz and that {R + z (o b + )} are independent of α t b , as follows from (2.16), and bounded the probability by first conditioning over the random variables different from α ω t b , so that the "interval" above is fixed.
The rest of the proof is devoted to the case q = 2, where we need to improve again. Using (2.17) twice, we have Define the events E 0 = {|α t b | ≤ y −1 }, Using an estimate from (5.32), we have . 8 The exponent of the first term is increased by 2 1+ς at each step, the second one by at least α 0 3(1+ς) , as for the last one, it already has the required decay. After finitely many steps, the exponent thus reaches βς 4 .
Proof of Theorem 1.8. Given p ≥ 1, choose ς such that p < βς 5 . By Lemma 5.10, we have I ⊆ σ ǫ ac (δ) for some δ > 0 and ǫ ≤ ǫ 0 . As p ≥ 1, given λ ∈ I, we have by the layer-cake representation, We may assume t δ ≥ 1 by taking a smaller x δ if necessary. Since this holds for any λ ∈ I and η ∈ (0, η 0 ), we get sup λ∈I sup η∈(0,η 0 ) x . This implies f 1 , G z f 2 = f 2 , G z f 1 for any real-valued f j supported on e(b). Hence, for any f = f 1 + if 2 supported in e, exist and are not both zero. Then for any f supported in e(b), In particular, if Im R ± λ (o b ) = 0 for all b ∈ B, then Im f, G λ f = 0 for all f ∈ L 2 (T). This lemma was stated without proof in [1, eq (A.15)].
Proof. First note that it suffices to prove this for real-valued f . In fact, for f = f 1 + if 2 , we then use (A.4) and deduce the result, since g + f 1 (λ) + g + f 2 (λ) = g + f (λ) as easily checked and similarly for g − f (λ). So assume f is real. Since b and λ are fixed, we shall denote R ± := R ± λ (o b ) and φ ± := φ ± λ;b , and drop b indices in x b and f b for simplicity.
Since f is real-valued, we get Now Re φ ± = C(x) ± (Re R ± )S(x) and Im φ ± = (± Im R ± )S(x). Hence, the term in curly brackets is On the other hand, which is exactly the expression at the end of (A.7). This completes the proof of (A.5). Finally, if Im R ± λ (o b ) = 0 for all b ∈ B, then Im f b , G λ f b = 0 for all f b supported in b, by (A.5). The same ideas show that Im f b , G λ f b ′ = 0 in this case. In fact, we don't need to go through all the above calculations, just note that in (A.6), we get Im φ ± = (± Im R ± )S = 0.
It follows that Im f, G λ f = 0 for all f ∈ L 2 (T).
To see the Herglotz property [17], note that if H max . On the other hand, the left-hand side can also be computed by integration by parts on every edge. All the boundary terms except the one at u cancel thanks to the self-adjoint conditions. We thus obtain V + z;u , H max z;u L 2 (T + u ) = V + z;u (u)(V + z;u ) ′ (u) − (V + z;u ) ′ (u)V + z;u (u).
Since V + z;u (u) = 1, this reduces to 2i Im(V + z;u ) ′ (u) = 2i Im . We thus showed Im R + z (u) = Im z V + z;u 2 T + u , implying the result by taking u = o b . The claim for R − z is similar: the preceding proof shows that in the twisted view, is Herglotz, and the negative sign is there to pass to the coherent view. These claims in turn show by (2.13) that z → G z (v, v) is Herglotz 9 To show that R + is Herglotz, we use the same approach. First, . On the other hand, the left-hand side is 2 Re V + z;u , H max T + u V + z;u . Integrating by parts yields A.2. A criterion for AC spectrum. We recall the following fact [29,Theorem XIII.20].
Suppose H is a self-adjoint operator on a Hilbert space H , with resolvent G z . Suppose there exists p > 1 such that for any ϕ in a dense subset of H , we have (A. 8) lim inf η↓0 b a | Im ϕ, G λ+iη ϕ | p dλ < ∞ .
Then H has purely absolutely continuous spectrum in (a, b).
This criterion also appeared later in [24] for p = 2. In [29], lim inf η↓0 is replaced by sup 0<η<1 , but one sees from the proof that the above statement holds.
Theorem A.6. Suppose a Schrödinger operator H T on a quantum tree satisfies the following: there exists p > 1 such that for any b ∈ T, Then H T has purely absolutely continuous spectrum in I.
In particular, if conditions (1.8) and (Green-p) hold, then P-a.e. operator H T has pure AC spectrum in I 1 .
. Since for any v ∈ V , y b ′ → G z (v, y b ′ ) is an eigenfunction with eigenvalue z, we have 10 where we used (2.17) repeatedly in the last step. On the other hand, by (2.13), where we used the fact that R ± z (v) is Herglotz. Hence, The other G z (v 0 ; v r ) appearing in (A.11) are bounded similarly. For b = b ′ , the first equality in (A.11) should be modified as we don't have an eigenfunction at the point x b = y b . In fact, assuming without loss that x b ≤ y b , we have This can be checked by explicit calculation from (A.2) and (2.13): these tell us that Here we passed from φ − (y b )φ + (L b ) to φ + (y b )φ − (L b ) as in Lemma A.2. Inserting this into the RHS of (A.12) gives In any case, for any b ∈ B, L b 0 |S z (x b )| p dx b ≤ c, uniformly in λ + iη ∈ (E 0 , E 1 ) + i(0, 1) (in fact, λ → S λ (x) is analytic). Recalling (A.10), (A.11), we have proved that where b ′′ is any edge with o b ′′ = t b ′ . Applying Cauchy-Schwarz and (A.9), we see that (A.8) is satisfied for any continuous ϕ of compact support. Hence, H T has pure AC spectrum in (E 0 , E 1 ). In particular, if (Green-p) holds for a random tree, then by Fatou's lemma and Fubini, we have where we applied (Green-p) in the last step. It follows that the lim inf on the left-hand side is finite for P-a.e. operator H T . Hence, P-a.e. H T has pure AC spectrum in I 1 .
where we denoted Z z k = Z z v if v has label k. In view of Lemma B.1, since I ⊂ Σ is chosen compact, the minimum exists and we have θ 0 > 0. We also let We may now state the main result of this appendix (corresponding to case 3 in the above discussion).