Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization
Nalini Anantharaman, Maxime Ingremeau, Mostafa Sabri, Brian Winn
aa r X i v : . [ m a t h - ph ] F e b QUANTUM ERGODICITY FOR EXPANDING QUANTUM GRAPHSIN THE REGIME OF SPECTRAL DELOCALIZATION
NALINI ANANTHARAMAN, MAXIME INGREMEAU, MOSTAFA SABRI, BRIAN WINN
Abstract.
We consider a sequence of finite quantum graphs with few loops, so thatthey converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree.We assume these quantum trees are spectrally delocalized in some interval I , in the sensethat their spectrum in I is purely absolutely continuous and their Green’s functions arewell controlled near the real axis. We furthermore suppose that the underlying sequenceof discrete graphs is expanding. We deduce a quantum ergodicity result, showing thatthe eigenfunctions with eigenvalues lying in I are spatially delocalized. Introduction
In their far-reaching work [21], Kottos and Smilansky suggested that the ideas andresults of quantum chaos should apply to quantum graphs. By quantum graphs, we meana metric graph, equipped with a differential operator and suitable boundary conditions ateach vertex. We refer the reader to Section 2.1 for a more precise definition.In this paper, we study an analogue on quantum graphs of one of the most famousproperties of quantum chaotic systems, namely quantum ergodicity. In its original context[26, 13, 29], quantum ergodicity says that, on a compact Riemannian manifold whose geo-desic flow is ergodic, the eigenfunctions of the Laplace-Beltrami operator become equidis-tributed in the high-frequency limit .On a fixed quantum graph (with Kirchhoff boundary conditions), it was shown in [14]that quantum ergodicity generically does not hold in the high-frequency limit, unless thegraph is very simple (homeomorphic to an interval or a circle).However, instead of studying the asymptotics of eigenfunctions on a fixed quantumgraph, one may study quantum ergodicity on sequences of quantum graphs whose sizegoes to infinity. Some positive results appeared in [9], [17], [20] [11] for several families ofquantum graphs, while it was shown in [8] that quantum ergodicity does not hold on stargraphs. We refer the reader to the introduction of [19] for an up-to-date survey of theserecent developments.All the preceding results study the high frequency behaviour of eigenfunctions. In[19], we proved a quantum ergodicity result for regular equilateral quantum graphs Q N converging to the regular equilateral tree T q (in the sense of Benjamini-Schramm), inthe bounded energy regime , more precisely for energies lying in some bounded interval I ⊂ σ ac ( H T q ), assuming the underlying discrete graphs are expanders. Our aim in thispaper is to extend this result to the case of non-regular non-equilateral quantum graphssatisfying δ -conditions ( § P supported on the set of quantum trees. We introduced the notion ofBenjamini-Schramm convergence for quantum graphs in [5], in analogy to the case ofdiscrete graphs. We also assume the underlying discrete graphs are expanders. Mathematics Subject Classification.
Primary 58J51. Secondary 34B45, Q1Q10.
Key words and phrases.
Quantum ergodicity, quantum graphs, delocalization, trees.
Our main assumption is that in the energy interval we consider, the spectrum at thelimit is absolutely continuous. More precisely, we need a good control over the Green’sfunction near the real axis (see hypothesis (Green) below). Our results thus convert spectral delocalization at the limit (AC spectrum) into spatial delocalization for the eigen-functions (i.e. quantum ergodicity). We give in § Q N of growing quantumgraphs, the aim is to show that for any orthonormal basis of eigenfunctions ( ψ ( N ) j ) on Q N ,the probability measure | ψ ( N ) j ( x ) | d x approaches the uniform measure |Q N | d x when N islarge enough. We only aim to prove this for most eigenfunctions in an interval I , so weconsider Ces`aro means. More precisely, denoting by N N ( I ) the number of eigenvalues of Q N in I , we set to prove that(1.1) lim N →∞ N N ( I ) X λ ( N ) j ∈ I (cid:12)(cid:12)(cid:12)(cid:12) h ψ ( N ) j , f N ψ ( N ) j i L ( G N ) − h f N i λ ( N ) j (cid:12)(cid:12)(cid:12)(cid:12) = 0for any uniformly bounded observable f N ∈ L ∞ ( G N ), where G N is the underlying metricgraph. Since h ψ j , f ψ j i = R G N f ( x ) | ψ j ( x ) | d x , if we had h f i λ ( N ) j = |G N | R G N f ( x ) d x , i.e.,if h f i were the uniform averages independently of λ ( N ) j this would show that | ψ j ( x ) | d x approaches |G N | d x in some weak sense.It turns out that such perfect uniform distribution can only be true in very special cases,cf. [19]. In fact, since we consider a regime of bounded energies (lying in a fixed I , not thehigh frequency regime), we expect the potential we put on the edges to have some influenceover the probability of finding the wavefunction in various places of the graph, which isgiven by | ψ ( N ) j ( x ) | d x . The true “limiting measure” is thus not the uniform measure ingeneral, but one with a possibly non-constant density. The density is very satisfactory as itis directly related to the spectral density of the limiting quantum tree. In fact, our resultsshow in a weak sense that | ψ ( N ) j ( x ) | d x tends to the measure Im ˜ g λ ( N ) j +i0 N (˜ x, ˜ x ) R G N Im ˜ g λ ( N ) j +i0 N (˜ y, ˜ y ) d y d x ,where ˜ g zN (˜ x, ˜ x ) is the Green’s function of the universal cover of Q N (˜ g zN (˜ x, ˜ x ) approachesthe Green’s function of the limiting tree when N → ∞ , see Appendix C). Accordingly,the mean h f N i λ ( N ) j above will actually depend on the energy λ ( N ) j .We now discuss the main steps of the proof:(1) In Sections 4 and 5 we reduce (1.1) to proving that analogous Ces`aro means definedon the discrete graph (which we call quantum variances ) vanish as N −→ ∞ . In thisprocess the L scalar product h ψ j , f ψ j i = R G N | ψ j ( x ) | f ( x ) d x is replaced by ℓ scalarproducts of the form P v ∈ V N | ψ j ( v ) | K f,j ( v ) or P v ∈ V N P w ∼ v ψ j ( v ) ψ j ( w ) M f,j ( v, w ),for some auxiliary (energy-dependent) observables K f,j , M f,j built from f . Suchdiscretization philosophy is well established in the quantum graphs literature, espe-cially when the quantum graph is equilateral, in which case the restrictions ψ j | V N become eigenfunctions of some nice adjacency matrix. It is known however thatwhen the graph is not equilateral, the discretization produces a complicated energy-dependent Schr¨odinger operator. In this paper we circumvent this problem by using Note that in the special case where ˜ g z ( x, x ) is independent of x we get the uniform measure |G N | d x .Roughly speaking, the general quotient detects the inhomogeneities in the limit object, a tree in our setting. UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 3 non-backtracking eigenfunctions f j , f ∗ j living on the directed edges of the graph, anidea that already proved fruitful in discrete graphs [3, 4], and it is quite remarkablethat it also works for quantum graphs. This new construction is explained in Section 4.The eigenvalue equation for f j , f ∗ j implies that the corresponding quantum variance isinvariant under simple averaging operators, weighted by the Green’s functions of thequantum universal cover (Proposition 4.2). In the usual proof of quantum ergodicityon manifolds, we would instead be using the invariance of eigenfunctions under thewave propagator.(2) We can bound the quantum variance by Hilbert-Schmidt norms. For this we follow thegeneral scheme of [4], but the procedure is complicated by two problems: first, neitherthe restrictions ψ j | V of the eigenfunctions to the vertices, nor the non-backtrackingeigenfunctions f j , f ∗ j form an orthonormal basis; second we have less a priori boundson the auxiliary observables K f,j , M f,j . This calls for several technical innovations(Sections 6–7).(3) Applying the averaging operators from step (1), and developing the Hilbert-Schmidtnorms (Section 8) reduces the proof to some contraction estimates on a family ofsub-stochastic operators. These estimates require a careful analysis involving theexpanding properties of the graphs (Section 9). Compared to [4], this part containsseveral novelties which are necessary due to the more complicated recursion relationssatisfied by the Green’s functions in the quantum setting (Section 3).Each of the preceding steps is not “exact” in the sense that it holds modulo error terms.To control them, we derive in Appendix C some important implications of Benjamini-Schramm convergence and spectral delocalization. Acknowledgments.
N.A. was supported by Institut Universitaire de France, by the ANRproject GeRaSic ANR-13-BS01-0007 and by USIAS (University of Strasbourg Institute ofAdvanced Study).M.I. was supported by the Labex IRMIA during part of the realization of this project.M.S. was supported by a public grant as part of the
Investissement d’avenir project,reference ANR-11-LABX-0056-LMH, LabEx LMH. He thanks the Universit´e Paris Saclayfor excellent working conditions, where part of this work was done.2.
Main results
Quantum graphs.
Let G = ( V, E ) be a graph with vertex set V and edge set E .For each vertex v ∈ V , we denote by d ( v ) the degree of v . If v , v ∈ V , we write v ∼ v if { v , v } ∈ E . We let B = B ( G ) be the set of oriented edges (or bonds), so that | B | = 2 | E | .We assume that there is at most one edge between two vertices, so we will view B as asubset of V × V . If b ∈ B , we shall denote by ˆ b the reverse bond. We write o b for theorigin of b and t b for the terminus of b . We will also write e ( b ) ∈ E for the edge obtainedby forgetting the orientation of b .For us, a quantum graph Q = ( V, E, L, W, α ) is the data of: • A connected combinatorial graph (
V, E ). • A map L : E → (0 , ∞ ). If b ∈ B , we denote L b := L ( e ( b )). • A potential W = ( W b ) b ∈ B ∈ L b ∈ B C ([0 , L b ]; R ) satisfying for x ∈ [0 , L b ],(2.1) W b b ( L b − x ) = W b ( x ) . • Coupling constants α = ( α v ) v ∈ V ∈ R V .The underlying metric graph is defined by G := { x = ( b, x b ); b ∈ B, x b ∈ [0 , L b ] } / ≃ , NALINI ANANTHARAMAN, MAXIME INGREMEAU, MOSTAFA SABRI, BRIAN WINN where ( b, x b ) ≃ ( b ′ , x ′ b ′ ) if b ′ = ˆ b and x ′ b ′ = L b − x b . In the sequel, we will sometimes write x ∈ b to indicate that x = ( b, x b ) ∈ G . Condition (2.1) then simply ensures that W iswell-defined on G .In general a function f : G −→ R can be identified with a collection of maps ( f b ) b ∈ B such that f b ( L b − · ) = f ˆ b ( · ). We say that f is supported on e for some e ∈ E if f b ≡ e ( b ) = e . In the sequel, we will often write f ( x b ) instead of f (( b, x b )) or f b ( x b ).If each f b is positive and measurable, we define R G f ( x )d x := P b ∈ B R L b f b ( x b )d x b . Wemay then define the spaces L p ( G ) for p ∈ [1 , ∞ ] in the natural way (see below).In the sequel, we will always make the following assumptions: (Data) There exist 0 < m < M and D ∈ N such that for all v ∈ V and all b ∈ B :3 ≤ d ( v ) ≤ D | α v | ≤ Mm ≤ L b ≤ M W b ∈ Lip([0 , L b ]) and max ( k W b k ∞ , Lip( W b )) ≤ M W b ( L b − · ) = W b ( · ) . This last assumption says that the potential on each edge is symmetric (in particular,this is the case if W ≡ I ) is the set of Lipschitz-continuous functions on I and Lip( f ) is the Lipschitzconstant of f .Let Q be a quantum graph. Consider the Hilbert space L ( G ) := n ( f b ) b ∈ B ∈ M b ∈ B L (0 , L b ) (cid:12)(cid:12)(cid:12) f b b ( L b − · ) = f b ( · ) and X b ∈ B k f b k L (0 ,L b ) < ∞ o and its subset H ( G ) := n ( f b ) b ∈ B ∈ M b ∈ B H (0 , L b ) (cid:12)(cid:12)(cid:12) f b b ( L b − · ) = f b ( · ) and X b ∈ B k f b k H (0 ,L b ) < ∞ o . We say that a function f ∈ H ( G ) satisfies the δ -boundary conditions at v ∈ V if(2.2) ∀ b, b ′ ∈ B such that o b = o b ′ = v, we have f b (0) = f b ′ (0) =: f ( v ) , P b ∈ B, o b = v f ′ o b (0) = α v f ( v ) . The set of such functions is denoted by: H Q := n f ∈ H ( G ) (cid:12)(cid:12) ∀ v ∈ V, f satisfies (2.2) o . We then define an operator H Q acting on ψ = ( ψ b ) b ∈ B ∈ H Q by(2.3) ( H Q ψ b )( x b ) = − ψ ′′ b ( x b ) + W b ( x b ) ψ b ( x b ) . Our assumption (Data) implies in particular that the operator H Q : H Q → L ( G ) isthen self-adjoint [7, Theorem 1.4.19].We say that the quantum graph is finite if V and E are finite. In this case we denote L ( Q ) = P e ∈ E L ( e ). If Q is finite, H Q has compact resolvent ([7, Theorem 3.1.1]). So for In the sequel, all the scalar products in Hilbert spaces will be linear in the right variable, and anti-linearin the left one: h zu, z ′ v i = zz ′ h u, v i . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 5 finite Q N , there exists an orthonormal basis ( ψ ( N ) j ) j ≥ of L ( G N ) made of eigenfunctionsof H Q N . We denote by (cid:0) λ ( N ) j (cid:1) j ≥ the corresponding eigenvalues. Recall that we write N N ( I ) := n j ≥ | λ ( N ) j ∈ I o . Since our Schr¨odinger operators have real coefficients, we may—and will—restrict ourattention to real valued eigenfunctions. This restriction was already present in the work[4] on discrete graphs. It is only necessary in Appendix B.2. More precisely, what we needto assume is that ψ ( N ) j ( o b ) ψ ( N ) j ( t b ) ∈ R for any N , j and b ∈ B N .2.2. Eigenfunctions on the edges.
Let Q be a quantum graph. Given an oriented edge b ∈ B and γ ∈ C , let C γ,b ( x b ) and S γ,b ( x b ) be a basis of solutions of the equation(2.4) − ψ ′′ ( x b ) + W b ( x b ) ψ ( x b ) = γψ ( x b )satisfying (cid:18) C γ,b (0) S γ,b (0) C ′ γ,b (0) S ′ γ,b (0) (cid:19) = (cid:18) (cid:19) . If W b ≡
0, these are the familiar cosine and sine functions, hence our notation. Note thatany solution ψ of (2.4) satisfies (cid:18) ψ ( L b ) ψ ′ ( L b ) (cid:19) = M γ ( b ) (cid:18) ψ (0) ψ ′ (0) (cid:19) where M γ ( b ) = (cid:18) C γ,b ( L b ) S γ,b ( L b ) C ′ γ,b ( L b ) S ′ γ,b ( L b ) (cid:19) . For all b ∈ B and all z ∈ C , we have(2.5) S γ,b = S γ,b , since both functions satisfy the same differential equation with the same boundary condi-tions.For any x b ∈ [0 , L b ], the maps γ C γ,b ( x b ) and γ S γ,b ( x b ) are holomorphic functions.A proof of this fact can be found, for instance, in [24, Chapter 1].Similarly, using the symmetry of the potential, we note that for any γ ∈ C , we have the“trigonometric” relations (similar to the usual ones satisfied by cos and sin)(2.6) S γ,b ( L b ) C γ,b ( x b ) − C γ,b ( L b ) S γ,b ( x b ) = S γ,b ( L b − x b ) ,S ′ γ,b ( L b ) C γ,b ( x b ) − C ′ γ,b ( L b ) S γ,b ( x b ) = C γ,b ( L b − x b ) , since the two pairs of functions satisfy the same equation with the same boundary con-ditions (see [19, § S γ ( x b ) , C γ ( x b ) for S γ,b ( x b ) , C γ,b ( x b ) to lighten notations.2.3. Main result.
Let Q N = ( V N , E N , L N , W N , α N ) be a sequence of quantum graphs,each with N vertices. We denote by G N = ( V N , E N ) the underlying discrete graphs.Our first two assumptions concern the geometry of G N = ( V N , E N ): (EXP) The sequence ( G N ) forms an expander family. That is, if A G N is the adjacencymatrix on G N , then the operator P N = d N A G N , has a uniform spectral gap in ℓ ( V N ).More precisely, the eigenvalue 1 of P N is simple, and the spectrum of P N is contained in[ − β, − β ] ∪ { } , where β > N . (BST) For all r >
0, lim N →∞ |{ x ∈ V N : ρ G N ( x ) < r }| N = 0 , Recall that the adjacency matrix acts on f ∈ ℓ ( V ) by (cid:0) A f (cid:1) ( v ) = P w ∼ v f ( w ). NALINI ANANTHARAMAN, MAXIME INGREMEAU, MOSTAFA SABRI, BRIAN WINN where ρ G N ( x ) is the injectivity radius at x , i.e. the largest ρ such that the ball B G N ( x, ρ ) isa tree. Here, because of our uniformity assumption (Data) , it does not matter to choosea discrete distance or a continuous one for this property.We will in addition suppose that Q N converges in the sense of Benjamini-Schramm tosome probability measure P on the set of rooted graphs satisfying (Data) — as we mayalways do, up to extracting a subsequence [5, Corollary 3.6]. Assumption (BST) becomesequivalent to asking that P is supported on the set of quantum trees, i.e. quantum graphswithout cycles. In a more probabilistic language, Q N converges to a random rootedquantum tree.The next two hypotheses can be seen as a condition of spectral delocalization . Indeed,they imply that P -almost all quantum trees have purely absolutely continuous spectrumin I , see [6, Theorem A.6]. (Green) There exists a bounded open interval I ⊂ R such that for all s > λ ∈ I ,η ∈ (0 , E P (cid:18)(cid:12)(cid:12)(cid:12) Im ˆ R + λ +i η ( o b ) (cid:12)(cid:12)(cid:12) − s + (cid:12)(cid:12)(cid:12) Im ˆ R + λ +i η ( o ˆ b ) (cid:12)(cid:12)(cid:12) − s (cid:19) < ∞ , where ˆ R + is the Weyl-Titchmarsh function (defined in Section 3.1 below).This assumption implies in particular that the Green’s functions of the Schr¨odingeroperator on the infinite tree has finite moments; see § C.2.As the proof will show, we actually only need (Green) to hold for all 0 < s < s forsome finite s which can in principle be made explicit and is not too big; we chose theabove formulation for comfort.Our last assumption is that I avoids the “Dirichlet spectrum”. (Non-Dirichlet) Let D = [ N [ b ∈ B N { λ ∈ R : S λ ( L b ) = 0 } . Then we assume I ∩ D = ∅ , so any compact I ⊂ I is isolated from all Dirichlet values. In the applications we have inmind, D is a discrete subset of R , or an ǫ -neighborhood of a discrete subset.It follows that there exists C Dir > λ ∈ I , we have | S λ ( L b ) | ≥ C Dir . By continuity, this implies the existence of C ′ Dir > < η Dir ≤ λ ∈ I , η ∈ [0 , η Dir ],(2.7) | Re S λ +i η ( L b ) | ≥ C ′ Dir . We also note that S γ ( L b ) = 0 for any γ ∈ C + = { z : Im z > } , since otherwise theself-adjoint operator Df = − f ′′ + W b f on [0 , L b ] with Dirichlet boundary conditions wouldhave a complex eigenvalue γ corresponding to S γ ( x ). Theorem 2.1.
Let Q N be a sequence of quantum graphs satisfying (Data) for each N ,such that (EXP) , (BST) , (Green) and (Non-Dirichlet) hold true on the interval I .Fix an interval I such that I ⊂ I .Let ( ψ ( N ) j ) j ∈ N be an orthonormal basis of eigenfunctions of H Q N . Then for any sequenceof functions f N ∈ L ∞ ( G N ) satisfying k f N k ∞ ≤ , we have (2.8) lim η ↓ lim N →∞ N N ( I ) X λ ( N ) j ∈ I (cid:12)(cid:12)(cid:12)(cid:12) h ψ ( N ) j , f N ψ ( N ) j i L ( G N ) − h f N i γ ( N ) j (cid:12)(cid:12)(cid:12)(cid:12) = 0 , UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 7 where h ψ j , f N ψ j i L ( G N ) = R G N f N ( x ) | ψ j ( x ) | d x , γ ( N ) j = λ ( N ) j + i η and (2.9) h f N i γ j = R G N f N ( x ) Im ˜ g γ j N ( x, x ) d x R G N Im ˜ g γ j N ( x, x ) d x . Here, ˜ g γN ( x, y ) is the Green kernel of the operator H e Q N , defined on the universal cover of Q N . It will be defined precisely in Section 3.3. We will see in Section 3.1 that its imaginarypart is always positive. In the special case of regular equilateral quantum graphs treatedin [19], the universal covering quantum tree is the regular equilateral quantum tree T q ,independent of N , so the Green function ˜ g γN ( v, w ) = G γ T q ( v, w ) itself is independent of N . Remark 2.2.
By the Cauchy-Schwarz inequality, if f N ≥
0, we have (cid:18)Z G N ( f N ( x )) / d x (cid:19) ≤ (cid:18)Z G N f N ( x ) Im ˜ g γ j N ( x, x ) d x (cid:19) (cid:18)Z G N (cid:0) Im ˜ g γ j N ( x, x ) (cid:1) − d x (cid:19) . By Corollary C.8, there exists
C > N such thatsup λ ∈ I ,η ∈ (0 ,η Dir ) lim sup N −→∞ (cid:12)(cid:12)(cid:12)(cid:12) N Z G N (cid:0) Im ˜ g γN ( x, x ) (cid:1) ± d x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. This implies that, when f N is non-negative,lim inf N −→∞ h f N i γ j ≥ C (cid:18) N Z G N ( f N ( x )) / d x (cid:19) . For example, if f N = χ Λ N with | Λ N | = αN , this gives a lower bound α C >
0. As Λ N isarbitrary (for fixed cardinality), Theorem 2.1 is really a delocalization result: for most ψ j ,we cannot have | ψ j ( x ) | concentrating on a portion of G N of cardinality o ( N ). Quantum ergodicity for integral operators.
In a weak sense, the previous theorem assertsthat | ψ ( N ) j ( x ) | behaves asymptotically like Im ˜ g λjN ( x,x ) R G N Im ˜ g λjN ( x,x ) d x . More generally, we have aquantum ergodicity result involving integral operators. The aim here is to study theeigenfunction correlator ψ ( N ) j ( x ) ψ ( N ) j ( y ).In general, an integral operator K on L ( G ) takes the form( K ψ ) b ( x b ) = X b ′ ∈ B Z L b ′ K b,b ′ ( x b , y b ′ ) ψ b ′ ( y b ′ ) d y b ′ , with the condition that for all b, b ′ ∈ B and almost all x b ∈ [0 , L b ], y b ′ ∈ [0 , L b ′ ], we have K ˆ b,b ′ ( L b − x b , y b ′ ) = K b,b ′ ( x b , y b ′ )and similarly for the second argument. In the sequel, we will denote by B k the set ofnon-backtracking paths of length k (see (3.16) below for a precise definition), and workwith the spaces of operators (indexed by k ∈ N ) K k := n K : L ( G ) −→ L ( G ) | ∀ b, b ′ ∈ B, K b,b ′ ≡ b , . . . , b k ) ∈ B k with b = b, b k = b ′ o . Thus, K k is the space of operators with kernel supported on edges connected by a non-backtracking path of length k . Note that any integral operator with a bounded compactlysupported kernel can be written as a finite linear combination of operators in K k forvarious k . When we want to insist that these operators live on the graph Q N indexed by N , we will denote this space by K ( N ) k . NALINI ANANTHARAMAN, MAXIME INGREMEAU, MOSTAFA SABRI, BRIAN WINN
Theorem 2.3.
Let k ≥ . Under the assumptions of Theorem 2.1, let ( K N ) N ∈ N be asequence of operators with K N ∈ K ( N ) k , such that | K N,b,b ′ ( x, y ) | ≤ for every N ∈ N .Then (2.10) lim η ↓ lim N →∞ N N ( I ) X λ ( N ) j ∈ I (cid:12)(cid:12)(cid:12)(cid:12) h ψ ( N ) j , K N ψ ( N ) j i L ( G N ) − hK N i γ ( N ) j (cid:12)(cid:12)(cid:12)(cid:12) = 0 , where h ψ j , K ψ j i = P ( b ; b k ) ∈ B k R L b R L bk K ( x b , y b k ) ψ j ( x b ) ψ j ( y b k ) d x b d y b k , γ ( N ) j = λ ( N ) j +i η ,and hKi γ = P ( b ; b k ) R R K ( x b , y b k ) Im ˜ g γ ( x b , y b k ) d x b d y b k R G N Im ˜ g γ ( x, x ) d x . This says that in a weak sense, when N gets large, the eigenfunction correlator ψ j ( x ) ψ j ( y )looks like the quotient of spectral densities Im ˜ g λjN ( x,y ) R G N Im ˜ g λjN ( x,x ) d x on the universal cover.2.4. Examples.
N-lifts.
An important example is when Q N is some (connected) N -lift of a compactquantum graph Q . In other words, the underlying graph G N is an N -fold covering over G and the data is lifted naturally L ( v,w ) = L ( π N v,π N w ) , W ( v,w ) = W ( π N v,π N w ) , α v = α π N v ,where π N : G N −→ G is the covering projection.It is known that N -lifts – when picked randomly – are typically connected and mostof their points have a large injectivity radius – see [10, Lemma 24], [12, Lemma 9]. Moreprecisely, condition (BST) holds generically. It is also known that they are typicallyexpanders; see [16, 25]. Thus, our assumptions are generic.It is known that such ( Q N ) converge in the Benjamini-Schramm sense to a deterministiclimit, namely the universal covering tree T = e Q with a random root (see [5]). Moreprecisely Q N converges to the random rooted quantum tree defined by the measure P = 1 P b ∈ B L b X b ∈ B Z L b δ [ T , ˜ L , ˜ W , ˜ α , (˜ b,x b )] d x b , where ( L , W , α ) is the data on the base graph G and T = e G is the combinatorial treeunderlying T . In particular E P ( | Im ˆ R ± λ +i η ( o b ) | − s + | Im ˆ R ± λ +i η ( o ˆ b ) | − s ) = 2 P b ∈ B L b X b ∈ B L b | Im ˆ R ± λ +i η ( o b ) | − s . Also note that in this example, D = ∪ b ∈ B { λ ∈ R : S λ ( L b ) = 0 } .We showed in [6] that the spectrum of H T consists of bands of pure AC spectra alongwith a possible set of discrete eigenvalues (outside the bands). We also showed that withinthe bands, the limits ˆ R ± λ +i0 ( o b ) exist, are finite and satisfy Im ˆ R ± λ +i0 ( o b ) >
0. It followsthat (Green) is satisfied on any compact I ⊂ I , where I is some AC band.Theorem 2.3 thus tells us that such ( Q N ) are quantum-ergodic. This result can beregarded as a non-regular, non-equilateral generalization of [19]. We refer to [27, Chapter 5] for more background on coverings of finite graphs.
UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 9
Random quantum graphs.
We may also consider weak random perturbations Q ω N of the previous example. We leave the precise definition to [5, Section 8.4], but essentiallyone endows the graphs with random independent, identically-distributed (i.i.d.) lengths L ωe and i.i.d. coupling constants α ωv . Note that here only the combinatorial graph G N covers G , the data on each Q ω N is entirely random, so each has its own universal cover.Assuming (BST) holds (which is true generically as previously mentioned), we calculatedthe limit measure in [5]. In particular, we get E P ( | Im ˆ R ± λ +i η ( o b ) | − s + | Im ˆ R ± λ +i η ( o ˆ b ) | − s ) = 2 | B | E ( L ωb ) X b ∈ B E ( L ωb | Im ˆ R ± λ +i η ( o b ) | − s ) , where E is the expectation with respect to the random data ( L ωb , α ω ) and the distributionsare assumed to be the same for each b ∈ B . In other words, this is a random perturbationof an equilateral model on Q N (more general situations can be considered).We showed in [6] that if the perturbation is weak enough, then the bands of AC spectraremain stable, and (Green) holds in such bands. Note that here we assume there is no edgepotential W . So D = ∪ n ≥ [ π n ( L − ǫ ) , π n ( L + ǫ ) ], where ǫ is the small disorder window around theunperturbed length L . We technically need the coupling constants to be nonnegative witha H¨older distribution. This can be seen as a result of Anderson (spectral) delocalization,strengthening earlier results in [1].Theorem 2.3 implies that almost surely, the eigenfunctions of Q ω N are quantum er-godic. In other words, Anderson (spatial) delocalization holds, in addition to spectraldelocalization. 3. Preliminary constructions and notation
The Green function on a quantum tree.
Let T = ( V, E, L, W, α ) be a quantumtree, i.e. a quantum graph such that (
V, E ) is a tree, with underlying metric graph T .We describe here the functional equations satisfied by the Green function on T , due tothe topological fact that trees are disconnected by removing a point. While such relationsare well-known for discrete laplacians on trees, they have been less exploited for quantumtrees. This paragraph builds on the work of Aizenman-Sims-Warzel [1].If b ∈ B ( T ) we denote T ± b the two subtrees obtained by removing b , more precisely t b ∈ T + b while o b ∈ T − b . Let T ± b be the induced quantum trees and x b ∈ [0 , L b ]. If x = ( b, x b ) is the corresponding point in T , we define T + x as the quantum tree [ x b , t b ] ∪ T + b see [6] for a more precise definition. The quantum tree T − x is defined in a similar fashion.Let us define H max T ± x on T ± x to be the Schr¨odinger operator − ∆ + W with domain D ( H max T ± x ), the set of ψ ∈ H ( T ± x ) satisfying δ -conditions on inner vertices of T ± x . Notethat H max T ± x is not self-adjoint, due to the absence of boundary condition at the root x .By [1, Theorem 2.1], for any γ ∈ C + = { z ∈ C ; Im z > } , there are unique eigen-functions V + γ ; x ∈ D ( H max T + x ), U − γ ; x ∈ D ( H max T − x ) of H max T ± x , for the eigenvalue γ , satisfying U − γ ; x ( x ) = V + γ ; x ( x ) = 1.One can use the functions U − γ , V + γ to construct the Green’s function G γ of H T , see [6,Lemma 2.1]. For our purposes, we use them to define the Weyl-Titchmarch functions [1]as follows: if x ∈ T + o ∩ T − v ,(3.1) R + γ ( x ) = ( V + γ ; o ) ′ ( x ) V + γ ; o ( x ) and R − γ ( x ) = − ( U − γ ; v ) ′ ( x ) U − γ ; v ( x ) . This does not depend on o, v . Given an oriented edge b = ( o b , t b ), we define(3.2) ζ γ ( b ) = G γ ( o b , t b ) G γ ( o b , o b ) . This quotient of Green kernels will appear in the definition of the non-backtrackingeigenfunctions. See [6, §
2] for more comments on this quantity.Given an oriented edge b , let N + b be the set of outgoing bonds from b , and let N − b bethe set of incoming bonds to b i.e.(3.3) N + b := n b ′ ∈ B ; o b ′ = t b , b ′ = ˆ b o N − b := n b ′ ∈ B ; t b ′ = o b , b ′ = ˆ b o . (Later these definitions will apply to more general graphs than trees.)The following lemma gives a quantum graph analog for the classical recursive identitiesof Green’s functions on discrete trees. It tells us that the functions ζ γ , C γ and S γ canbe used as building blocks to understand the function G γ . In particular, (3.10) is thewell-known multiplicative property of the Green function on a tree. See [6, Section 2] fora proof, and Appendix A for a complement. Lemma 3.1.
Let γ ∈ C + . We have the following relations between ζ γ and the WTfunctions R ± γ : (3.4) ζ γ ( b ) = C γ ( L b ) + R + γ ( o b ) S γ ( L b ) , ζ γ (ˆ b ) = S ′ γ ( L b ) + R − γ ( t b ) S γ ( L b ) , (3.5) R + γ ( t b ) = S ′ γ ( L b ) S γ ( L b ) − S γ ( L b ) ζ γ ( b ) , R − γ ( o b ) = C γ ( L b ) S γ ( L b ) − S γ ( L b ) ζ γ (ˆ b ) . Moreover, (3.6) 1 ζ γ ( b ) S γ ( L b ) + X b + ∈N + b ζ γ ( b + ) S γ ( L b + ) = X b + ∈N + b C γ ( L b + ) S γ ( L b + ) + S ′ γ ( L b ) S γ ( L b ) + α t b , (3.7) 1 ζ γ ( b ) − ζ γ (ˆ b ) = S γ ( L b ) G γ ( t b , t b ) , ζ γ (ˆ b ) ζ γ ( b ) = G γ ( o b , o b ) G γ ( t b , t b ) , (3.8) − G γ ( o b , o b ) = R + γ ( o b ) + R − γ ( o b ) and (3.9) X b + ∈N + b C γ ( L b + ) S γ ( L b + ) + S ′ γ ( L b ) S γ ( L b ) + α t b = X t b ′ ∼ t b ζ γ ( b ′ ) S γ ( L b ′ ) + 1 G γ ( t b , t b ) , where b ′ = ( t b , t b ′ ) . Given a non-backtracking path ( v ; v k ) ∈ T , if b j = ( v j − , v j ) , then (3.10) G γ ( v , v k ) = G γ ( v , v ) ζ γ ( b ) · · · ζ γ ( b k ) = G γ ( v k , v k ) ζ γ (ˆ b ) · · · ζ γ (ˆ b k ) . Finally, for any path ( v ; v k ) ∈ T , (3.11) G γ ( v , v k ) = G γ ( v k , v ) . The following lemma is an important result on the properties of the Weyl-Titchmarshfunctions (3.1) and the fact that they are involved in “currents” passing through the edgesfrom some fixed arbitrary source ∗ (the “current” is I λ ∗ ( b ) = | G λ +i0 ( ∗ , o b ) | Im R + λ +i0 ( o b )). Lemma 3.2.
The functions F ( γ ) = R + γ ( o b ) , R − γ ( t b ) and G γ ( v, v ) are Herglotz functions: Im F ( γ ) ≥ for γ ∈ C + . Moreover, we have the following “current” relations: (3.12) X b + ∈N + b Im R + γ ( o b + ) ≤ Im R + γ ( o b ) | ζ γ ( b ) | and X b − ∈N − b Im R − γ ( t b − ) ≤ Im R − γ ( t b ) | ζ γ (ˆ b ) | . Equality holds in both cases if Im γ = 0 , whenever defined. UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 11
More precisely, we have (3.13) X b + ∈N + b Im R + γ ( o b + ) = Im R + γ ( o b ) | ζ γ ( b ) | − Im γ | ζ γ ( b ) | Z L b | ξ γ + ( x b ) | d x b , (3.14) X b − ∈N − b Im R − γ ( t b − ) = Im R − γ ( t b ) | ζ γ (ˆ b ) | − Im γ | ζ γ (ˆ b ) | Z L b | ξ γ − ( x b ) | d x b , where (3.15) ξ γ + ( x b ) = V + γ ; o ( x b ) V + γ ; o ( o b ) = C γ ( x b ) + R + γ ( o b ) S γ ( x b ) ξ γ − ( x b ) = U − γ ; v ( x b ) U − γ ; v ( t b ) = C γ ( L b − x b ) + R − γ ( t b ) S γ ( L b − x b ) . See Appendix A for a proof.3.2.
Operators on graphs.
Let now Q be a finite quantum graph (typically one in ourfamily Q N ) with vertex set V and bond set B . When Q = Q N the corresponding notionswill be indexed by N (e.g. B N , V N ), but we will often tend to drop the index N from thenotation.In the study of quantum ergodicity, we need to go back and forth between “classicalobservables” (i.e. functions on a classical phase space) and “quantum observables” (i.e.operators on a Hilbert space). Here this will be done in a simple-minded way: startingfrom a function on the set of non-backtracking paths, we explain how to build an operatoron ℓ ( B ) or ℓ ( V ).If k ≥
1, we denote by B k the set of non-backtracking paths in B k of length k :(3.16) B k := (cid:8) ( b , . . . , b k ) ∈ B k ; ∀ i = 1 , . . . , k − , t b i = o b i +1 and t b i +1 = o b i (cid:9) . If b ∈ B , we will also writeB b k := (cid:8) ( b , . . . , b k ) ∈ B k such that ( b , . . . , b k ) ∈ B k +1 (cid:9) B k,b := (cid:8) ( b − k , . . . , b − ) ∈ B k such that ( b − k , . . . , b ) ∈ B k +1 (cid:9) . Note that B b = N + b and B ,b = N − b .In the sequel, we will often write ( b ; b k ) instead of ( b , . . . , b k ) to lighten the notation.For all k ≥
1, we also define H k := C B k . If K ∈ H k , then K is a map from B k to C , and we extend it to a map B k → C by zeroon B k \ B k . This extension will still be denoted by K .If K ∈ H k , we define an operator K B : ℓ ( B ) −→ ℓ ( B ) by(3.17) ∀ f ∈ ℓ ( B ) , ∀ b ∈ B, ( K B f )( b ) = X ( b ; b k ) ∈ B b k − K ( b ; b k ) f ( b k ) . In particular, if K ∈ H , then K B is a diagonal operator.For k = 0, we will write B := V , and H := C V .For all k ≥
0, we also define K G : ℓ ( V ) −→ ℓ ( V ) by ∀ h ∈ ℓ ( V ) , ∀ v ∈ V, ( K G h )( v ) := X ( b ; b k ) ∈ B k o b = v K ( b ; b k ) h ( t b k ) if k ≥ K G h )( v ) := ( Kh )( v ) = K ( v ) h ( v ) if k = 0 . Green functions notation.
In the paper we consider a variety of quantum graphs,and we need to adopt a notation for the Green function of each of them: the sequence Q N , their universal covers e Q N , the limiting random quantum tree T .Let us first define notations pertaining to universal covers. Let Q be a quantum graph.Let ˜ G = ( ˜ V , ˜ E ) be the universal cover of the combinatorial graph G . We endow ˜ G withthe lifted data ˜ L b := L πb , ˜ W b := W πb and ˜ α v = α πv , where π : ˜ G → G is the coveringmap. This yields a quantum tree e Q := ( ˜ V , ˜ E, ˜ L, ˜ W , ˜ α ), which is called the universal cover of Q . The underlying metric graph of e Q will be denoted by e G . We then have a naturalprojection π : e G −→ G .Throughout the paper, if v, w ∈ V N and z ∈ C \ R , we will write g γN ( v, w ) := ( H Q N − γ ) − ( v, w )for the Green function of the compact quantum graph Q N .Let ˜ g γN be the Green’s functions of H e Q N . Throughout the paper we will encounterquantities of the form ˜ g γN ( o b i , t b j ) where ( b , . . . , b k ) is a fixed non-backtracking path in G N and i, j ≤ k . We define this as follows.Given ( b ; b k ) ∈ B k ( Q N ), choose any lift ˜ b ∈ B ( e Q N ) and let (˜ b ; ˜ b k ) ∈ B ˜ b k − ( e Q ) be thepath such that π (˜ b , . . . , ˜ b k ) = ( b , . . . , b k ). Then we define˜ g γN ( o b , t b k ) := ( H g Q N − γ ) − ( o ˜ b , t ˜ b k ) . This depends on the full path ( b , . . . , b k ) (although not apparent in our notation),however it does not depend on the choice of the lift (˜ b , . . . , ˜ b k ). We define ˜ g γN ( o b i , t b j ) :=( H g Q N − γ ) − ( o ˜ b i , t ˜ b j ) for i, j ≤ k , where (˜ b , . . . , ˜ b k ) is the lift we fixed. The definitionextends naturally to ˜ g γN ( x, y ) with x ∈ b i and y ∈ b j .Throughout the paper, we always let for b ∈ B N ,(3.18) ζ γ ( b ) := ˜ g γN ( o b , t b )˜ g γN ( o b , o b )thus suppressing the index N , which should cause no confusion.Similarly, the Weyl-Titchmarsh functions (3.1) denoted by R ± γ ( x ) will stand (withoutindex N ) for the WT-functions of the universal covering tree e Q N .For the Benjamini-Schramm limiting random tree T , we use the notation G γ ( x, y ) = ( H T − γ ) − ( x, y )for x, y ∈ T . For ζ and the WT-functions, we simply add a hat. More precisely, we letˆ ζ γ ( b ) := G γ ( o b , t b ) G γ ( o b , o b )for b ∈ B ( T ), and similarly denote the Weyl-Titchmarsh functions of T by ˆ R ± γ ( x ).3.4. A scalar product expression for boundary values of eigenfunctions.
For each γ ∈ C and b ∈ B N , let us defineΣ ( γ ; b ) := Z L b | S γ ( x b ) | d x b Σ ( γ ; b ) := Z L b S γ ( L b − x b ) S γ ( x b )d x b . Note that, by the Cauchy-Schwarz inequality, Σ ( γ ; b ) − | Σ ( γ ; b ) | >
0. As the lowerbound only depends on L b , W b , γ , we have Σ ( γ ; b ) − | Σ ( γ ; b ) | ≥ c > γ, L, W ) ∈ ( I + i[0 , × Lip M [m , M], where Lip M [m , M] denotes the set of (
L, f ), where
UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 13 m ≤ L ≤ M , and f is a Lipschitz function on [0 , L ] with both norm and Lipschitz constant ≤ M.For each γ ∈ C and each b ∈ B N , we denote by S bγ, + and S bγ, − the functions on G N defined respectively as x b S γ ( x b ) and x b S γ ( L b − x b ) on the edge b , and vanishingon the other edges.If ψ γ satisfies H Q ψ γ = γψ γ , and if the potentials are symmetric, then we have ψ γ ( x b ) = S γ ( L b − x b ) S γ ( L b ) ψ γ ( o b ) + S γ ( x b ) S γ ( L b ) ψ γ ( t b ) , so that S γ ( L b ) h S bγ, + , ψ γ i = ψ γ ( o b )Σ ( γ ; b ) + ψ γ ( t b )Σ ( γ ; b ) S γ ( L b ) h S bγ, − , ψ γ i = ψ γ ( o b )Σ ( γ ; b ) + ψ γ ( t b )Σ ( γ ; b ) . We therefore have(3.19) ψ γ ( o b ) = S γ ( L b ) | Σ ( γ ; b ) | − Σ ( γ ; b ) D Σ ( γ ; b ) S bγ, + − Σ ( γ ; b ) S bγ, − , ψ γ E =: h Y bγ , ψ γ i L ( G N ) ψ γ ( t b ) = S γ ( L b ) | Σ ( γ ; b ) | − Σ ( γ ; b ) D Σ ( γ ; b ) S bγ, − − Σ ( γ ; b ) S bγ, + , ψ γ E =: h Z bγ , ψ γ i L ( G N ) Note that, thanks to hypothesis (Data) , we havesup N max b ∈ B N sup γ ∈ I +i[0 , max( k Y bγ k L ( G N ) , k Z bγ k L ( G N ) ) ≤ C I, M . That γ Z bγ is not analytic poses a technical obstacle. Since we will be using holo-morphic tools, we prefer to use c Σ ( γ ; b ) := Z L b S γ ( x b ) d x b , c Σ ( γ ; b ) := Z L b S γ ( L b − x b ) S γ ( x b )d x b , in other words the analytic functions that coincide with Σ , Σ on the real line.By Cauchy-Schwarz and by continuity, we may find η I , c I > N , b ∈ B N , γ ∈ Ω I := I + i[ − η I , η I ], we haveRe (cid:16)c Σ ( γ ; b ) − c Σ ( γ ; b ) (cid:17) > c I . Therefore, we may define, for γ ∈ Ω I , the functions b Z bγ := S γ ( L b ) c Σ ( γ ; b ) − c Σ ( γ ; b ) (cid:16)c Σ ( γ ; b ) S bγ, − − c Σ ( γ ; b ) S bγ, + (cid:17) . Note that Ω I ∋ γ b Z bγ ∈ L ( G ) is holomorphic, coincides with Z bγ when γ ∈ I , and(3.20) (cid:13)(cid:13)(cid:13) b Z bγ − Z bγ (cid:13)(cid:13)(cid:13) L ( G N ) ≤ (cid:13)(cid:13)(cid:13) b Z bγ − b Z b Re( γ ) (cid:13)(cid:13)(cid:13) L ( G N ) + (cid:13)(cid:13)(cid:13) Z bγ − Z b Re( γ ) (cid:13)(cid:13)(cid:13) L ( G N ) ≤ C ′ I η uniformly in γ ∈ Ω I , b ∈ B N , N ∈ N .3.5. Notation for the remainders.
In all the paper, we will be dealing with quantitiesdepending on N , and on a complex parameter γ , and we will use the following notation.Let f N : C + −→ C be a sequence of functions. We will write that f N = O N → + ∞ ,γ (1) ifsup η ∈ (0 ,η Dir ) lim sup N −→∞ sup λ ∈ I | f N ( λ + i η ) | < ∞ , with η Dir as in (2.7).Similarly, we will write f N = O N → + ∞ ,γ (Im γ ) if f N ( γ )Im γ = O N → + ∞ ,γ (1). In fact, most oftime the imaginary part of γ will be denoted η , so we will write f N = O N → + ∞ ,γ ( η ). If f N depends on an additional parameter κ , we write f N = O ( κ ) N → + ∞ ,γ (1).If the quantity f N we consider depends on N and η , but not on λ , we will write f N = O N → + ∞ ,η (1) or f N = O N → + ∞ ,η ( η ), with the same definition. Remark 3.3.
Our main statements, Theorems 2.1 and 2.3 say that some quantity, dividedby N N ( I ) goes to zero as N −→ + ∞ followed by η ↓
0. We will recall in Appendix C,and more precisely in (C.2) and (C.3) that, under the assumptions we make, there existconstants C , C > N large enough, C N ≤ N N ( I ) ≤ C N. Therefore, in the course of the proof, when trying to show that a quantity divided by N N ( I ) goes to zero, we will sometimes replace N N ( I ) by N .4. Non-backtracking eigenfunctions
The quantum variance (2.8) involves functions living on the metric graph G N . Throughthe main part of the paper, we shall prefer to work with quantum variances defined onthe combinatorial graph G N . It is shown in Section 5 how to pass from one to the other.Such discretization is generally nontrivial for non-equilateral quantum graphs. We willshow however that we can construct functions on the directed edges B N which, quitemiraculously, are eigenfunctions of a simple non-backtracking operator denoted below by ζ γ B . This reduction from continuous to discrete will use the quantum Green’s functionsidentities derived in Section 3.1, and may be relevant to other problems on quantumgraphs.Let Q be a quantum graph, whose set of oriented edges is denoted by B (later, thefollowing construction will be applied to Q = Q N , and all the objects depend on N ).From now on, we fix η > ζ γ was definedin Section 3.3 using the universal cover of Q .Let ψ j be an eigenfunction of H Q with eigenvalue λ j . We define f j , f ∗ j ∈ C B by(4.1) f j ( b ) = ψ j ( t b ) S λ j ( L b ) − ζ γ j ( b ) ψ j ( o b ) S λ j ( L b ) , f ∗ j ( b ) = ψ j ( o b ) S λ j ( L b ) − ζ γ j ( b b ) ψ j ( t b ) S λ j ( L b )where γ j = λ j + i η and ζ γ j ( b ) is as in (3.18).Note that ψ j ( o b ) and ψ j ( t b ) can be recovered from f j ( b ) and f ∗ j ( b ) as follows: ψ j ( o b ) = S λj ( L b )1 − ζ γj ( b ) ζ γj ( b b ) (cid:0) f ∗ j ( b ) + ζ γ j ( b b ) f j ( b ) (cid:1) and ψ j ( t b ) = S λj ( L b )1 − ζ γj ( b ) ζ γj ( b b ) (cid:0) f j ( b ) + ζ γ j ( b ) f ∗ j ( b ) (cid:1) . Theseexpressions are well defined, since by (3.7) we have1 − ζ γ ( b ) ζ γ (ˆ b ) = S γ ( L b ) ζ γ ( b ) G γ ( t b , t b ) = S γ ( L b ) ζ γ (ˆ b ) G γ ( o b , o b ) , which does not vanish using (3.6).Recall that N + b was introduced in (3.3). We define the non-backtracking operator B : C B → C B by ( B f )( b ) = X b + ∈N + b f ( b + ) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 15
We observe that ψ j ( t b ) = ψ j ( o b ) C λ j ( L b ) + ψ ′ j ( o b ) S λ j ( L b ), so X b + ∈N + b ψ j ( t b + ) S λ j ( L b + ) = ψ j ( t b ) X b + ∈N + b C λ j ( L b + ) S λ j ( L b + ) + X b + ∈N + b ψ ′ j ( o b + )= ψ j ( t b ) X b + ∈N + b C λ j ( L b + ) S λ j ( L b + ) + ψ ′ j ( t b ) + α t b ψ j ( t b )= ψ j ( t b ) X b + ∈N + b C λ j ( L b + ) S λ j ( L b + ) + S ′ λ j ( L b ) ψ j ( t b ) − ψ j ( o b ) S λ j ( L b ) + α t b ψ j ( t b ) , where we used the δ -conditions and the fact that (cid:18) ψ λ ( o b ) ψ ′ λ ( o b ) (cid:19) = M λ ( b ) − (cid:18) ψ λ ( t b ) ψ ′ λ ( t b ) (cid:19) .We thus get(4.2) ( B f j )( b ) = ψ j ( t b ) (cid:18) X b + ∈N + b (cid:20) C λ j ( L b + ) S λ j ( L b + ) − ζ γ j ( b + ) S λ j ( L b + ) (cid:21) + S ′ λ j ( L b ) S λ j ( L b ) + α t b (cid:19) − ψ j ( o b ) S λ j ( L b )= 1 ζ γ j ( b ) f j ( b ) + O ψ j ,η ( b ) , with O ψ j ,η ( b ) = ψ j ( t b ) O γ j ( b ) where, by (3.6), we have O γ j ( b ) = X b + ∈N + b (cid:20) C λ j ( L b + ) − ζ γ j ( b + ) S λ j ( L b + ) − C γ j ( L b + ) − ζ γ j ( b + ) S γ j ( L b + ) (cid:21) + S ′ λ j ( L b ) S λ j ( L b ) − S ′ γ j ( L b ) S γ j ( L b ) + 1 ζ γ j ( b ) S γ j ( L b ) − ζ γ j ( b ) S λ j ( L b ) . Similarly, since f ∗ j = ιf j , where ι is the edge-reversal, and since B ∗ = ι B ι , we get B ∗ f ∗ j = ιζ γj f ∗ j + ιO ψ j ,η ( b ), with ιO ψ j ,η ( b ) = ψ j ( o b ) ιO γ j ( b ).Note that, by Corollary C.8 and analyticity of C z ( L b ) , S z ( L b ), we have for all s > N k O γ j k sℓ s ( B N ) = O ( s ) N → + ∞ ,γ ( η ) . If K = K γ ∈ H k for some k ≥
1, is some operator, possibly depending on γ ∈ C + , wedefine its non-backtracking quantum variance byVar I nb ,η ( K ) := 1 N N ( I ) X λ j ∈ I (cid:12)(cid:12)(cid:12) h f ∗ j , K λ j +i η B f j i (cid:12)(cid:12)(cid:12) (4.4) = 1 N N ( I ) X λ j ∈ I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ( b ,...,b k ) ∈ B k f ∗ j ( b ) K λ j +i η ( b , . . . , b k ) f j ( b k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.5)The advantage of this non-backtracking variance is that it is invariant under a spectralaveraging operator, whose kernel is relatively simple due to non-backtracking.More precisely, we have by (4.2) and its analog for f ∗ j that h f ∗ j , K B f j i = h ( ιζ γ j B ∗ ) n − r f ∗ j , K B ( ζ γ j B ) r f j i + error(4.6)for any n ≥ ≤ r ≤ n , where the error depends on O ψ j ,η and will be developedin (4.9) and (4.11). But before starting the calculation, we would like to express the“sandwich” ( B ιζ γ j ) n − r K B ( ζ γ j B ) r as a new observable in phase space . Very roughly, this heuristic can be likened to the classical proof of quantum ergodicity on manifolds[13, 29] where we replace the sandwich e − i t ∆ a e i t ∆ by Op( a ◦ G t ), with G t the geodesic flow, using Egorov’s First note that [( ζ γ B ) r f ]( b ) = X ( b ,...,b r +1 ) ∈ B b r ζ γ ( b ) · · · ζ γ ( b r ) f ( b r +1 )[( ιζ γ B ∗ ) k f ]( b ) = X ( b − k +1 ,...,b ) ∈ B r,b ζ γ (ˆ b ) · · · ζ γ (ˆ b − k +2 ) f ( b − k +1 ) . Recalling (3.17) this yields:
Proposition 4.1.
If we define R γ j n,r : H k −→ H n + k by (4.7) ( R γ j n,r K )( b ; b n + k )= ζ γ j (ˆ b ) · · · ζ γ j (ˆ b n − r +1 ) K ( b n − r +1 ; b n − r + k ) ζ γ j ( b n − r + k ) · · · ζ γ j ( b n + k − ) , we have h ( ιζ γ j B ∗ ) n − r f ∗ j , K B ( ζ γ j B ) r f j i = h f ∗ j , ( R γ j n,r K ) B f j i . Thus, R γ j n,r K is the “observable” we seek.We now derive the expression of the error in (4.6) more precisely. We have(4.8) X ( b ; b n + k ) ∈ B n + k f ∗ j ( b )( R γ j n,r K )( b ; b n + k ) f j ( b n + k )= X ( b ; b n + k − ) ∈ B n + k − h f ∗ j ( b ) ζ γ j (ˆ b ) · · · ζ γ j (ˆ b n − r +1 ) K ( b n − r +1 ; b n − r + k ) × ζ γ j ( b n − r + k ) · · · ζ γ j ( b n + k − ) · [ ζ γ j B f j ]( b n + k − ) i = X ( b ; b n + k − ) ∈ B n + k − f ∗ j ( b )( R γ j n − ,r − K )( b ; b n − k )[ f j + ζ γ j O ψ j ,η ]( b n + k − )= h f ∗ j , ( R γ j n − ,r − K ) B [ f j + ζ γ j O ψ j ,η ] i . where we used (4.2). Iterating this equation r times, we obtain(4.9) h f ∗ j , ( R γ j n,r K ) B f j i = h f ∗ j , ( R γ j n − r, K ) B f j i + r X ℓ =1 h f ∗ j , ( R γ j n − ℓ,r − ℓ K ) B ζ γ j O ψ j ,η i . By a computation similar to (4.8), we obtain that for all m > X ( b ; b m + k ) ∈ B m + k f ∗ j ( b )( R γ j m,s K )( b ; b m + k ) f j ( b m + k )= X ( b ,...,b m + k ) ∈ B m + k − (cid:0) f ∗ j ( b ) + ιζ γ j ιO ψ j ,η ( b ) (cid:1) ( R γ j m − ,s K )( b ; b m + k ) f j ( b m + k )= h f ∗ j + ιζ γ j ιO ψ j ,η , ( R γ j m − ,s K ) B f j i . Applying (4.10) n − r times to the first term in the left-hand side of (4.9), we obtain(4.11) h f ∗ j , ( R γ j n − r, K ) B f j i = h f ∗ j , K B f j i + n − r X ℓ ′ =1 h ιζ γ j ιO ψ j ,η , ( R γ j n − r − ℓ ′ , K ) B f j i . As (4.9) and (4.11) hold for any r ≤ n , we thus get the desired (approximate) invarianceof the quantum variance under the averaging operator R γn,r : theorem. We see that even if a ( x ) is a function, one must consider the phase space observable a ◦ G t ( x, ξ ).In our case, even if K ( f N ) ∈ H is a function, the new observable R γn,r K will lie in H n . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 17
Proposition 4.2. (4.12) Var I nb ,η ( K ) ≤ Var I nb ,η (cid:18) n n X r =1 R γn,r K (cid:19) + E n,η ( K ) , where (4.13) E n,η ( K ) = 1 n n X r =1 N N ( I ) X λ j ∈ I (cid:12)(cid:12)(cid:12)(cid:12) r X ℓ =1 h f ∗ j , ( R γ j n − ℓ,r − ℓ K ) B ζ γ j O ψ j ,η i + n − r X ℓ ′ =1 h ιζ γ j ιO ψ j ,η , ( R γ j n − r − ℓ ′ , K ) B f j i (cid:12)(cid:12)(cid:12)(cid:12) . The quantity E n,η ( K ) is negligible thanks to the following lemma proven in Section 6.1. Lemma 4.3.
For any n ∈ N , k ≥ and any K ∈ H k possibly depending on γ , butsatisfying (5.1) , we have E n,η ( K ) = O ( n ) N → + ∞ ,η ( η ) . Reduction to non-backtracking variances
From now on, all the constructions take place on the graph Q N , and N varies and goesto + ∞ . So it should be understood that all objects depend on N , although it is not alwaysapparent in the notation.In this section, we will show that the statement of Theorem 2.3 can be reduced toan estimate on the non-backtracking variances (4.4). To this end, we will start from anintegral operator K ∈ K k , and build several new operators J γ K ∈ H k ′ from it. These J γ K will depend on the parameter γ ∈ C + (and N ), but this dependence will always satisfythe following hypothesis. Definition 5.1.
Let k ≥ C + ∋ γ K γ = K γN ∈ H k . We say that K γ satisfieshypothesis (Hol) if • For all 0 < η < η Dir and ( b , . . . , b k ) ∈ B k , the map R ∋ λ K λ +i η ( b , . . . , b k )has an analytic extension K zη ( b , . . . , b k ) to the strip { z : | Im z | < η / } . • For all s >
0, we have(5.1) sup η ∈ (0 ,η Dir ) lim sup N → + ∞ sup η ∈ ( − η , η ) sup λ ∈ I N X ( b ,...,b k ) ∈ B k | K λ +i η η ( b , . . . , b k ) | s < + ∞ . • For almost all λ ∈ I , for all s > t ∈ ( − / , / η ↓ lim sup N → + ∞ N X ( b ,...,b k ) ∈ B k | K λ + t i η η ( b , . . . , b k ) − K λ +i η ( b , . . . , b k ) | s = 0 . Beware that, unless γ K γ ( b , . . . , b k ) is analytic, the quantities K λ +i η +i η ( b , . . . , b k )and K λ +i η η ( b , . . . , b k ) are in general different for η = 0. Remark 5.2.
Note that if K γ and J γ satisfy (Hol) , so does their sum and product. If K γ satisfies (Hol) , then K γ also satisfies (Hol) . Indeed, its restriction to a horizontal lineis real-analytic, so it can be extended holomorphically to a strip by an extension satisfyingthe desired properties.Last but not least, thanks to Corollary C.8 and Proposition C.9, the functions in theclass L γk introduced in Definition C.2 all satisfy (Hol) . Given a linear operator F γ : ℓ ( V ) −→ ℓ ( V ) possibly depending on γ ∈ C + , we defineVar I η ( F γ ) := 1 N N ( I ) X λ j ∈ I |h ˚ ψ j , F λ j +i η ˚ ψ j i ℓ ( V ) | , where ˚ ψ j is the restriction of ψ j to the vertices, which is well-defined thanks to (2.2)˚ ψ j ( v ) = ψ j ( v ) . Let us write Ψ γ,v ( w ) := Im ˜ g γN ( v, w ) and define, for K ∈ H k (possibly depending on γ ∈ C + ):(5.3) h K i γ := 1 P v ∈ V Ψ γ,v ( v ) X ( b ,...,b k ) ∈ B k K ( b , . . . , b k )Ψ γ,o b ( t b k ) if k ≥ h K i γ := 1 P v ∈ V Ψ γ,v ( v ) X v ∈ V K ( v )Ψ γ,v ( v ) if k = 0 . Finally, we denote by = ∈ H the constant function equal to one on every vertex.In the sequel, we will also denote by ˆ the function on G N which is constant equal to one.As a first step towards the reduction to non-backtracking variances, we control thequantities appearing in Theorems 2.1 and 2.3 by some discrete variances of the formVar I η . Proposition 5.3. (1)
Let f N ∈ L ∞ ( G N ) be a sequence of functions with k f N k ∞ ≤ . Wemay build ( J γf,p ) p =1 ,..., , each belonging to H or H and satisfying (Hol) , such that lim η ↓ lim sup N →∞ N N ( I ) X λ j ∈ I |h ψ j , f ψ j i − h f i γ j | ≤ C X p =1 lim η ↓ lim sup N →∞ Var I η (cid:16) ( J γf,p ) G − h J γf,p i γ (cid:17) . (2) Let k ≥ and let K N ∈ K k be a sequence of non-backtracking integral operatorssatisfying |K N ( x, y ) | ≤ for all x, y ∈ G N .We may build ( J γ K ,p ) p =1 ,..., , each belonging to H k p for k − ≤ k p ≤ k , and satisfying (Hol) , such that lim η ↓ lim sup N →∞ N N ( I ) X λ j ∈ I |h ψ j , K ψ j i−hKi γ j | ≤ C X p =1 lim η ↓ lim sup N →∞ Var I η (cid:16) ( J γ K ,p ) G − h J γ K ,p i γ (cid:17) . The idea of the proof is quite simple: expand the eigenfunction ψ j in the basis ofsolutions S λ , C λ given in § h f i γ j into convenient pieces, so thateach operator in the discrete variance has zero mean. The scheme is quite similar to theequilateral case [19]. We give the technical details in Appendix B.1Our next step is to show that the discrete variance of J − h J i γ can be bounded bysome non-backtracking variances, plus some terms which will be negligible. Before statingthe proposition, we need to introduce some notation. In the sequel, we will often write h J i γ instead of h J i γ in the computations of the variances.If d ( x ) is the degree of x ∈ V , we denote P = d A G . Let N γ ( x ) = Im ˜ g γ (˜ x, ˜ x ) , P γ = dN γ P N γ d . We define S T,γ : ℓ ( V ) → ℓ ( V ) and e S T,γ : ℓ ( V ) → ℓ ( V ) by S T,γ J = 1 T T − X s =0 ( T − s ) P sγ J and e S T,γ J = 1 T T X s =1 P sγ J .
UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 19
We also define L γ : ℓ ( V ) → ℓ ( B ) and E γ : ℓ ( V ) → ℓ ( V ) by( L γ J )( b ) = S γ ( L b ) | ˜ g γ ( t b , t b ) | (1 + ζ γ ( b ) ζ γ ( b b ))Re S γ ( L b ) | ζ γ ( b ) | (cid:18) J ( t b ) N γ ( o b ) − J ( o b ) N γ ( t b ) (cid:19) , ( E γ J )( o b ) = X t b ∼ o b (cid:20) Im S γ ( L b ) Re ˜ g γ ( t b , t b )Re S γ ( L b ) (cid:16) J ( t b ) N γ ( o b ) − J ( o b ) N γ ( t b ) (cid:17)(cid:21) . Proposition 5.4. (1)
Let J ∈ H . For any T ∈ N , we have (5.4) Var Iη ( J − h J i γ ) ≤ Var I nb ,η [ L γ d − S T,γ ( J − h J i γ )]+ Var Iη [ E γ d − S T,γ ( J − h J i γ )] + Var Iη ( e S T,γ ( J − h J i γ )) . (2) Let J ∈ H k satisfy (Hol) . There exists P , P ∈ N depending only on k , and operators ( L γp J ) ≤ p ≤ P and ( F γp J ) ≤ p ≤ P with L γp J ∈ H and F γp J ∈ H k p for some ≤ k p ≤ k ,all satisfying (Hol) , such that Var Iη ( J G − h J i γ ) ≤ P X p =1 Var Iη ( L γp J − h L γp J i γ ) + P X p =1 Var I nb ,η (cid:0) F γp J (cid:1) + O N → + ∞ ,γ ( η ) . This proposition will be applied to the J = J γf,p , J γ K ,p appearing in Proposition 5.3.The proof is given in Appendix B.2.To finish this section, we will show that the error variances Var Iη [ E γ d − S T,γ ( J − h J i γ )]and Var Iη ( e S T,γ ( J − h J i γ )) from (5.4) vanish asymptotically. Note that if we apply part(1) of Proposition 5.4 to the first sum of variances in part (2), then all that remains willbe to control non-backtracking variances.If K ∈ C V ∼ = H , we define the “weighted Hilbert-Schmidt norm”(5.5) k K k γ := 1 | V | X v ∈ V | K ( v ) | N γ ( x ) . Then we have the following bound proven in Section 6.1. An analogous bound for non-backtracking variances will be given in Proposition 7.1. These two propositions tell us thequantum variance(s) are dominated by weighted Hilbert-Schmidt norms.
Lemma 5.5.
Let K γ ∈ H satisfy hypothesis (Hol) from Definition 5.1. We have lim η ↓ lim sup N →∞ Var Iη ( K γ ) . lim η ↓ lim sup N →∞ Z I k K λ +i η k λ +i η d λ. This lemma makes it easy to deal with Var Iη [ E γ d − S T,γ ( J − h J i γ )]. Indeed, for anyfixed T ∈ N , E γ d − S T,γ ( K γ − h K γ i γ ) ∈ H satisfies (Hol) by Remark 5.2. Therefore,from the expression of E γ , which involves Im S γ ( L b ) = O ( η ), we see using (5.1) that(5.6) lim η ↓ lim N →∞ Var Iη (cid:0) E γ d − S T,γ ( K γ − h K γ i γ (cid:1) = 0 . The error Var Iη ( e S T,γ ( J − h J i γ )) is dealt with thanks to the following lemma. Lemma 5.6.
Let K ∈ H satisfy (Hol) . We then have lim T →∞ sup η ∈ (0 ,η Dir ) lim sup N →∞ sup λ ∈ I (cid:13)(cid:13) e S T,γ ( K γ − h K γ i γ ) (cid:13)(cid:13) γ = 0 . Proof.
Let us write ( Y γ K )( v ) = d ( v ) N γ ( v ) P w ∈ V N γ ( w ) K ( w ) P w ∈ V d ( w ) , so that Y γ K = h N γ K i U h d i U dN γ , where h J i U := N P v ∈ V J ( v ) is the average with respect to the uniform scalar product.Noting that for any s ∈ N , we have P sγ = dN γ P s N γ d , we get (cid:13)(cid:13) e S T,γ K γ − Y γ K γ (cid:13)(cid:13) γ = 1 N X v ∈ V (cid:12)(cid:12)(cid:12) T T X s =1 d ( v ) (cid:16) P s N γ K γ d (cid:17) ( v ) − h N γ K γ i U h d i U d ( v ) (cid:12)(cid:12)(cid:12) ≤ DN T T X s =1 (cid:13)(cid:13)(cid:13)(cid:13) P s (cid:16) N γ K γ d − h N γ K γ i U h d i U (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) ℓ ( V,d ) ≤ DN T T X s =1 (1 − β ) s (cid:13)(cid:13)(cid:13)(cid:13) N γ K γ d − h N γ K γ i U h d i U (cid:13)(cid:13)(cid:13)(cid:13) ℓ ( V,d ) ≤ DβN T (cid:13)(cid:13)(cid:13)(cid:13) N γ K γ d − h N γ K γ i U h d i U (cid:13)(cid:13)(cid:13)(cid:13) ℓ ( V,d ) ≤ DβN T k N γ K γ k ℓ ( V,d ) , where we used the fact that N γ K γ d − h N γ K γ i U h d i U is orthogonal to constants in the space ℓ ( V N , d N ), the assumption (EXP) , and the fact that d ≥ K γ satisfies (Hol) , we know by Remark 5.2 that N γ K γ satisfies hypothesis (Hol) , so that N k N γ K γ k ℓ ( V ) = O N → + ∞ ,γ (1). Therefore,(5.7) lim T →∞ sup η ∈ (0 ,η Dir ) lim sup N →∞ sup λ ∈ I (cid:13)(cid:13) e S T,γ K γ − Y γ K γ (cid:13)(cid:13) γ = 0 . Now, the result follows by applying (5.7) to ˚ K γ := K γ − h K γ i γ , and by noting that Y γ ˚ K γ = 0, since Y γ h K γ i γ = h K γ i γ h N γ i U h d i U dN γ and, by definition, h K γ i γ = h N γ K γ i U h N γ i U . (cid:3) Combining Propositions 5.3 and 5.4, Lemma 5.5, Lemma 5.6 and equation (5.6) weobtain the following corollary.
Corollary 5.7.
Let Q N be a sequence of quantum graphs satisfying (Data) for each N ,such that (EXP) , (BST) , (Green) and (Non-Dirichlet) hold true on the interval I .Suppose that for any K γ ∈ H k satisfying (Hol) , k ≥ , we have (5.8) lim η ↓ lim N →∞ Var I nb ,η ( K γ ) = 0 . Then, if f N ∈ L ∞ ( G N ) is a sequence of functions satisfying k f N k ∞ ≤ , (2.8) holds.Furthermore, for any k ≥ , if K N ∈ K k is a sequence non-backtracking integral operatorsatisfying |K N ( x, y ) | ≤ for all x, y ∈ G N , then (2.10) holds. The rest of the paper will be devoted to proving (5.8), thus establishing Theorem 2.3.6.
Contour integrals and complex analysis
In this section, we shall prove Lemmas 4.3 and 5.5 and develop tools from complexanalysis that will be used later on. The quantities we wish to estimate are expressed assums over the eigenvalues of the quantum graph Q N , which are the poles of the Greenfunction g N . Thanks to Cauchy’s formula, these sums will be expressed as contour integralsinvolving the Green functions. The manipulation of the unknown eigenfunctions ψ ( N ) j isthus replaced by manipulation of g N . Later on, this will be replaced by the Green function˜ g N of the universal cover, and finally we will use that it converges in distribution to theGreen function of the limiting random tree T .If χ ∈ C ∞ c ( R ), we will denote by ˜ χ an almost analytic extension of χ , i.e., a smoothfunction ˜ χ : C C such that ˜ χ ( z ) = χ ( z ) for z ∈ R , ∂ ˜ χ∂z ( z ) = O ((Im z ) ), and(6.1) supp ˜ χ ⊂ { z ; Re z ∈ supp χ } . Here ∂∂z = ( ∂∂x + i ∂∂y ). For instance, one can take ˜ χ ( x + i y ) = χ ( x ) + i yχ ′ ( x ) − y χ ′′ ( x ).We refer the reader to [15, § UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 21
Recall that if K γ ∈ H k satisfies (Hol) , then for any 0 < η < η Dir and ( b , . . . , b k ) ∈ B k , λ K λ +i η ( b , . . . , b k ) admits a holomorphic extension to {| Im z | < η / } , which isdenoted by K zη ( b , . . . , b k ). Proposition 6.1.
Let χ ∈ C ∞ c ( I ) . Define ˜ χ as above. Let k, k ′ ∈ N and let K γ ∈ H k ,and K ′ γ ∈ H k ′ satisfy (Hol) . Then for any < η < min( η Dir , η I ) , we have (6.2) 1 N X j ≥ X ( b ; b k ) X ( b ′ ; b ′ k ′ ) b ′ = b χ ( λ j ) K γ j ( b ; b k ) K ′ γ j ( b ′ ; b ′ k ′ ) ψ j ( o b k ) ψ j ( o b ′ k ′ )= − N X ( b ; b k ) X ( b ′ ; b ′ k ′ ) b ′ = b π i Z Γ η / ˜ χ ( z ) g zN ( o b k , o b ′ k ′ ) K zη ( b ; b k ) K ′ zη ( b ′ ; b ′ k ′ )d z + O N → + ∞ ,η ( η ) , where Γ η is the boundary of Ω η = I + i[ − η, η ] .The same formula holds if o b k and/or o b ′ k ′ are replaced by t b k and/or t b ′ k ′ in both terms.Proof. Step 1 : From a sum to an integral
Let v, w ∈ V N , and let h be a holomorphic function in a strip { z ∈ C ; | Im z | < η Dir } .The function h may depend on N , v, w .Using (3.19), and noting that Z bλ = Z bλ for λ ∈ R , we have X j ≥ χ ( λ j ) h ( λ j ) ψ j ( v ) ψ j ( w ) = X j ≥ χ ( λ j ) h ( λ j ) h Z b v λ j , ψ j ih Z b w λ j , ψ j i , where we chose b v = b w such that v = t b v and w = t b w . This is possible since d ( v ) , d ( w ) ≥ (Data) .We may define holomorphic functions b Z bz as in the end of Section 3.4, on some open setΩ I := I + i[ − η I , η I ] which does not depend on N ∈ N or b ∈ B N .Let 0 < η < min( η I , η Dir ). Consider the rectangle Ω η := I + i[ − η , η ] and the curveΓ η := ∂ Ω η . Cauchy’s integral formula f ( λ ) = π i R Γ f ( z ) z − λ d z for analytic f generalizes to f ( λ ) = 12 π i (cid:18) Z Γ η f ( z ) z − λ d z + Z Ω η ∂f /∂zz − λ d z ∧ d z (cid:19) for λ ∈ Ω η , where d z ∧ d z = −
2i d x d y , see e.g. [18, Theorem 1.2.1].We apply this result to λ = λ j and f ( z ) = ˜ χ ( z ) h ( z ) h b Z b v z , ψ j ih b Z b w z , ψ j i . Noting that z ∈ Ω I h ( z ) h b Z b v z , ψ j ih b Z b w z , ψ j i is holomorphic, we obtain χ ( λ j ) h ( λ j ) ψ j ( v ) ψ j ( w ) = 12 π i (cid:18) Z Γ η ˜ χ ( z ) h ( z ) h b Z b v z , ψ j ih b Z b w z , ψ j i z − λ j d z + Z Ω η h ( z ) h b Z b v z , ψ j ih b Z b w z , ψ j i z − λ j ∂ ˜ χ∂z d z ∧ d z (cid:19) . Next, we want to sum this expression over j . Since ( ψ j ) is an orthonormal basis of L ( G N ), we obtain that X j ≥ h b Z b v z , ψ j ih b Z b w z , ψ j i λ j − z = X j ≥ h b Z b v z , ψ j ih ψ j , b Z b w z i λ j − z = h b Z b v z , ( H Q N − z ) − b Z b w z i . Therefore, we have X j χ ( λ j ) h ( λ j ) ψ j ( v ) ψ j ( w ) = − π i (cid:16) Z Γ η ˜ χ ( z ) h ( z ) h b Z b v z , ( H Q N − z ) − b Z b w z i d z + Z Ω η h ( z ) h b Z b v z , ( H Q N − z ) − b Z b w z i ∂ ˜ χ∂z d z ∧ d z (cid:17) . Now, we know that (cid:12)(cid:12) h b Z b v z , ( H Q N − z ) − b Z b w z i (cid:12)(cid:12) ≤ C Im z , where C does not depend on N ∈ N or on v, w .We deduce that (cid:12)(cid:12)(cid:12)(cid:12) X j χ ( λ j ) h ( λ j ) ψ j ( v ) ψ j ( w ) − − π i Z Γ η ˜ χ ( z ) h ( z ) h b Z b v z , ( H Q N − z ) − b Z b w z i d z (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ η Z Ω η | h ( z ) | d z ∧ d z. Step 2 : Using the properties of Z z We would like to replace the b Z z by Z z in theprevious formula, for the following reason. Since b v = b w , the map y ( H Q N − z ) − ( x, y )is an eigenfunction on b w , so that(6.3) (cid:0) ( H Q N − z ) − Z b w z (cid:1) ( x ) = ( H Q N − z ) − ( x, w )by (3.19). The map x ( H Q N − z ) − ( x, w ) is an eigenfunction on b v , so that, by (3.19)again, we have h Z b v z , ( H Q N − z ) − Z b w z i = ( H Q N − z ) − ( v, w ) . To estimate the cost of replacing b Z z by Z z , we write(6.4) D b Z b v z , ( H Q N − z ) − b Z b w z E = D Z b v z , ( H Q N − z ) − Z b w z E + D b Z b v z − Z b v z , ( H Q N − z ) − Z b w z E + D Z b v z , ( H Q N − z ) − (cid:16) b Z b w z − Z b w z (cid:17)E + D b Z b v z − Z b v z , ( H Q N − z ) − (cid:16) b Z b w z − Z b w z (cid:17)E By (3.20), the last term is easy to estimate (cid:12)(cid:12)(cid:12)D b Z b v z − Z b v z , ( H Q N − z ) − (cid:16) b Z b w z − Z b w z (cid:17)E(cid:12)(cid:12)(cid:12) ≤ C I , M (Im z ) k ( H Q N − z ) − k≤ C I , M | Im z | . As to the second term on the right-hand side of (6.4), we use (3.20), the Cauchy-Schwarzinequality and (6.3), to see that its modulus is bounded by some constant times | Im z | (cid:13)(cid:13) ( H Q N − z ) − ( · , w ) (cid:13)(cid:13) L ( b v ) := | Im z | (cid:18)Z L bv (cid:12)(cid:12) ( H Q N − z ) − ( x b v , w ) (cid:12)(cid:12) d x b v (cid:19) / . Since ( H Q N − z ) − ( x b v , w ) = S z ( L b v − x b v ) S z ( L b v ) g zN ( o b v , w ) + S z ( x b v ) S z ( L b v ) g zN ( t b v , w ) , we deduce from (Data) and (Non-Dirichlet) that the second term is bounded by C I, M ,C Dir | Im z | ( | g zN ( o b v , w ) | + | g zN ( v, w ) | ) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 23
We have a similar estimate for the third term. Therefore, we have(6.5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j χ ( λ j ) h ( λ j ) ψ j ( v ) ψ j ( w ) − − π i Z Γ η ˜ χ ( z ) h ( z )( H Q N − z ) − ( v, w )d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cη (cid:20) Z Ω η | h ( z ) | d z ∧ d z + Z Γ η | h ( z ) | (cid:0) | g zN ( o b v , w ) | + 2 | g zN ( v, w ) | + | g zN ( v, o b w ) | + 1 (cid:1) d z (cid:21) . Step 3 : Using (Hol)
Take η = η for η ∈ (0 , η I ). For each ( b ; b k ), ( b , b ′ k ′ ), weapply (6.5) with h ( z ) = K zη ( b ; b k ) K ′ zη ( b ′ ; b ′ k ′ ).When summing over ( b ; b k ), ( b , b ′ k ′ ) and dividing by N , the first term in the remainderis bounded by C ′ η sup λ ∈ I ,η ∈ ( − η , η ) N X ( b ; b k ) X ( b ′ ; b ′ k ′ ) b ′ = b | K λ +i ηη ( b ; b k ) K ′ λ +i ηη ( b ′ ; b ′ k ′ ) | , which is a O N → + ∞ ,η ( η ) by the Cauchy-Schwarz inequality and (5.1).Concerning the second term, using Cauchy-Schwarz and (5.1), it can be bounded by C ′ η (cid:18) sup λ ∈ I ,η ∈ ( − η , η ) N X ( b ; b k ) X ( b ′ ; b ′ k ′ ) b ′ = b (cid:0) | g zN ( o ˇ b k , o b ′ k ′ ) | + | g zN ( o b k , o b ′ k ′ ) | + | g zN ( o b k , o ˇ b ′ k ′ ) | + 1 (cid:1)(cid:19) / , where ˇ b k , ˇ b ′ k ′ are chosen so that t ˇ b k = o b k and t ˇ b ′ k ′ = o b ′ k ′ but ˇ b k = ˇ b ′ k ′ . Applying Corol-lary C.8 to F z ( b ; b k ) = P ( b ′ ; b ′ k ′ ) b ′ = b ( ... ), we deduce this is O N → + ∞ ,η ( η ). Proposition 6.1follows. (cid:3) We deduce the following corollary, which we will use several times.
Corollary 6.2.
Let χ ∈ C ∞ c ( I ) , let K γ ∈ H k , and K ′ γ ∈ H k ′ satisfy (Hol) . Then N X j ≥ X ( b ; b k ) X ( b ′ ; b ′ k ′ ) b ′ = b χ ( λ j ) K γ j ( b ; b k ) K ′ γ j ( b ′ ; b ′ k ′ ) ψ j ( o b k ) ψ j ( o b ′ k ′ ) = O N → + ∞ ,η (1) . The same result holds if o b k and/or o b ′ k ′ are replaced by t b k and/or t b ′ k ′ .Proof. By Proposition 6.1, up to a term which is O N → + ∞ ,η ( η ) the modulus of the quan-tity we want to estimate is bounded by CN sup λ ∈ I X ( b ; b k ) X ( b ′ ; b ′ k ′ ) b ′ = b (cid:12)(cid:12)(cid:12)(cid:12) g λ ± i η N ( o b k , o b ′ k ′ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) K λ ± i η η ( b ; b k ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) K ′ λ ± i η η ( b ′ ; b ′ k ′ ) (cid:12)(cid:12)(cid:12)(cid:12) . Using H¨older’s inequality, (5.1) and Corollary C.8 this is O N → + ∞ ,γ (1). (cid:3) Remark 6.3.
The very same proof as that of Proposition 6.1 gives us that, if K γ ∈ H satisfies (Hol) , then up to an error O N → + ∞ ,η ( η ) we have1 N X j ≥ X v ∈ V χ ( λ j ) K γ j ( v ) | ψ j ( v ) | = − N X v ∈ V π i Z Γ η / ˜ χ ( z ) g zN ( v, v ) K zη ( v )d z, and, in particular, as in Corollary 6.2,(6.6) 1 N X j ≥ X v ∈ V χ ( λ j ) K γ j ( v ) | ψ j ( v ) | = O N → + ∞ ,η (1) . Applications.
The previous results are used here to prove Lemmas 4.3 and 5.5, andwill be used again in the proof of Proposition 7.1 later on.
Proof of Lemma 4.3.
Let χ ∈ C ∞ c ( I ) be positive and equal to one on I .For simplicity, we will consider only the term r = n , ℓ = n in (4.13). All the otherterms can be treated in the same fashion. By (C.3) and Cauchy-Schwarz, this term is1 N N ( I ) X λ j ∈ I |h f ∗ j , K B ζ γ j O ψ j ,η i| . (cid:18) N X j ≥ χ ( λ j ) k f ∗ j k (cid:19) / (cid:18) N X j ≥ χ ( λ j ) k K B ζ γ j O ψ j ,η k (cid:19) / . We have k f ∗ j k = P b ∈ B | ψ j ( o b ) − ζ γj (ˆ b ) ψ j ( t b ) | S λj ( L b ) . Applying Corollary 6.2, with k = k ′ = 1, K ′ = Id, and K taking values K γ ( b ) = S γ ) ( b ) , K γ ( b ) = ζ γ (ˆ b ) S γ ) ( b ) , K γ ( b ) = ζ γ (ˆ b ) S γ ) ( b ) , K γ ( b ) = | ζ γ (ˆ b ) | S γ ) ( b ) , which all satisfy (Hol) thanks to Remark 5.2, we deduce that the firstfactor is O N → + ∞ ,η (1). Concerning the second sum, it can be written1 N X j ≥ X ( b ; b k ) X ( b ′ ; b ′ k ) b ′ = b χ ( λ j ) K ( b ; b k ) K ( b ′ ; b ′ k ) ζ γ j ( b k ) O η ( b k ) ζ γ j ( b ′ k ) O η ( b ′ k ) ψ j ( t b k ) ψ j ( t b ′ k ) . Using Remark 5.2, we may bound this as in the proof of Corollary 6.2. Then the resultfollows from (4.3). (cid:3)
Proof of Lemma 5.5.
Let K γ ∈ H satisfy (Hol) and let M γ ( v ) := N γ ( v ) | g γN ( v, v ) | − / .We have by (C.3) and Cauchy-Schwarz,Var Iη (K γ ) = 1 N N ( I ) X λ j ∈ I |h ˚ ψ j , K γ j ˚ ψ j i| . (cid:18) N X λ j ∈ I k M − γ j ˚ ψ j k (cid:19) / (cid:18) N X λ j ∈ I k M γ j K γ j ˚ ψ j k (cid:19) / . Therefore, if χ ∈ C ∞ c ( I ) is positive, and equal to one on I , we haveVar Iη (K γ ) . (cid:18) N X j ≥ χ ( λ j ) X v ∈ V | ψ j ( v ) | | M γ j ( v ) | (cid:19)(cid:18) N X j ≥ X v ∈ V | M γ j ( v ) K γ j ( v ) | | ψ j ( v ) | (cid:19) . The first factor is O N → + ∞ ,γ (1), by (6.6), as | M γ | − satisfies (Hol) .Next, using Remark 6.3, up to a term O N → + ∞ ,η ( η ), the second factor is − N π i X v ∈ V N Z Γ η / ˜ χ ( z ) g zN ( v, v )( M KM K ) zη ( v )d z. Using (5.2), this may be replaced by − Nπ i P v ∈ V N R Γ η / ˜ χ ( z ) g zN ( v, v ) | M z ( v ) K z ( v ) | d z . In-deed, dominated convergence is applicable by (5.1). The modulus of this is bounded by πN P v ∈ V N R Γ η / | N z ( v ) K z ( v ) | d z . (cid:3) UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 25 Upper bound on the non-backtracking variance
Let k ≥
1. Given
K, K ′ ∈ H k and γ = λ + i η ∈ C + , we define the weighted scalarproduct(7.1) (cid:0) K, K ′ (cid:1) γ := 1 N X ( b ; b k ) ∈ B k Im R − γ ( t b ) · K ( b ; b k ) K ′ ( b ; b k ) · Im R + γ ( o b k ) , and k K k γ := ( K, K ) γ the associated norm. The aim of this section is to prove the followingproposition, which tells us that the non-backtracking quantum variance is dominated bythis weighted Hilbert-Schmidt norm. Proposition 7.1.
Let I ⊂ I , with I as in (Green) . There exists C I > such that forall k ≥ , and all K γ ∈ H k satisfying hypothesis (Hol) from Definition 5.1, we have lim η ↓ lim sup N →∞ Var I nb ,η ( K γ ) ≤ C I lim η ↓ lim sup N →∞ Z I k K λ +i η k λ +i η d λ. This proposition is analogous in appearance to Lemma 5.5, but is actually much moreinvolved because the constant C I does not depend on k , which is important for the nextsections; it is a lot easier to derive cruder bounds depending on k , by arguing as in § Proof.
Let χ ∈ C ∞ c ( I ), with χ ≡ I , 0 ≤ χ ≤
1, and let | α γ ( b ) | = Im R − γ ( t b ) .Denoting γ j = λ j + i η and using (C.3), we have(7.2) Var I nb ,η ( K γ ) = (cid:18) N N ( I ) X λ j ∈ I (cid:12)(cid:12)(cid:12) h f ∗ j , K λ j +i η B f j i (cid:12)(cid:12)(cid:12) (cid:19) ≤ C ( I ) (cid:18) N X λ j ∈ I k α − γ j f ∗ j k (cid:19)(cid:18) N X λ j ∈ I k α γ j K γ j B f j k (cid:19) ≤ C ( I ) (cid:18) N X j ≥ χ ( λ j ) k α − γ j f ∗ j k (cid:19)(cid:18) N X j ≥ χ ( λ j ) k α γ j K γ j B f j k (cid:19) . The first factor is O N −→ + ∞ ,γ (1) by application of Corollary 6.2, as in Section 6.1. Sowe focus on the second factor. Step 1: From a sum to an integral
Recalling (3.17) and definition (4.1), we have X j ≥ χ ( λ j ) k α γ j K γ j B f j k = X b ∈ B N X ( b ; b k ) , ( b ′ ; b ′ k ) ∈ B b k − X j ≥ χ ( λ j ) | α γ j ( b ) | × K γ j ( b ; b k ) (cid:16) ψ j ( t b k ) S λ j ( L b k ) − ζ γ j ( b k ) ψ j ( o b k ) S λ j ( L b k ) (cid:17) K γ j ( b ; b ′ k ) (cid:16) ψ j ( t b ′ k ) S λ j ( L b ′ k ) − ζ γ j ( b ′ k ) ψ j ( o b ′ k ) S λ j ( L b ′ k ) (cid:17) . By Remark 5.2, | α γ ( b ) | , K γ ( b ; b k ), and K γ ( b ; b ′ k ) satisfy (Hol) , so they have analyticextensions to the strip {| Im z | < η / } which we denote by f zα,η ( b ), K zη ( b ; b k ) and K zη ( b ; b ′ k ), respectively (note that in general K zη ( b ; b ′ k ) = K zη ( b ; b ′ k )). To lighten theexpressions a bit, we denote J zη ( b ; b k ; b ′ k ) := f zα,η ( b ) K zη ( b ; b k ) K zη ( b ; b ′ k ) S z ( L b k ) S z ( L b ′ k ) . Next, ζ λ +i η ( b ′ k ) = ζ λ − i η ( b ′ k ) can be extended holomorphically by ζ z − i η ( b ′ k ). We maythus apply Proposition 6.1 to obtain (7.3) X j ≥ χ ( λ j ) k α γ j K γ j B f j k = − π i X b ∈ B N X ( b ; b k ) , ( b ′ ; b ′ k ) ∈ B b k − Z Γ η ˜ χ ( z ) J zη ( b ; b k ; b ′ k ) × (cid:16) g zN ( t b k , t b ′ k ) − ζ z +i η ( b k ) g zN ( o b k , t b ′ k ) − ζ z − i η ( b ′ k ) g zN ( t b k , o b ′ k )+ ζ z +i η ( b k ) ζ z − i η ( b ′ k ) g zN ( o b k , o b ′ k ) (cid:17) d z + N O N → + ∞ ,η ( η ) . By (6.1), R Γ η reduces to R I − i η − R I +i η , which we denote by R Γ ′ η , η = η . Step 2: From finite graphs to infinite trees
The aim of this step is to use the fact that our graphs look locally like trees (assumption (BST) ) to replace the Green function g zN on Q N in (7.3) by ˜ g zN , the Green function ofthe universal covering tree e Q N .Given a rooted quantum tree [ Q , b ], and two paths p k = ( b ; b k ) and p ′ k = ( b ′ ; b ′ k ) inB b k − , let us introduce(7.4) f z ([ Q , b ] , p k , p ′ k ) := G z ( t b k , t b ′ k ) − ζ z +i η ( b k ) G z ( o b k , t b ′ k ) − ζ z − i η ( b ′ k ) G z ( t b k , o b ′ k )+ ζ z +i η ( b k ) ζ z − i η ( b ′ k ) G z ( o b k , o b ′ k ) e f z ([ Q , b ] , p k , p ′ k ) := e G z ( t b k , t b ′ k ) − ζ z +i η ( b k ) e G z ( o b k , t b ′ k ) − ζ z − i η ( b ′ k ) e G z ( t b k , o b ′ k )+ ζ z +i η ( b k ) ζ z − i η ( b ′ k ) e G z ( o b k , o b ′ k ) , where e G z ( o b k , t b ′ k ) is the Green’s function of the universal cover e Q of the given Q (so wefix some lift ˜ b and consider ˜ p k , ˜ p ′ k ∈ B ˜ b k − projecting to p k , p ′ k ; see § Q is a tree then f z ([ Q , b ] , p k , p ′ k ) = e f z ([ Q , b ] , p k , p ′ k ). Now1 N (cid:12)(cid:12)(cid:12) X b ∈ B N X p k ,p ′ k ∈ B b k − Z Γ ′ η ˜ χ ( z ) J zη ( b ; b k ; b ′ k ) × (cid:16) f z ([ Q N , b ] , p k , p ′ k ) − e f z ([ Q N , b ] , p k , p ′ k ) (cid:17) d z (cid:12)(cid:12)(cid:12) ≤ (cid:18) N X b ∈ B N X p k ,p ′ k ∈ B b k − Z Γ ′ η (cid:12)(cid:12) ˜ χ ( z ) J zη ( b ; b k ; b ′ k ) (cid:12)(cid:12) d z (cid:19) / × (cid:18) N X b ∈ B N X p k ,p ′ k ∈ B b k − Z Γ ′ η (cid:12)(cid:12)(cid:12) f z ([ Q N , b ] , p k , p ′ k ) − e f z ([ Q N , b ] , p k , p ′ k ) (cid:12)(cid:12)(cid:12) d z (cid:19) / . The first factor is controlled by (5.1). For the second factor, we note that if F z ([ Q , b ]) := X p k ,p ′ k ∈ B b k − (cid:12)(cid:12)(cid:12) f z ([ Q , b ] , p k , p ′ k ) − e f z ([ Q , b ] , p k , p ′ k ) (cid:12)(cid:12)(cid:12) then the same arguments leading to (C.6) show thatlim sup N → + ∞ N Z Γ ′ η X b ∈ B N F z ([ Q N , b ])d z ≤ C sup λ ∈ I E P ( F λ ± i η ) = 0 , since the last expectation runs over trees [ Q , ( b , x )] and thus F z ([ Q , ( b , x )]) = 0. UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 27
Combining all this with (7.3), we obtain that(7.5)lim sup N → + ∞ N X λ j ∈ I k α γ j K γ j B f j k ≤ lim sup N → + ∞ − N π i X b ∈ B N X p k ,p ′ k ∈ B b k − Z Γ ′ η ˜ χ ( z ) J zη ( b ; b k ; b ′ k ) × e f z ([ Q N , b ] , p k , p ′ k )d z + O ( η ) . Let us write B ,kN := { b ∈ B N : ρ G N ( o b ) ≤ k } , where ρ G N ( o b ) is the injectivity radiusof o b , and B ,kN := B N \ B ,kN . By (BST) , we have |B ,kN | = o ( N ). So by Cauchy-Schwarz,if Y N ( b ) = P p k ,p ′ k R ( . . . ), we have(7.6) (cid:12)(cid:12)(cid:12)(cid:12) N X b ∈B ,kN Y N ( b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) |B ,kN | N (cid:19) / (cid:18) N X b ∈ B N | Y N ( b ) | (cid:19) / −→ N → + ∞ P b ∈ B N above by P b ∈B ,kN . Step 3: Off-diagonal terms vanish
Here we mean to show that the terms with p k = p ′ k in (7.5) vanish. Suppose p ′ k =( b ′ ; b ′ k ) = ( b ; b k ) = p k . If b ∈ B ,kN , this implies t b k = t b ′ k . Using (3.10) on ( v , . . . , v s )where v = t b k , v = o b k , v s − = o b ′ k , v s = t b ′ k we obtain(7.7)˜ g zN ( t b k , t b ′ k ) − ζ z +i η ( b k )˜ g zN ( o b k , t b ′ k ) = ζ z ( b ′ k )˜ g zN ( t b k , o b ′ k ) − ζ z +i η ( b k ) ζ z ( b ′ k )˜ g zN ( o b k , o b ′ k ) . Similarly, using (3.11) then (3.10) on ( v s , . . . , v ), then (3.11), we have(7.8)˜ g zN ( t b k , t b ′ k ) − ζ z − i η ( b ′ k )˜ g zN ( t b k , o b ′ k ) = ζ z ( b k )˜ g zN ( o b k , t b ′ k ) − ζ z − i η ( b ′ k ) ζ z ( b k )˜ g zN ( o b k , o b ′ k ) . Let us first consider the part of Γ ′ η where Im z <
0. Recalling (7.4), if we use (7.7)along with the Cauchy-Schwarz inequality, we obtain(7.9) lim η ↓ lim sup N →∞ (cid:12)(cid:12)(cid:12) N X b ∈B ,kN X ( b ; b k ) X ( b ′ ; b ′ k ) =( b ; b k ) Z I ˜ χ ( λ − i η / J λ − i η η ( b ; b k ; b ′ k ) × e f λ − i η ([ Q N , b ] , p k , p ′ k )d λ (cid:12)(cid:12)(cid:12) ≤ C lim η ↓ lim sup N →∞ sup λ ∈ I h N X b ∈ B N X ( b ; b k ) , ( b ′ ; b ′ k ) ∈ B b k − (cid:12)(cid:12)(cid:12) J λ − i η η ( b ; b k ; b ′ k ) × (cid:16) | ˜ g λ − i η N ( t b k , o b ′ k ) | + | ζ λ + η ( b k )˜ g λ − i η N ( o b k , o b ′ k ) | (cid:17)(cid:12)(cid:12)(cid:12) i / × lim η ↓ Z I lim sup N →∞ (cid:26) N X b ∈ B N X ( b ; b k ) , ( b ′ ; b ′ k ) ∈ B b k − (cid:12)(cid:12)(cid:12) ζ λ − i η ( b ′ k ) − ζ λ − η ( b ′ k ) (cid:12)(cid:12)(cid:12) (cid:27) / d λ. Thanks to (5.1), the first factor is finite. Concerning the second factor, thanks toCorollary C.8, the integrand is bounded independently of η , and, by Proposition C.9,it goes to zero almost everywhere as η ↓
0. Therefore, by the dominated convergencetheorem, the double limit (7.9) is zero.For the part of Γ ′ η / with Im z >
0, we argue similarly, using (7.8) instead of (7.7).
From what precedes we thus obtainlim η ↓ lim sup N →∞ N X λ j ∈ I χ ( λ j ) k α γ j K γ j B f j k ≤ lim η ↓ lim sup N →∞ − πN i X b ∈B ,kN X ( b ; b k ) ∈ B b k − Z Γ ′ η ˜ χ ( z ) J zη ( b ; b k ; b k ) e f z ([ Q N , b ] , p k , p k )d z =: ( ⋆ )As in (7.6), we may replace P b ∈B ,kN by P b ∈ B . Step 4: Adjusting the energies.
To finish the proof of Proposition 7.1, there remainsto put ( ⋆ ) into final form by setting all the spectral parameters equal to λ +i η with λ ∈ I .Thanks to (5.2), we have for almost all λ ∈ I ,lim η ↓ lim sup N → + ∞ (cid:18) N X ( b ; b k ) ∈ B k (cid:12)(cid:12)(cid:12) ˜ χ ( λ ± i η J λ ± i η η ( b ; b k ; b k ) − ˜ χ ( λ ) (cid:12)(cid:12) K λ +i η ( b ; b k ) (cid:12)(cid:12) S λ ( L b k ) | α λ +i η ( b ) | (cid:12)(cid:12)(cid:12) (cid:19) / = 0 . Using (5.1) and Remark 5.2, we may therefore apply the dominated convergence theoremto deduce that( ⋆ ) = lim η ↓ lim sup N →∞ πN X ( b ; b k ) ∈ B k Z I h χ ( λ ) | K λ +i η ( b ; b k ) | | α λ +i η ( b ) | × Im n e f λ + i η ([ Q N , b ] , p k , p k ) S λ ( L b k ) oi d λ, where we used that e f λ − i η ( p k , p k ) = e f λ + i η ( p k , p k ), as readily checked.Recalling (7.4), we note that the energies in e f λ + i η S λ are not homogeneous. We thus useProposition C.9 and dominated convergence to replace ˜ g λ + i η ˜ g λ +i η , ζ λ + 5i η S λ ζ λ +i η S λ +i η , ζ λ − η S λ ζ λ − i η S λ − i η . Let γ := λ + i η . Using (3.11), we get in this fashion( ⋆ ) = lim η ↓ lim sup N →∞ πN X ( b ; b k ) ∈ B k Z I χ ( λ ) (cid:12)(cid:12) K γ ( b ; b k ) (cid:12)(cid:12) | α γ ( b ) | × Im h ˜ g γN ( t b k , t b k ) S γ ( L b k ) − (cid:16) ζ γ ( b k ) S γ ( L b k ) (cid:17) ˜ g γN ( o b k , t b k ) S γ ( L b k ) + (cid:12)(cid:12)(cid:12) ζ γ ( b k ) S γ ( L b k ) (cid:12)(cid:12)(cid:12) ˜ g γN ( o b k , o b k ) i d λ ≤ lim η ↓ lim sup N →∞ N X ( b ; b k ) ∈ B k Z I (cid:12)(cid:12) K γ ( b ; b k ) (cid:12)(cid:12) | α γ ( b ) | Im R + γ ( o b k )d λ = lim η ↓ lim sup N →∞ Z I k K λ +i η k λ +i η d λ, where the last inequality is by (A.3). Recalling (7.2), this completes the proof. (cid:3) Estimating the Hilbert-Schmidt norm
We are now in Step (3) of the proof as described on page 3 of the Introduction. Wecombine the invariance of the quantum variance (Proposition 4.2), Lemma 4.3, and thedomination of the quantum variance by a weighted Hilbert-Schmidt norm (Proposition
UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 29 n ∈ N , any k ∈ N and any family K γ of operators in H k satisfying (Hol) :(8.1) lim η ↓ lim sup N →∞ Var I nb ,η ( K γ ) . lim η ↓ lim sup N →∞ Z I (cid:13)(cid:13)(cid:13) n n X r =1 R λ +i η n,r K λ +i η (cid:13)(cid:13)(cid:13) λ +i η d λ. We now estimate k n P nr =1 R γn,r K γ k γ by developing the scalar product. Let r ≥ r ′ , sothat n − r ≤ n − r ′ . By (4.7) and (7.1), we have (cid:16) R γn,r K γ , R γn,r ′ K γ (cid:17) γ = 1 N X ( b ; b n + k ) ∈ B n + k Im R − γ ( t b ) | ζ γ (ˆ b ) · · · ζ γ (ˆ b n − r +1 ) | · ζ γ (ˆ b n − r +2 ) · · · ζ γ (ˆ b n − r ′ +1 ) K γ ( b n − r ′ +1 ; b n − r ′ + k ) K γ ( b n − r +1 ; b n − r + k ) · ζ γ ( b n − r + k ) · · · ζ γ ( b n − r ′ + k − ) | ζ γ ( b n − r ′ + k ) · · · ζ γ ( b n + k − ) | Im R + γ ( o b n + k ) . To simplify this expression, we use (3.13), (3.14) repeatedly to obtain(8.2) (cid:16) R γn,r K γ , R γn,r ′ K γ (cid:17) γ = 1 N X ( b n − r +1 ; b n − r ′ + k ) ∈ B k + r − r ′ Im R − γ ( t b n − r +1 ) K γ ( b n − r +1 ; b n − r + k ) · ζ γ ( b n − r + k ) · · · ζ γ ( b n − r ′ + k − ) · ζ γ (ˆ b n − r +2 ) · · · ζ γ (ˆ b n − r ′ +1 ) · K γ ( b n − r ′ +1 ; b n − r ′ + k ) Im R + γ ( o b n − r ′ + k ) − E n,r,r ′ ( η , K γ ) , where E n,r,r ′ is an error term defined by E n,r,r ′ ( η , K γ ) = η N n − r +1 X s =2 X ( b s ; b n + k ) Z L bs | ξ γ − ( x b s ) | d x b s · | ζ γ (ˆ b s +1 ) · · · ζ γ (ˆ b n − r +1 ) | · ζ γ (ˆ b n − r +2 ) · · · ζ γ (ˆ b n − r ′ +1 ) K γ ( b n − r ′ +1 ; b n − r ′ + k ) K γ ( b n − r +1 ; b n − r + k ) · ζ γ ( b n − r + k ) · · · ζ γ ( b n − r ′ + k − ) · | ζ γ ( b n − r ′ + k ) · · · ζ γ ( b n + k − ) | Im R + γ ( o b n + k )+ η N n + k − X s ′ = n − r ′ + k X ( b n − r +1 ; b s ′ ) Im R − γ ( t b n − r +1 ) ζ γ (ˆ b n − r +2 ) · · · ζ γ (ˆ b n − r ′ +1 ) · K γ ( b n − r ′ +1 ; b n − r ′ + k ) K γ ( b n − r +1 ; b n − r + k ) ζ γ ( b n − r + k ) · · · ζ γ ( b n − r ′ + k − ) · | ζ γ ( b n − r ′ + k ) · · · ζ γ ( b s ′ − ) | Z L bs ′ | ξ γ + ( x b s ′ ) | d x b s ′ , with the ξ γ ± as in (3.15).Thanks to Remark 5.2 and (5.1), we have that for any n ∈ N , any r, r ′ ≤ n (8.3) E n,r,r ′ ( η , K γ ) = O ( n ) N → + ∞ ,γ ( η ) . Introduce the operator acting on H k ,( A γ K γ )( b ; b k ) = X b k +1 ∈N + bk ζ γ (ˆ b ) ζ γ ( b k )Im R + γ ( o b k ) Im R + γ ( o b k +1 ) K γ ( b ; b k +1 ) . Recall that all the operators and the quantities we manipulate here depend on N , although thisdependence is not explicit in our notations. Calculating ( A r − r ′ γ K γ )( b n − r +1 ; b n − r + k ), we find that (8.2) takes the form (cid:16) R γn,r K γ , R γn,r ′ K γ (cid:17) γ = (cid:16) K γ , A r − r ′ γ K γ (cid:17) γ − E n,r,r ′ ( η , K γ ) . We finally introduce the operator( S u γ K )( b ; b k ) = | ζ γ ( b k ) | Im R + γ ( o b k ) u γ ( b k ) X b k +1 ∈N + bk Im R + γ ( o b k +1 ) K ( b ; b k +1 ) , where u γ ( b ) = ζ γ ( b ) ζ γ ( b ) − has modulus one. When we want to remember that S u γ actson H k ≃ C B k , we will denote it by S ( k ) u γ .The advantage of this operator over A γ is that, if we forget the multiplication by u γ ,then it is sub-stochastic: S ≤ , using (3.12). To link it to A γ , introduce( Z γ K )( b ; b k ) = ζ γ (ˆ b ) · · · ζ γ (ˆ b k ) ζ γ (ˆ b ) · ζ γ (ˆ b k ) · ˜ g γ ( o b k , o b k ) K γ ( b ; b k ) . In particular, if k = 1, ( Z γ K )( b ) = ˜ g γ ( o b ,o b ) ζ γ (ˆ b ) K ( b ) and if k = 2, ( Z γ K )( b , b ) =˜ g γ ( o b , o b ) K ( b , b ).We observe that the multiplication operator Z γ conjugates S u γ and A γ : using (3.7), Z γ S u γ Z − γ = A γ . Let us introduce the following measure on B k (8.4) µ γk ( b ; b k ) = Im R − γ ( t b ) | ζ γ (ˆ b ) | · | ζ γ (ˆ b ) · · · ζ γ (ˆ b k ) | · | ˜ g γ ( o b k , o b k ) | · Im R + γ ( o b k ) | ζ γ (ˆ b k ) | . Then, if
K, K ′ ∈ H k , we have h Z γ K, Z γ K ′ i γ = N h K, K ′ i ℓ (B k ,µ γk ) . Hence, hR γn,r K γ , R γn,r ′ K γ i γ = 1 N h Z − γ K γ , S r − r ′ u γ Z − γ K γ i ℓ (B k ,µ γk ) − E n,r,r ′ ( η , K γ ) . From now on, we will write K ′ γ := Z − γ K γ . Note that by Remark 5.2, if K γ satisfies (Hol) , then so does K ′ γ .Note we also have µ γk ( b ; b k ) = Im R − γ ( t b ) | ζ γ ( b ) | | ˜ g γ ( t b , t b ) | | ζ γ ( b ) · · · ζ γ ( b k ) | R + γ ( o bk ) | ζ γ ( b k ) | using(3.10) or (3.7). In particular, we see using (3.12) that(8.5) X b k ∈N + bk − µ γk ( b ; b k ) ≤ µ γk − ( b ; b k − ) and X b ∈N − b µ γk ( b ; b k ) ≤ µ γk − ( b ; b k ) . In particular,(8.6) µ γk (B k ) = X ( b ; b k ) ∈ B k µ γk ( b ; b k ) ≤ X b ∈ B µ γ ( b ) = µ γ (B ) . Note that this inequality becomes an equality for γ ∈ R (whenever both sides are defined). Remark 8.1.
Clearly µ γk ( b ; b k ) ± belongs to L γk . It follows from Corollary C.8 that forany s ∈ R , we have(8.7) 1 N X b ∈ B µ γ ( b ) s = O ( s ) N → + ∞ ,γ (1) , UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 31
Let us come back to the problem of estimating (cid:13)(cid:13)(cid:13) n P nr =1 R γn,r K γ (cid:13)(cid:13)(cid:13) γ . Writing E ′ n ( η , K γ ) := 1 n n X r,r ′ =1 E n,r,r ′ ( η , K γ ) , we have (cid:13)(cid:13)(cid:13) n n X r =1 R γn,r K γ (cid:13)(cid:13)(cid:13) γ = 1 n n X r =1 kR γn,r K γ k γ + 2 n n − X r =1 n X r ′ = r +1 Re (cid:16) R γn,r K γ , R γn,r ′ K γ (cid:17) γ = 1 n (cid:13)(cid:13) K γ (cid:13)(cid:13) γ + 2 N n n − X r =1 n X r ′ = r +1 Re h K ′ γ , S r ′ − ru γ K ′ γ i ℓ (B k ,µ γk ) − Re E ′ n ( η , K γ )= 1 n (cid:13)(cid:13) K γ (cid:13)(cid:13) γ + 2 N n n − X r =1 k − X j =1 Re h K ′ γ , S ju γ K ′ γ i ℓ (B k ,µ γk ) + 2 N n n − k X r =1 n − r X j = k Re h K ′ γ , S ju γ K ′ γ i ℓ (B k ,µ γk ) − Re E ′ n ( η , K γ ) ≤ k − n k K γ k γ + 2 N n (cid:12)(cid:12)(cid:12) n − k X r =1 n − r X j = k h K ′ γ , S ju γ K ′ γ i ℓ (B k ,µ γk ) (cid:12)(cid:12)(cid:12) + | E ′ n ( η , K γ ) | . (8.8)where, for the last inequality, we used that S ∗ is also sub-stochastic, cf. (9.2).The first term in (8.8) will go to zero as N → ∞ followed by n → ∞ thanks to (5.1).Therefore, combining (8.8) with (8.1) and (8.3), we obtain thatlim η ↓ lim sup N →∞ Var I nb ,η ( K γ ) ≤ lim n →∞ lim η ↓ lim sup N →∞ sup λ ∈ I C I N n (cid:12)(cid:12)(cid:12) n − k X r =1 n − r X j = k h K ′ γ , S ju γ K ′ γ i ℓ (B k ,µ γk ) (cid:12)(cid:12)(cid:12) ≤ lim n →∞ lim η ↓ lim sup N →∞ sup λ ∈ I C I N n (cid:12)(cid:12)(cid:12) n X j = k ( n − j ) h K ′ γ , S ju γ K ′ γ i ℓ (B k ,µ γk ) (cid:12)(cid:12)(cid:12) . We must therefore understand the contraction properties of S ju γ .In this expression, S ju γ acts on H k ; it would more adequately be written as (cid:0) S ( k ) u γ (cid:1) j . Wewill now explain why it suffices to understand the contraction properties of (cid:0) S (1) u γ (cid:1) j . Lemma 8.2.
There exists operators T , J ∗ : ℓ ( µ γk ) −→ ℓ ( µ γ ) with operator norm smallerthan one, such that for j ≥ k we have h K ′ γ , S ju γ K ′ γ i ℓ ( µ γk ) = D K ′ γ , (cid:0) S ( k ) u γ (cid:1) j − k +1 J T K ′ γ E ℓ ( µ γk ) (8.9) = D K ′ γ , J (cid:0) S (1) u γ (cid:1) j − k +1 T K ′ γ E ℓ ( µ γk ) = D J ∗ K ′ γ , (cid:0) S (1) u γ (cid:1) j − k +1 T K ′ γ E ℓ ( µ γ ) . (8.10) As a consequence, lim η ↓ lim sup N →∞ Var I nb ,η ( K γ ) ≤ lim n →∞ lim η ↓ lim sup N →∞ sup λ ∈ I C I N n (cid:12)(cid:12)(cid:12) n X j =1 ( n − j ) hJ ∗ K ′ γ , (cid:0) S (1) u γ (cid:1) j T K ′ γ i ℓ (B ,µ γ ) (cid:12)(cid:12)(cid:12) . Furthermore, if K γ satisfies (Hol) , then T K γ and J ∗ K γ also satisfy (Hol) . Proof.
Let us denote by J : ℓ ( µ γ ) −→ ℓ ( µ γk ) the map (cid:0) J φ (cid:1) ( b , . . . , b k ) = φ ( b k ). Using(8.5), we see that kJ k ℓ ( µ γ ) → ℓ ( µ γk ) ≤
1, so that its adjoint J ∗ : ℓ ( µ γk ) −→ ℓ ( µ γ ) satisfiesthe same bound. From Remark 5.2, if K γ satisfies (Hol) , then J ∗ K γ also satisfies (Hol) .Now, we have (cid:0) S k − u γ K ′ γ (cid:1) ( b ; b k ) = X ( b k +1 ; b k − ) ∈ B bkk − Λ( b k ; b k − ) K ′ γ ( b k ; b k − ) , where Λ( b k ; b k − ) = k − Y ℓ = k | ζ γ ( b ℓ ) | Im R + γ ( o b ℓ ) u γ ( b ℓ ) Im R + γ ( o b ℓ +1 ) . In particular, (cid:0) S k − u γ K ′ γ (cid:1) ( b ; b k ) depends only on b k , so we may define (T K ′ γ )( b k ) :=( S k − u γ K ′ γ )( b ; b k ). Then T K ′ γ satisfies (Hol) , and J T K ′ γ = S k − u γ K ′ γ .Note that S ( k ) u γ J = J S (1) u γ . This is just saying that, if a function on B k depends only onthe last variable, so will its image by S ( k ) u γ .This proves (8.9).Let us now check that for any K ∈ ℓ ( µ γk ), we have k T K k ℓ ( µ γ ) ≤ k K k ℓ ( µ γk ) . Notingthat we have µ γ ( b k ) | Λ( b k ; b k − ) | = µ γk ( b k ; b k − ) and, by (3.12), that X ( b k +1 ; b k − ) ∈ B bkk − | Λ( b k ; b k − ) | ≤ , we deduce that, for any K ∈ ℓ ( µ γk ), we have k T K k ℓ ( µ γ ) = X b k ∈ B µ γ ( b k ) (cid:12)(cid:12)(cid:12) X ( b k +1 ; b k − )B bkk − Λ( b k ; b k − ) K ( b k ; b k − ) (cid:12)(cid:12)(cid:12) (8.11) ≤ X b k ∈ B µ γ ( b k ) X ( b k +1 ; b k − )B bkk − | Λ( b k ; b k − ) || K ( b k ; b k − ) | = X ( b k ; b k − ) ∈ B k µ γk ( b k ; b k − ) | K ( b k ; b k − ) | = k K k ℓ ( µ γk ) . This concludes the proof. (cid:3)
Remark 8.3.
Let j ≥ k . Since S ju γ K = J S j − k +1 u γ T K , we also deduce that(8.12) kS ju γ k ℓ ( µ γk ) → ℓ ( µ γk ) ≤ kS j − k +1 u γ k ℓ ( µ γ ) → ℓ ( µ γ ) . In the sequel, we will work with ν γk := 1 µ γk (B k ) µ γk , which is a probability measure on B k .Then the bounds k T k ℓ ( µ γk ) → ℓ ( µ γ ) ≤ kJ ∗ k ℓ ( µ γk ) → ℓ ( µ γ ) ≤ k T k ℓ ( ν γk ) → ℓ ( ν γ ) ≤ s µ γk (B k ) µ γ (B ) ≤ kJ ∗ k ℓ ( ν γk ) → ℓ ( ν γ ) ≤ s µ γk (B k ) µ γ (B ) ≤ , where the ≤ UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 33
To lighten the notation, we will write in the sequel k f k ν = p h f, f i ν := k f k ℓ ( ν γ ) , while we write k f k = h f, f i for the usual ℓ ( B ) norm with respect to the uniform scalarproduct. If H ⊂ ℓ ( B ) is a linear subspace, we will denote by P H the orthogonal projectionon H with respect to the usual scalar product on B , and by P H,ν the orthogonal projectionwith respect to the ℓ ( ν γ ) scalar product.9. Spectral gaps and contraction
The aim of this section is to obtain bounds on kS ju γ k ℓ ( ν γ ) → ℓ ( ν γ ) . To this end, weintroduce the operator( S γ K )( b ) = | ζ γ ( b ) | Im R + γ ( o b ) X b ∈N + b Im R + γ ( o b ) K ( b ) , which is sub-stochastic as mentioned above. Note that(9.1) S u γ = M u γ S γ , where M u γ denotes the multiplication by u γ .Using (3.7), one checks that the adjoint of S γ in ℓ ( B, µ γ ) is given by( S ∗ γ K )( b ) = | ζ γ (ˆ b ) | Im R − γ ( t b ) X b ∈N − b Im R − γ ( t b ) K ( b ) , which is also sub-stochastic.Since S ∗ γ is sub-stochastic and ν is a probability measure, we have kS ∗ γ k ℓ ∞ ( ν ) → ℓ ( ν ) ≤ kS γ k ℓ ( ν ) → ℓ ∞ ( ν ) ≤
1. In particular, we have for any p ∈ [1 , + ∞ ](9.2) kS u γ k ℓ p ( ν ) → ℓ p ( ν ) = kS γ k ℓ p ( ν ) → ℓ p ( ν ) ≤ . Indeed, since ν is a probability measure, kS γ K k ℓ p ( ν ) ≤ kS γ K k ℓ ∞ ( ν ) ≤ k K k ℓ ( ν ) ≤ k K k ℓ p ( ν ) .We also let S : ℓ ( B ) −→ ℓ ( B ) be the (non-backtracking) transfer operator, defined by( S K )( b ) = 1 q ( t b ) X b ∈N + b K ( b ) , where q ( v ) = deg( v ) − S u γ can be understood as follows. Start with S actingon ℓ ( B ), and multiply it by positive weights to turn it into S γ acting on ℓ ( B, ν γ ); then,add some phases u γ , to turn it into S u γ . We will therefore start by recalling the contractingproperties of S ; we will then understand the effect of adding some positive weights; finally,we will understand how the phases affect the contraction properties.9.1. Contraction properties of the transfer operator.
Let us denote by F the setof functions on B which depend only on the origin: F := { f : B → C | ∀ b, b ′ ∈ B, o b = o b ′ = ⇒ f ( b ) = f ( b ′ ) } . We denote by F ⊥ the orthogonal complement of F for the usual scalar product on ℓ ( B ).Note that for the usual scalar product on ℓ ( B ), the orthogonal projector on F can bewritten ( P F K )( b ) = 1 d ( o b ) X b ′ ∈ B ; o b ′ = o b K ( b ′ ) . If f ∈ F , then kS f k = k f k , so that S enjoys no contraction properties on F . However,it is contracting on the orthogonal complement of F : Lemma 9.1.
For all f ∈ F ⊥ , we have kS f k ≤ k f k . Proof.
Since P F f = 0, we have S f ( b ) = − q ( t b ) f (ˆ b ), so S ∗ S f ( b ) = q ( o b ) P b − − q ( t b − ) f ( c b − ) = − q ( o b ) B f (ˆ b ) = q ( o b ) f ( b ). By assumption (Data) , q ( o b ) ≥ ∀ b , so kS f k ≤ k f k . (cid:3) We will also need the much deeper fact that S is contracting on the space ⊥ , theorthogonal complement in ℓ ( B ) of constant functions (which is a larger space than F ⊥ ).Namely, Theorem 1.1 in [2] says that there exists 0 < c ( D, β ) < D on the degrees of the vertices of the graph, and on the spectral gap β appearingin Hypothesis (EXP) such that(9.3) ∀ f ∈ ⊥ , kS f k ≤ (cid:0) − c ( D, β ) (cid:1) k f k . Contraction properties of S γ . The aim of this section is to prove analogues ofLemma 9.1 and the bound (9.3) for S γ acting on ℓ ( B, ν γ ). The main difficulties come, onone hand, from the fact that µ γ is not a priori bounded from above, and could have somepeaks; on the other hand, from the fact that the weights in S γ could tend to disconnectthe graph. We will therefore call “bad” an oriented edge, or a pair of oriented edges, whereone of these events happen: for any M >
0, we setBad( M ) := n b ∈ B | ν γ ( b ) > MN o Badp( M ) := n ( b, b ′ ) ∈ B | < ν γ ( b ) (cid:0) S ∗ γ S γ (cid:1) ( b, b ′ ) < M N o . Note that, if ( b, b ′ ) ∈ Badp( M ), we must have o b = o b ′ .To formulate the analogue of Lemma 9.1, we denote by P F,ν the orthogonal projectionon F in the space ℓ ( B, ν γ ), and by P F ⊥ ,ν := 1 − P F,ν the orthogonal projection onto F ⊥ .The following proposition says that the bad (pairs of) edges are the only obstruction forcontraction properties of S γ P F ⊥ ,ν . Proposition 9.2.
For any
M > , and any K ∈ H (possibly depending on γ ), we have kS γ P F ⊥ ,ν K k ν ≤ (cid:16) − M − (cid:17) k P F ⊥ ,ν K k ν + C N,M ( K ) , and (9.4) k P F ⊥ ,ν K k ν ≤ M (cid:0) k K k ν − kS γ K k ν + C N,M ( K ) (cid:1) , where C N,M ( K ) = 12 N M X ( b,b ′ ) ∈ Badp( M ) (cid:0) S ∗ S (cid:1) ( b, b ′ ) (cid:12)(cid:12) K ( b ) − K ( b ′ ) (cid:12)(cid:12) + 1 M X b ∈ Bad( M ) ν γ ( b ) (cid:12)(cid:12) K ( b ) − P F K ( b ) (cid:12)(cid:12) . To formulate the analogue of (9.3), we introduce another set of bad pairs of orientededges Badp ( M ) := n ( b, b ′ ) ∈ B | < ν γ ( b ) (cid:0) ( S ∗ γ ) S γ (cid:1) ( b, b ′ ) < M N o . Let P ⊥ ,ν denote the orthogonal projection onto ⊥ with respect to ℓ ( ν γ ). We have UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 35
Proposition 9.3.
For any
M > , and any K ∈ H (possibly depending on γ ), we have kS γ P ⊥ ,ν K k ν ≤ (cid:0) − c ( D, β ) M − (cid:1) k P ⊥ ,ν K k ν + C ′ N,M ( K ) , and (9.5) k P ⊥ ,ν K k ν ≤ M c ( D, β ) (cid:0) k K k ν − kS γ K k ν + C ′ N,M ( K ) (cid:1) , where C ′ N,M ( K ) = 12 N M X ( b,b ′ ) ∈ Badp ( M ) (cid:0) ( S ∗ ) S (cid:1) ( b, b ′ ) (cid:12)(cid:12) K ( b ) − K ( b ′ ) (cid:12)(cid:12) + 1 M X b ∈ Bad( M ) ν γ ( b ) (cid:12)(cid:12) K ( b ) − P K ( b ) (cid:12)(cid:12) . Let us start by proving Proposition 9.2
Proof of Proposition 9.2.
Let us write Q γ := S ∗ γ S γ , where the adjoint is taken for thescalar product of ℓ ( ν γ ). The operator Q γ is self-adjoint for this scalar product, so thatfor all b, b ′ ∈ B , we have(9.6) ν γ ( b ) Q γ ( b, b ′ ) = ν γ ( b ′ ) Q γ ( b ′ , b ) . We define D γ ( b ) := P b ′ ∈ B Q γ ( b, b ′ ) ≤
1, and M γ ( b, b ′ ) := D γ ( b ) δ b = b ′ − Q γ ( b, b ′ ). Then(9.6) implies the Dirichlet identity (9.7) 12 X b,b ′ ∈ B ν γ ( b ) Q γ ( b, b ′ ) | K ( b ) − K ( b ′ ) | = h K, M γ K i ν , and since Q γ ( b, b ′ ) = 0 ⇐⇒ ( o b = o b ′ ), this shows in particular that(9.8) K ∈ F ⇐⇒ h K, M γ K i ν = 0 ⇐⇒ M γ K = 0 , where the last equivalence comes from the fact that M γ is a self-adjoint non-negativeoperator. In particular we have(9.9) h K, M γ K i ν = h P F ⊥ ,ν K, M γ P F ⊥ ,ν K i ν . Thanks to (9.7) and (9.9), we have(9.10) k P F ⊥ ,ν K k ν − kS γ P F ⊥ ,ν K k ν = h P F ⊥ ,ν K, (Id − Q γ ) P F ⊥ ,ν K i ν ≥ h P F ⊥ ,ν K, M γ P F ⊥ ,ν K i ν = 12 X b,b ′ ∈ B ν γ ( b ) Q γ ( b, b ′ ) | K ( b ) − K ( b ′ ) | ≥ M N X b,b ′ ∈ B \ Badp( M ) (cid:0) S ∗ S (cid:1) ( b, b ′ ) | K ( b ) − K ( b ′ ) | , since (cid:0) S ∗ S (cid:1) ( b, b ′ ) ≤
1. As in (9.7), the last member of the inequality is then equal to1
M N h K, ( I − S ∗ S ) K i − M N X b,b ′ ∈ Badp( M ) (cid:0) S ∗ S (cid:1) ( b, b ′ ) | K ( b ) − K ( b ′ ) | . By Lemma 9.1, and using the analog of (9.9), we have(9.11) h K, ( I − S ∗ S ) K i = k P F ⊥ K k − kS P F ⊥ K k ≥ k P F ⊥ K k . Now, by definition of Bad( M ), we have(9.12) 1 N k P F ⊥ K k ≥ M X b ∈ B \ Bad( M ) ν γ ( b ) (cid:12)(cid:12)(cid:0) P F ⊥ K (cid:1) ( b ) (cid:12)(cid:12) = 1 M k P F ⊥ K k ν − M X b ∈ Bad( M ) ν γ ( b ) (cid:12)(cid:12)(cid:0) P F ⊥ K (cid:1) ( b ) (cid:12)(cid:12) . Finally, k P F ⊥ ,ν K k ν = k P F ⊥ ,ν K − P F ⊥ ,ν P F K k ν = k P F ⊥ ,ν P F ⊥ K k ν ≤ k P F ⊥ K k ν . Com-bining this and (9.10), (9.11) and (9.12), we obtain the first part of the statement.Note that, in all the computations following (9.10), we have shown that h K, M γ K i ν ≥ M k P F ⊥ ,ν K k ν − C N,M ( K ) . But since h K, M γ K i ν = h K, ( D γ − Q γ ) K i ν ≤ k K k ν − kS γ K k ν , equation (9.4) follows. (cid:3) The proof of Proposition 9.3 is exactly the same as the previous proof, using Q γ :=( S ∗ γ ) S γ instead of Q γ , and using the bound (9.3) instead of Lemma 9.1.Let us now estimate the quantities C N,M and C ′ N,M .9.3.
Estimates on the bad terms.Proposition 9.4.
For all t ∈ N , and all K ∈ H (possibly depending on γ ), we have (9.13) C N,M ( S tu γ K ) ≤ N M ( M )) / (cid:18) X b ∈ B ν γ ( b ) (cid:19) / k K k ℓ ( ν γ ) + 2 M ( ν γ (Bad( M ))) / (cid:20) k K k ℓ ( ν γ ) + (cid:18) X b ∈ B | ( P F ν γ ) ( b ) | ν γ ( b ) (cid:19) / k K k ℓ ( ν γ ) (cid:21) where ( P F ν )( b ) = P b ′ P F ( b, b ′ ) ν ( b ′ ) = d ( o b ) P o b ′ = o b ν ( b ′ ) . Similarly, (9.14) C ′ N,M ( S tu γ K ) ≤ N M ( ( M )) / (cid:18) X b ∈ B ν γ ( b ) (cid:19) / k K k ℓ ( ν γ ) + 2 M ( ν γ (Bad( M ))) / (cid:20) k K k ℓ ( ν γ ) + (cid:18) N X b ∈ B ν γ ( b ) (cid:19) / k K k ℓ ( ν γ ) (cid:21) . Proof.
Recall that C N,M ( · ) is made of two terms: we will start by estimating12 N M X ( b,b ′ ) ∈ Badp( M ) (cid:0) S ∗ S (cid:1) ( b, b ′ ) (cid:12)(cid:12) S tu γ K ( b ) − S tu γ K ( b ′ ) (cid:12)(cid:12) ≤ N M X ( b,b ′ ) ∈ Badp( M ) (cid:0) S ∗ S (cid:1) ( b, b ′ ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) = 2 N M X b ∈ B n ( b ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) , where n ( b ) := X b ′ ∈ B ( b,b ′ ) ∈ Badp( M ) ( S ∗ S ) ( b, b ′ ) . Using the Cauchy-Schwarz inequality twice, we have
UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 37 X b ∈ B n ( b ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) = X b ∈ B n ( b ) p ν γ ( b ) q ν γ ( b ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) ≤ (cid:18) X b ∈ B n ( b ) (cid:19) / (cid:18) X b ∈ B ν γ ( b ) (cid:19) / (cid:18) X b ∈ B ν γ ( b ) |S tu γ K ( b ) | (cid:19) / . Now, by H¨older’s inequality, X b ∈ B n ( b ) = X b ∈ B (cid:18) X b ′ ∈ B ( b,b ′ ) ∈ Badp( M ) (cid:0) S ∗ S (cid:1) ( b, b ′ ) (cid:19) ≤ X b ∈ B (cid:18) X b ′ ∈ B ( b,b ′ ) ∈ Badp( M ) (cid:19)(cid:18) X b ′ ∈ B (cid:0) ( S ∗ S )( b, b ′ ) (cid:1) / (cid:19) . (9.15)But P b ′ ∈ B (cid:0) ( S ∗ S )( b, b ′ ) (cid:1) / ≤
1, since S and S ∗ are stochastic. Therefore, we have X b ∈ B n ( b ) ≤ M ) . On the other hand, we have (cid:18) X b ∈ B ν γ ( b ) |S tu γ K ( b ) | (cid:19) / = kS tu γ K k ℓ ( ν γ ) ≤ k K k ℓ ( ν γ ) , by (9.2). Therefore, the first term making up C N,M ( S tu γ K ) is bounded by the first termin the right-hand side of (9.13).Let us now consider the second term in C N,M ( S tu γ K ), namely1 M X b ∈ Bad( M ) ν γ ( b ) (cid:12)(cid:12) S tu γ K ( b ) − P F S tu γ K ( b ) (cid:12)(cid:12) ≤ M X b ∈ Bad( M ) ν γ ( b ) (cid:0) |S tu γ K ( b ) | + | P F S tu γ K ( b ) | (cid:1) . We have X b ∈ Bad( M ) ν γ ( b ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) ≤ ( ν γ (Bad( M ))) / (cid:18) X b ∈ B ν γ ( b ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) (cid:19) / ≤ ( ν γ (Bad( M )) / k K k ℓ ( ν γ ) by (9.2) for p = 4. Similarly, X b ∈ Bad( M ) ν γ ( b ) (cid:12)(cid:12) P F S tu γ K ( b ) (cid:12)(cid:12) ≤ ( ν γ Bad( M )) / (cid:18) X b ∈ B ν γ ( b ) (cid:12)(cid:12) P F S tu γ K ( b ) (cid:12)(cid:12) (cid:19) / . Using H¨older’s inequality and the fact that P F is stochastic as in (9.15), we have X b ∈ B ν γ ( b ) (cid:12)(cid:12) P F S tu γ K ( b ) (cid:12)(cid:12) ≤ X b ∈ B ν γ ( b ) X b ′ ∈ B P F ( b, b ′ ) (cid:12)(cid:12) S tu γ K ( b ′ ) (cid:12)(cid:12) ≤ (cid:18) X b ′ ∈ B | ( P F ν γ ) ( b ′ ) | ν γ ( b ′ ) (cid:19) / (cid:18) X b ∈ B ν γ ( b ) (cid:12)(cid:12) S tu γ K ( b ) (cid:12)(cid:12) (cid:19) / ≤ (cid:18) X b ′ ∈ B | ( P F ν γ ) ( b ′ ) | ν γ ( b ′ ) (cid:19) / k K k ℓ ( ν γ ) . The very same proof gives (9.14), by noting that P ν γ = | B N | ≤ N . (cid:3) Corollary 9.5.
Let Q N satisfy Hypotheses (BST) , (Data) and (Green) . For any s > and any M > , we have, for any K γ ∈ H satisfying (5.1)sup t ∈ N C N,M (cid:0) S tu γ K γ (cid:1) = O ( s ) N → + ∞ ,γ (1) M − s sup t ∈ N C ′ N,M (cid:0) S tu γ K γ (cid:1) = O ( s ) N → + ∞ ,γ (1) M − s , where the O ’s above depend on s , but not on M .Proof. We have to bound the different quantities appearing in the right-hand side of (9.13)and (9.14).Thanks to (5.1) and to Remark 8.1, we know that k K γ k ℓ α ( ν γ ) = (cid:18) N P b µ γ ( b ) (cid:19) /α (cid:18) N X b µ γ ( b ) | K γ ( b ) | α (cid:19) /α = O N → + ∞ ,γ (1) . Next, by Remark 8.1,1 N α +1 X b ∈ B ν γ ( b )) α = (cid:18) P b ∈ B µ γ ( b ) N (cid:19) α × N X b ∈ B µ γ ( b )) α = O N → + ∞ ,γ (1) . Similarly, X b ∈ B | ( P F ν γ ) ( b ) | ν γ ( b ) = X b ∈ B ν γ ( b ) 1 d ( o b ) (cid:18) X o b ′ = o b ν γ ( b ′ ) (cid:19) ≤ X b ∈ B ν γ ( b ) X o b ′ = o b ν γ ( b ′ ) ≤ D (cid:18) X b ∈ B ν γ ( b ) (cid:19) / (cid:18) X b ∈ B ν γ ( b ) (cid:19) / = D N P b ∈ B µ γ ( b ) (cid:18) N X b ∈ B µ γ ( b ) (cid:19) / (cid:18) N X b ∈ B µ γ ( b ) (cid:19) / . By Remark 8.1, each factor is a O N → + ∞ ,γ (1).Recalling (9.13), we have obtained that C N,M ( S tu γ K γ ) ≤ (cid:20) M (cid:18) M ) N (cid:19) / + 1 M ( ν γ (Bad( M ))) / (cid:21) O N → + ∞ ,γ (1)and we may get a similar bound for C ′ N,M ( S tu γ K γ ). Therefore, the result follows from thefollowing lemma. (cid:3) Lemma 9.6.
Let Q N be a sequence of quantum graphs satisfying Hypotheses (BST) , (Data) and (Green) . For any s > , we have ν γ (Bad( M )) = O ( s ) N → + ∞ ,γ (1) M − s M ) N = O ( s ) N → + ∞ ,γ (1) M − s , ( M ) N = O ( s ) N → + ∞ ,γ (1) M − s . Proof.
We have ν γ (Bad( M )) = ν γ (cid:0)(cid:8) b ∈ B | ν γ ( b ) > MN (cid:9)(cid:1) , so by Markov’s inequality, ν γ (Bad( M )) ≤ N s M s X b ∈ B ν γ ( b ) s +1 = M − s (cid:18) Nµ γ ( B ) (cid:19) s +1 N X b ∈ B µ γ ( b ) s +1 . By Remark 8.1, this quantity is a O N → + ∞ ,γ (1) M − s .For the second estimate, recall thatBadp( M ) ⊂ n ( b, b ′ ) ∈ B , o b ′ = o b | (cid:0) ( S ∗ γ S γ )( b, b ′ ) ν γ ( b ) (cid:1) − > M N o . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 39
Therefore, by Markov inequality, we have M ) ≤ M N ) s X b,b ′ ∈ Bo b = o b ′ (cid:2) ( S ∗ γ S γ )( b, b ′ ) ν γ ( b ) (cid:3) − s = 1 M s (cid:18) µ γ ( B ) N (cid:19) s X b ∈ B X b ′ ,o b ′ = o b (cid:2) ( S ∗ γ S γ )( b, b ′ ) µ γ ( b ) (cid:3) − s . Using Remark 5.2, we see that F γ ( b ) = P o b ′ = o b [( S ∗ γ S γ )( b, b ′ ) µ γ ( b )] − s satisfies (Hol) .Combining this fact with Remark 8.1, we deduce that M ) N = O ( s ) N → + ∞ ,γ (1) M − s .For the last estimate we note that if ( S ∗ γ S γ )( b, b ′ ) >
0, then there exist b ′′ , b , b ′ suchthat b ′′ b b and b ′′ b ′ b ′ , where b b means t b = o b . So now F γ ( b )sums over t b ′ having the same grandparent as t b instead of same parent; we conclude asbefore. (cid:3) Remark 9.7. If K γ ∈ H satisfies | K γ ( b ) | ≤ N ∈ N , all b ∈ B N and all γ ∈ I × (0 , k K γ k ℓ p ( ν γ ) ≤ k K γ k ∞ ν γ ( B ) /p ≤
1, so C ′ N,M ( K γ ) ≤ C N,M := 2
N M ( ( M )) / (cid:18) X b ∈ B ν γ ( b ) (cid:19) / + 2 M ( ν γ (Bad( M ))) / (cid:20) (cid:18) N X b ∈ B ν γ ( b ) (cid:19) / (cid:21) . The proof of Corollary 9.5 tells us that C N,M = O ( s ) N → + ∞ ,γ (1) M − s .9.4. Contraction properties of S u γ . In the previous subsections we showed that S γ , S γ are contractions on proper subspaces. The following proposition says that S u γ is contract-ing on the full space, unless the phases u γ satisfy very special relations. This improvementis reminiscent of Wielandt’s theorem (see for instance [23, Chapter 8]), saying that addingphases to a matrix with positive entries will strictly diminish its spectral radius unless thephases satisfy very special conditions.Wielandt’s theorem is insufficient for us as we need more precise information, namelya contraction on the norm of S u γ instead of its spectral radius, which should moreover beuniform as the graph becomes large and γ approaches the real axis. This will require acareful analysis of the various operators.Let us write ω γ ( b ) := − | ζ γ ( b ) | Im R + γ ( o b ) X b ′ ∈N + b Im R + γ ( o b ′ )= − Im γ Im R + γ ( o b ) Z L b | ξ γ + ( x b ) | d x b , where the second equality comes from (3.13).From the definition of ξ γ + ( x b ) and from Remark 5.2, one sees that b R L b | ξ γ + ( x b ) | d x b satisfies (Hol) , so that(9.16) k ω γ k ν = O N → + ∞ ,γ (Im γ ) . Proposition 9.8.
Let γ ∈ C + , M > , ε ∈ (0 , / , and let Q be a quantum graph in thefamily ( Q N ) . Then one of the following two options holds: (i) Either we have, for any K γ ∈ H satisfying (5.1) , (9.17) kS u γ K γ k ν ≤ (1 − ε ) k K γ k ν + ˜ C N,M ( K γ ) , where ˜ C N,M ( K γ ) = max j =0 , , (cid:8) C N,M ( S ju γ K γ ) , C ′ N,M ( S ju γ K γ ) (cid:9) ; (ii) or there exists θ : V −→ R and constants c ∈ C with | c | ≤ , and C ( β, D ) dependingonly on the spectral gap β and the maximal degree D of the graph such that (cid:13)(cid:13)(cid:13) u γ ( b ) − c e i [ θ ( o b ) − θ ( t b )] (cid:13)(cid:13)(cid:13) ν ≤ C ( β, D ) M h ε / + k ω γ k ν + k ω γ k ν i + C ( β, D ) M C N,M , with C N,M as in Remark 9.7.
In the sequel, we will write(9.18) C = C ( M, ε, γ ) = (cid:16) C ( β, D ) M h ε / + k ω γ k ν + k ω γ k ν i + C ( β, D ) M C N,M (cid:17) / . If we are in case (i) of the previous proposition, we may iterate (9.17) to obtain thefollowing bound
Corollary 9.9.
Let γ ∈ C + , M > , ε ∈ (0 , / , and let Q be a quantum graph in thefamily ( Q N ) . Suppose we are in case (i) in Proposition 9.8. Then for any K γ ∈ H satisfying (5.1) and any ℓ ≥ , we have kS ℓu γ K γ k ν ≤ (1 − ε ) ℓ k K γ k ν + ℓ max ≤ j ≤ ℓ ˜ C N,M ( S ju γ K γ ) . Heuristics of the proof of Proposition 9.8.
Before proving the result, we would liketo give a heuristics of why, in the second alternative, u γ must take this special form. Todo this, we consider the case when γ ∈ R supposing that all the quantities we deal withare well-defined on the real axis, and we suppose for simplicity that there exists K γ suchthat kS u γ K γ k ν = k K γ k ν , which is the case in which (i) is the furthest from being satisfied.By (9.2), we would have kS ju γ K γ k ν = k K γ k ν for j = 1 ,
2. In particular, kS γ K γ k ν = k K γ k ν . By (9.8), this would imply that K γ ∈ F . Similarly, we would have S u γ K γ ∈ F and S u γ K γ ∈ F . But, since γ ∈ R and K γ ∈ F , we would have by (3.12) that ( S γ K γ )( b ) = K γ ( t b ), so that ( S u γ K γ ) ( b ) = ( S u γ K γ ) ( o b ) = u γ ( b ) K γ ( t b ) , and, similarly, (cid:0) S u γ K γ (cid:1) ( o b ) = u γ ( b ) ( S u γ K γ ) ( t b ) . Since | u γ | = 1, we get in particular |S uγ K γ ( o b ) || K γ ( t b ) | = |S uγ K γ ( o b ) ||S uγ K γ ( t b ) | = 1.Writing ( S uγ K γ )( o b ) |S uγ K γ ( o b ) | =: e i θ ( o b ) , ( S uγ K γ ) ( o b ) | S uγ K γ ( o b ) | =: e i θ ′ ( o b ) , K γ ( t b ) | K γ ( t b ) | =: e i θ ′′ ( t b ) , we obtain that u γ ( b ) = e i θ ( o b ) − i θ ′′ ( t b ) = e i θ ′ ( o b ) − i θ ( t b ) . In particular, for all b ∈ B , we would have θ ′ ( o b ) − θ ( o b ) = θ ( t b ) − θ ′′ ( t b ).This quantity must be equal to a constant c ∈ R , because the graph is not bipartite for N large enough (since ( G N ) is expanding). Therefore, we would have(9.19) u γ ( b ) = e i c e i θ ( o b ) − i θ ( t b ) . This shows we are indeed in case (ii).Let us continue these heuristics and show moreover that e i c = 1 (this supplement toProposition 9.8 is the object of § P j S ju γ K .Suppose for contradiction that u γ ( b ) = e i θ ( o b ) − i θ ( t b ) . Writing ζ γ ( b ) = ρ γ ( b )e i ϕ ( b ) , wededuce that − ϕ ( b ) = θ ( o b ) − θ ( t b ). In particular, ϕ (ˆ b ) = − ϕ ( b ) mod π . If the graph is non-bipartite, it contains an odd cycle. As the quantity takes equal values for b , b having same origin, and also for b , b having same terminus, it follows that it must be constant on thiscycle. From this, we readily see that it must be constant on the whole graph. UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 41
By (3.7), we then havee − i ϕ ( b ) h ρ γ ( b ) − e i( ϕ ( b )+ ϕ (ˆ b )) ρ γ (ˆ b ) i = S γ ( L b )˜ g γ ( t b , t b ) . Since the term between brackets is real and the phase of the right-hand side dependsonly on t b (as S γ is real), we deduce that e − ϕ ( b ) does not depend on o b . Therefore,since e ϕ (ˆ b ) = e − ϕ ( b ) , and since the graph is non bipartite, we deduce that e − ϕ ( b ) isa real constant. This constant must be one, otherwise we would have e i θ ( o b ) = − e i θ ( t b ) ,contradicting non-bipartiteness. Therefore, 2 ϕ ( b ) ≡
0. But this would imply that ζ γ isalways real, thus contradicting (Green) , in view of (3.4). Remark 9.10.
One may wonder if (9.17) always holds. The answer is no in general.For example take G N a family of expanders satisfying (BST) , like the ones built in [22].Consider quantum graphs on G N with all lengths equal, Kirchhoff boundary conditions,and zero potential. This was the model studied in [19]. Then all the oriented edges playexactly the same role, so u γ does not depend on b . In this case kS u γ K γ k ν = kS γ K γ k ν ,which only contracts on subspaces, thus violating (9.17). Indeed, here we are in case (ii).9.4.2. Proof of Proposition 9.8.
Let K γ ∈ H satisfy (5.1), and let γ ∈ C + . When wedo not have kS u γ K γ k ν = k K γ k ν , but only kS u γ K γ k ν ≈ k K γ k ν , then we can still say that S ju γ K γ is close to being in F for j = 0 , ,
2. However, we cannot apply directly the previousargument, in which we divided by |S ju γ K γ ( v ) | , since these could be very small, and couldcause the different remainders to become large.Therefore, we will have to show the stronger fact that S ju γ K γ is close to a function in F of constant modulus. To this end, we will use several times the following lemma, whichsimply says that if K is close to being in F and is close to having constant modulus, thenit is close to being a function in F with constant modulus. Lemma 9.11.
Let us write f K := P F,ν K . We have (9.20) (cid:13)(cid:13)(cid:13) K − k K k ν f K | f K | (cid:13)(cid:13)(cid:13) ν ≤ k K − f K k ν + 2 k P ⊥ ,ν | K |k ν , with the convention that f K | f K | = 1 when f K vanishes. Note that the terms in the right-hand side of (9.20) can be estimated by (9.4) and (9.5).
Proof.
We have, by the triangle inequality (cid:13)(cid:13)(cid:13) K − k K k ν f K | f K | (cid:13)(cid:13)(cid:13) ν ≤ k K − f K k ν + (cid:13)(cid:13)(cid:13) f K − k K k ν f K | f K | (cid:13)(cid:13)(cid:13) ν . Since dividing by f K | f K | does not change the k · k ν norm, we have (cid:13)(cid:13)(cid:13) f K − k K k ν f K | f K | (cid:13)(cid:13)(cid:13) ν = (cid:13)(cid:13) | f K | − k K k ν (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) | K | − | f K | (cid:13)(cid:13) ν + (cid:13)(cid:13) | K | − k K k ν (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) K − f K (cid:13)(cid:13) ν + (cid:13)(cid:13) | K | − k K k ν (cid:13)(cid:13) ν . Let us write C K = P | K | . We have (cid:13)(cid:13) | K |− k K k ν (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) | K |− C K (cid:13)(cid:13) ν + (cid:12)(cid:12) k K k ν − C K (cid:12)(cid:12) ≤ (cid:13)(cid:13) | K | − C K (cid:13)(cid:13) ν = 2 k P ⊥ ,ν | K |k ν . Putting these inequalities together gives the result. (cid:3) Proof of Proposition 9.8.
Suppose that (i) does not hold: we can find K γ such that(9.21) kS u γ K γ k ν > (1 − ε ) k K γ k ν + ˜ C N,M ( K γ ) . Step 1.
By (9.2), we have kS γ S ju γ K γ k ν = kS j +1 u γ K γ k ν ≥ kS u γ K γ k ν and k K γ k ν ≥kS ju γ K γ k ν for j = 0 , ,
2. Thus, (9.21) implies that for j = 0 , , (cid:13)(cid:13) S ju γ K γ (cid:13)(cid:13) ν − kS γ S ju γ K γ k ν < ε kS ju γ K γ k ν − ˜ C N,M ( K γ ) . Similarly, since kS γ |S ju γ K |k ν ≥ kS j +2 u γ K γ k ν , (9.21) implies that for j = 0 , , kS ju γ K γ k ν − (cid:13)(cid:13) S γ |S ju γ K γ | (cid:13)(cid:13) ν < ε kS ju γ K γ k ν − ˜ C N,M ( K γ ) . Let δ M := M , δ ′ M := M c ( D,β ) . For j = 0 , ,
2, we write f j := P F,ν S ju γ K γ . We now applyLemma 9.11 to S ju γ K γ , and use (9.4), (9.5), (9.22) and (9.23) to obtain(9.24) (cid:13)(cid:13)(cid:13) S ju γ K γ − kS ju γ K γ k ν f j | f j | (cid:13)(cid:13)(cid:13) ν ≤ k P F ⊥ ,ν S ju γ K γ k ν + 8 k P ⊥ ,ν |S ju γ K γ |k ν ≤ δ M (cid:0) ε kS ju γ K γ k ν + C N,M ( S ju γ K γ ) − ˜ C N,M ( K γ ))+ 8 δ ′ M (cid:0) ε kS ju γ K γ k ν + C ′ N,M ( S ju γ K γ ) − ˜ C N,M ( K γ )) ≤ ε ( δ M + δ ′ M ) kS ju γ K γ k ν , thanks to the definition of ˜ C N,M ( K γ ). Step 2.
In this step, we use the fact that S u γ has a simple action on F .Let us write g j := kS ju γ K γ k ν f j | f j | , and R j := S ju γ K γ − g j . Then (cid:0) S j +1 u γ K γ (cid:1) ( b ) = u γ ( b ) (cid:0) S γ g j (cid:1) ( b ) + (cid:0) S u γ R j (cid:1) ( b )= u γ ( b ) g j ( t b ) + u γ ( b ) ω γ ( b ) g j ( t b ) + (cid:0) S u γ R j (cid:1) ( b ) . But (cid:0) S j +1 u γ K γ (cid:1) ( b ) = g j +1 ( o b ) + R j +1 ( b ). So we obtain for j = 0 , u γ ( b ) = g j +1 ( o b ) g j ( t b ) + r j ( b ) , where r j = g j (cid:0) R j +1 − S u γ R j (cid:1) − u γ ω γ . By (9.24) and (9.2), k r j k ν ≤ k R j +1 k ν + kS u γ R j k ν kS ju γ K γ k ν + k ω γ k ν ≤ √ ε ( δ M + δ ′ M ) / + k ω γ k ν . Therefore,(9.26) g ( t b ) g ( t b ) = g ( o b ) g ( o b ) + r ′ ( b ) , where k r ′ k ν ≤ √ ε ( δ M + δ ′ M ) / + 2 k ω γ k ν . Step 3.
In this last step, we show that g ( o b ) ≈ c g ( o b ) to deduce the result.Let us write h ( b ) := g ( o b ) g ( o b ) , h ( b ) = g ( t b ) g ( t b ) , h ( b ) = g ( o b ) g ( o b ) and h ( b ) = g ( t b ) g ( t b ) . Note that,by (9.2) and (9.22), we have 1 − ε < k h j k ν ≤ j = 0 , , , S γ h = S γ (cid:0) h + ω γ h (cid:1) = S γ (cid:0) h + r ′ + ω γ h (cid:1) = h + r ′′ , where r ′′ = ω γ h + S γ r ′ + S γ ω γ h . In particular, k r ′′ k ν ≤ k r ′ k ν + 2 k ω γ k ν .We deduce from (9.5) and Remark 9.7 that k P ⊥ ,ν h k ν ≤ δ ′ M (cid:0) k h k ν − k h + r ′′ k ν + C ′ N,M ( h ) (cid:1) ≤ δ ′ M (cid:0) − k h k ν + 2 k r ′′ k ν − k r ′′ k ν + C N,M (cid:1) ≤ δ ′ M (cid:0) ε + 2 k r ′′ k ν + C N,M (cid:1) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 43
We may thus write h ( b ) = c + r ′′′ ( b ), where c := P ,ν h , so that | c | ≤
1, and k r ′′′ k ν ≤ δ ′ M (cid:0) ε + 2 k r ′′ k ν + C N,M (cid:1) . It follows that u γ ( b ) = c g ( o b ) g ( t b ) + R ( b ) , where R ( b ) = r ( b ) + g ( o b ) g ( t b ) r ′′′ ( b ). Using the estimates we have on r and r ′′′ , and recallingthat ε ≤ √ ε (as ε < ), we see that we may find a constant C ( β, D ) > k R k ν ≤ C ( β, D ) M h ε / + k ω γ k ν + k ω γ k ν i + C ( β, D ) M C N,M . Recalling that | g ( o b ) | = | g ( t b ) | , we write e i θ ( v ) := g ( v ) | g ( v ) | , which gives us the result. (cid:3) Properties of the phases u γ . In the previous subsection, we did not use the preciserelation between u γ and ζ γ . Now, we are going to use (3.7) to show that, if case (ii) ofProposition 9.8 occurs then c cannot get very close to unity.Recall that the quantity C was defined in (9.18). Proposition 9.12.
For any large
L > there exists δ = δ ( N, γ, L ) > satisfying lim inf η ↓ lim inf N → + ∞ δ ≥ CL − , such that if case (ii) of Proposition 9.8 is satisfied, then | − c | ≥ δ − C . Proof.
Let r := | − c | . Note that there exists C I > N and of b suchthat for all γ ∈ I + i [0 , | Im S γ ( L b ) | ≤ C I Im γ. By (2.7), we have | Re( S γ ( L b )) | ≥ C ′ Dir > N and all b ∈ B N .Let us write ζ γ ( b ) = ρ γ ( b )e i φ γ ( b ) , so that u γ ( b ) = e − φ γ ( b ) . Step 1: e φ γ ( b ) depends (almost) only on t b . By assumption, we have(9.28) (cid:13)(cid:13)(cid:13) e − φ γ ( b ) − e i θ ( o b ) − i θ ( t b ) (cid:13)(cid:13)(cid:13) ν ≤ C + r (cid:13)(cid:13)(cid:13) e − φ γ (ˆ b ) − e i θ ( t b ) − i θ ( o b ) (cid:13)(cid:13)(cid:13) ν ≤ C + r , so that (cid:13)(cid:13)(cid:13) − e − φ γ ( b ) − φ γ (ˆ b ) (cid:13)(cid:13)(cid:13) ν = (cid:13)(cid:13)(cid:13) e φ γ ( b ) − e − φ γ (ˆ b ) (cid:13)(cid:13)(cid:13) ν ≤ C + 2 r . Let us write ǫ ( b ) = ( − i φ γ ( b ) − i φ γ (ˆ b ) ) ≥ − − i φ γ ( b ) − i φ γ (ˆ b ) ) < , so that | e − i φ γ ( b ) − i φ γ (ˆ b ) − ǫ ( b ) | ≤ | − e − φ γ ( b ) − φ γ (ˆ b ) | . Since | e − i φ γ ( b ) − i φ γ (ˆ b ) − ǫ ( b ) | ≤
2, wededuce that for any s ≥ (cid:13)(cid:13)(cid:13) ǫ ( b ) − e − i φ γ ( b ) − i φ γ (ˆ b ) (cid:13)(cid:13)(cid:13) ℓ s ( ν ) = (cid:13)(cid:13)(cid:13) ǫ ( b ) − e i φ γ ( b )+i φ γ (ˆ b ) (cid:13)(cid:13)(cid:13) ℓ s ( ν ) ≤ C + r ) /s . The first part of (3.7) can be rewritten ase − i φ γ ( b ) (cid:16) ρ γ ( b ) − ρ γ (ˆ b )e i φ γ ( b )+i φ γ (ˆ b ) (cid:17) = S γ ( L b )˜ g γ ( t b , t b ) . Therefore, we have e − i φ γ ( b ) (cid:16) ρ γ ( b ) − ρ γ (ˆ b ) ǫ ( b ) (cid:17) = S γ ( L b )˜ g γ ( t b , t b ) + R , with k R k ℓ s ( ν ) ≤ C + r ) /s k ζ γ k ℓ s ( ν ) . Let us write G ( b ) := ˜ g γ ( t b , t b ) = | ˜ g γ ( t b , t b ) | e i ψ γ ( t b ) , then recalling (9.27),e i φ γ ( b ) − i ψ γ ( t b ) Re( S γ ( L b )) | ˜ g γ ( t b , t b ) | = (cid:16) ρ γ ( b ) − ρ γ (ˆ b ) ǫ ( b ) (cid:17) + R , where k R k ℓ s ( ν ) ≤ k R k ℓ s ( ν ) + C I | Im γ | (cid:13)(cid:13) G − (cid:13)(cid:13) ℓ s ( ν ) .Taking the imaginary parts, we obtainsin (cid:0) φ γ ( b ) − ψ γ ( t b ) (cid:1) Re( S γ ( L b )) | ˜ g γ ( t b , t b ) | = Im( R )( b ) , so that (cid:12)(cid:12)(cid:12) sin (cid:0) φ γ ( b ) − ψ γ ( t b ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ | ˜ g γ ( t b , t b ) | C ′ Dir | Im( R )( b ) | . Let δ γ ( b ) ∈ π Z be such that φ γ ( b ) − ψ γ ( t b ) + δ γ ( b ) ∈ (cid:2) − π , π (cid:3) . We have (cid:12)(cid:12)(cid:12) sin (cid:0) φ γ ( b ) − ψ γ ( t b ) (cid:1)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) sin (cid:0) φ γ ( b ) − ψ γ ( t b ) + δ γ ( b ) (cid:1)(cid:12)(cid:12)(cid:12) ≥ π (cid:12)(cid:12) φ γ ( b ) − ψ γ ( t b ) + δ γ ( b ) (cid:12)(cid:12) . Therefore, (cid:12)(cid:12) φ γ ( b ) − ψ γ ( t b ) + δ γ ( b ) (cid:12)(cid:12) ≤ π · | ˜ g γ ( t b , t b ) | C ′ Dir | Im( R )( b ) | , As | e i x − e i t | ≤ | x − t | , we get,(9.29) e φ γ ( b ) = e ψ γ ( t b ) + R ( b ) , with | R ( b ) | ≤ π | ˜ g γ ( t b ,t b ) | C ′ Dir | R ( b ) | . In particular, we have k R k ν ≤ πC ′ Dir k G k ℓ ( ν ) k R k ℓ ( ν ) . Step 2: e φ γ ( b ) is almost equal to one. Using (9.28) and (9.29), we obtain that(9.30) e i θ ( t b ) = e i θ ( o b ) − ψ γ ( o b ) + R ( b ) , with k R k ν ≤ k R k ν + C + r .Let us write f ( b ) = e i θ ( o b ) , f ( b ) = e i θ ( t b ) , f ( b ) = e i θ ( o b ) − ψ γ ( o b ) , f ( b ) = e i θ ( t b ) − ψ γ ( t b ) .We have S γ f = S γ (cid:0) f + ω γ f (cid:1) = S γ (cid:0) f + R + ω γ f (cid:1) = f + ω γ f + S γ (cid:0) R + ω γ f (cid:1) , so that kS γ f k ν ≥ − k R k ν − k ω γ k ν . It follows that k f k ν − kS γ f k ν = ( k f k ν + kS γ f k ν )( k f k ν − kS γ f k ν ) ≤ k R k ν + 4 k ω γ k ν . From (9.5) and Remark 9.7, we thus get for any
L > k P ⊥ ,ν f k ν ≤ L c ( D, β ) (cid:0) k R k ν + 4 k ω γ k ν + C N,L (cid:1) , so there exists s ∈ C such that(9.31) f ( b ) = s + R ( b ) , with | s | ≤ k R k ν ≤ L c (2 k R k ν + 4 k ω γ k ν + C N,M ).Since also f (ˆ b ) = s + R (ˆ b ), we deduce from (9.28) that e ϕ γ ( b ) = | s | + R ( b ), where k R k ν ≤ k R k ν + k R ˆ R k ν + r + C , where ˆ R ( b ) = R (ˆ b ). By (9.31), | ˆ R ( b ) | ≤
2, so k R k ν ≤ k R k ν + r + C . Writing s ( b ) := ( | s | if Re(e i ϕ γ ( b ) ) > −| s | if Re(e i ϕ γ ( b ) ) ≤ , UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 45 we always have | e i ϕ γ ( b ) + s ( b ) | ≥
1, so that | e i ϕ γ ( b ) − s ( b ) | ≤ | e ϕ γ ( b ) − s ( b ) | = | R ( b ) | . Inparticular, we get (cid:12)(cid:12) Im e i ϕ γ ( b ) (cid:12)(cid:12) ≤ | R ( b ) | and thus k Im e i ϕ γ ( b ) k ν ≤ k R k ν .Putting it all together, we obtain (cid:13)(cid:13) Im e i φ γ (cid:13)(cid:13) ν ≤ L c D,β (cid:16) πC ′ Dir k G k ℓ ( ν ) (cid:2) C + r ) / k ζ γ k ℓ ( ν ) + C I Im γ k G − k ℓ ( ν ) (cid:3) + 2( C + r ) + 4 k ω k ν + C N,L (cid:17) + 2( r + C ) . On the other hand, by the Cauchy-Schwarz inequality, (cid:13)(cid:13) Im ζ γ (cid:13)(cid:13) ν = (cid:13)(cid:13) | ζ γ | Im e i φ γ (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) | ζ γ | (cid:13)(cid:13) ν (cid:13)(cid:13) (Im e i φ γ ) (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) | ζ γ | (cid:13)(cid:13) ν (cid:13)(cid:13) Im e i φ γ (cid:13)(cid:13) ν , so 1 ≤ (cid:13)(cid:13) Im ζ γ (cid:13)(cid:13) ν · (cid:13)(cid:13) | Im ζ γ | − (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) | Im ζ γ | − (cid:13)(cid:13) ν · (cid:13)(cid:13) | ζ γ | (cid:13)(cid:13) ν · (cid:13)(cid:13) Im e i φ γ (cid:13)(cid:13) ν . Thanks to Remarks 5.2, 9.7 and 8.1 and to (9.16), we deduce that1 ≤ L h(cid:0) r + C (cid:1) / + Im γ + L − s i × O N → + ∞ ,γ (1) , where we bounded ( r + C ) α ≤ ( r + C ) / for α = 1 ,
2, which holds if r + C ≤ r + C > r + C ) / ≥ L O N → + ∞ ,γ (1) − Im γ − L − s , so that r ≥ (cid:16) L O N → + ∞ ,γ (1) − Im γ − L − s (cid:17) − C . Taking s = 3, the claim follows. (cid:3) End of the proof
Recall that by Lemma 8.2, our aim is to estimate, for operators K ′ γ ∈ H k satisfying (Hol) , the quantitylim n →∞ lim sup η ↓ lim sup N →∞ sup λ ∈ I µ γ (B ) N n (cid:12)(cid:12)(cid:12) n X j =1 ( n − j ) hJ ∗ K ′ γ , S ju γ T K ′ γ i ℓ (B ,ν γ ) (cid:12)(cid:12)(cid:12) . From now on, we take M arbitrarily large, and take ε M := M − , n M = M .We will now consider each alternative that can happen in Corollary 9.9. First alternative : Suppose that case (i) of Proposition 9.8 is satisfied for ε = ε M .We may apply Corollary 9.9, (8.13) and (9.2), to get for all j ∈ N (10.1) (cid:13)(cid:13) S ju γ T K ′ γ (cid:13)(cid:13) ν γ ≤ (1 − ε M ) ⌊ j ⌋ k K ′ γ k ν γk + j max ≤ i ≤ j ˜ C N,M ( S iu γ T K ′ γ ) . By Corollary 9.5, we have sup i ˜ C N,M ( S iu γ T K ′ γ ) = M − O N → + ∞ ,γ (1). We deduce fromthis and (8.13) that(10.2) µ γ (B ) N n M (cid:12)(cid:12)(cid:12) n M X j =1 ( n M − j ) hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ (cid:12)(cid:12)(cid:12) ≤ µ γ (B ) N n M k K ′ γ k ν γk n M X j ′ =1 (1 − ε M ) ⌊ j ⌋ + n M N k K ′ γ k ν γk µ γ (B ) max ≤ j ≤ n M ˜ C N,M ( S ju γ T K ′ γ )= 1 M O N → + ∞ ,γ (1) , where we estimated P j (1 − ε ) ⌊ j/ ⌋ ≤ · / P j (1 − ε ) j ≤ ε − , as ε ≤ , then we used(5.1) and Remark 8.1. Second alternative : Suppose that case (ii) of Proposition 9.8 is satisfied, still for ε = ε M , so that the addition of phases u γ did not improve the contraction of kS ju γ K k .Then we will not be able to control individual scalar products as in (10.1), however phaseswill still help to control the scalar products in mean by inducing cancellations.Let us write θ ( b ) := θ ( o b ) and θ ( b ) = θ ( t b ). Given K ∈ H , we get (cid:0) S γ e − i θ K (cid:1) ( b ) = (cid:0) e − i θ S γ K (cid:1) ( b ) . Therefore, (cid:13)(cid:13) S u γ K − c e i θ S γ e − i θ K (cid:13)(cid:13) ν = (cid:13)(cid:13)(cid:0) u γ − c e i( θ − θ ) (cid:1) S γ K (cid:13)(cid:13) ν ≤ (cid:13)(cid:13) u γ − c e i( θ − θ ) (cid:13)(cid:13) ℓ ( ν ) (cid:13)(cid:13) S γ K (cid:13)(cid:13) ℓ ( ν ) ≤ √ (cid:13)(cid:13) u γ − c e i( θ − θ ) (cid:13)(cid:13) / ℓ ( ν ) (cid:13)(cid:13) S γ K (cid:13)(cid:13) ℓ ( ν ) ≤ p C ( M, ε M , γ ) · k K k ℓ ( ν ) , where we used | u γ − c e i( θ − θ ) | ≤ C is as in (9.18). Thanks to (9.2), we get for all j ≥ (cid:13)(cid:13) S ju γ K − c j e i θ S jγ e − i θ K (cid:13)(cid:13) ν = (cid:13)(cid:13)(cid:13) j X i =1 S j − iu γ ( S u γ − c e i θ S γ e − i θ )( c e i θ S γ e − i θ ) i − · K (cid:13)(cid:13)(cid:13) ν ≤ j p C · k K k ℓ ( ν ) . Using (9.2) again, it follows that for all m and t ,(10.3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t X j =1 hJ ∗ K ′ γ , S j + mu γ T K ′ γ i ν − t X j =1 c j hJ ∗ K ′ γ , e i θ S jγ e − i θ S mu γ T K ′ γ i ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ t p C · kJ ∗ K ′ γ k ℓ ( ν ) · k T K ′ γ k ℓ ( ν ) ≤ t p C · k K ′ γ k ℓ ( ν ) · k K ′ γ k ℓ ( ν ) where the bound on T : ℓ → ℓ follows as in (8.11)–(8.13), using H¨older’s inequality.As we will see below, the size t of packets should be chosen so that t √C is small as M gets large. Remembering that C . f ( β, D )( M / ε / + O ( η )) and ε = M − , we take t = M α with 0 < α < /
4. We now write (cid:12)(cid:12)(cid:12) n M X j =1 ( n M − j ) hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n M X r =1 n M − r X j =1 hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) n M X r =1 ⌊ nM − rt ⌋− X a =0 t ( a +1) X j =1+ at hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ (cid:12)(cid:12)(cid:12) + n M t k K ′ γ k ν , where we estimated | P n M r =1 P n M − rj = t ⌊ nM − rt ⌋ hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ | ≤ n M t k K ′ γ k ν .Note that P t ( a +1) j =1+ at h· , S ju γ ·i = P tj =1 h· , S j + atu γ ·i . So using (10.3), (cid:12)(cid:12)(cid:12) n M X r =1 ⌊ nM − rt ⌋− X a =0 t ( a +1) X j =1+ at hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) n M X r =1 ⌊ nM − rt ⌋− X a =0 t X j =1 c j hJ ∗ K ′ γ , e i θ S jγ e − i θ S atu γ T K ′ γ i ν (cid:12)(cid:12)(cid:12) + n M · n M t · t p C · k K ′ γ k ν k K ′ γ k ℓ ( ν ) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 47
Therefore,(10.4) µ γ (B ) N n M (cid:12)(cid:12)(cid:12) n M X j =1 ( n M − j ) hJ ∗ K ′ γ , S ju γ T K ′ γ i ν γ (cid:12)(cid:12)(cid:12) ≤ µ γ (B ) N n M (cid:12)(cid:12)(cid:12) n M X r =1 ⌊ nM − rt ⌋− X a =0 D J ∗ K ′ γ , e i θ t X j =1 c j S jγ e − i θ S atu γ T K ′ γ E ν (cid:12)(cid:12)(cid:12) + µ γ (B ) N · t (cid:16)p C k K ′ γ k ν γk k K ′ γ k ℓ ( ν ) + n − M k K ′ γ k ν (cid:17) . Thanks to (5.1) and Remark 8.1, the last term goes to zero for our choice of t , as N → ∞ followed by η ↓ M → ∞ .Now, we decompose e − i θ S atu γ T K ′ γ = P ,ν e − i θ S atu γ T K ′ γ + P ⊥ ,ν e − i θ S atu γ T K ′ γ .Consider the first term. Recall that ( S γ − )( b ) = ω γ ( b ). We deduce that S jγ P ,ν e − i θ S atu γ T K ′ γ = P ,ν e − i θ S atu γ T K ′ γ + R j , where R j = c γ P j − k =0 S kγ ω γ if P ,ν e − i θ S atu γ T K ′ γ = c γ . By (9.16) and (9.2), k R j k ν = jη O N → + ∞ ,γ (1) . With S jγ gone, we obtain a geometric sum, so we get (cid:28) J ∗ K ′ γ , e i θ t X j =1 c j S jγ P ,ν e − i θ S atu γ T K ′ γ (cid:29) ν γ = e i θ D J ∗ K ′ γ , P ,ν e − i θ S atu γ T K ′ γ E ν γ c − c t +10 − c + t O N → + ∞ ,γ ( η ) . By Proposition 9.12, we have | − c | ≥ δ − C , where lim inf η ↓ lim inf N →∞ δ ≥ CL − . Take L = M / and recall that C . ( M / ε / M + O ( η ) + M − s +1 ) ≈ M − / + O ( η ) + M − s +1 .For s = 2 we thus get | − c | & M − / as N → ∞ followed by η ↓
0. Using (8.13), (9.2),(5.1) and Remark 8.1, we thus get(10.5) (cid:12)(cid:12)(cid:12)(cid:12) µ γ (B ) N n M n M X r =1 ⌊ nM − rt ⌋− X a =0 (cid:28) J ∗ K ′ γ , e i θ t X j =1 c j S jγ P ,ν e − i θ S atu γ T K ′ γ (cid:29) ν γ (cid:12)(cid:12)(cid:12)(cid:12) ≤ t (cid:16) M / + t η (cid:17) O N →∞ ,γ (1) . Finally, let us deal with the term P ⊥ ,ν e − i θ S atu γ T K ′ γ . By (3.14), we have S ∗ γ = + ˜ ω γ ,where ˜ ω γ ( b ) = − Im γ Im R − γ ( t b ) R L b | ξ γ − ( x b ) | d x b . So by iteration, S ∗ lγ = + P l − s =0 S ∗ sγ ˜ ω γ . Hence, h , S lγ J i ν = h , J i ν + h P l − s =0 S ∗ sγ ˜ ω γ , J i ν . Denoting Z l J := ˜ ω γ P l − s =0 S sγ J , we see that if J ⊥ , then ( S lγ J − Z l J ) ⊥ . Consequently, by Proposition 9.3, for any L , kS r +1) γ J k ν ≤ kS γ ( S rγ − Z r ) J k ν + kZ r J k ν ≤ (cid:0) − c ( D, β ) L − (cid:1) / k ( S rγ − Z r ) J k ν + C ′ N,L ,r ( J ) / + kZ r J k ν . where C ′ N,L ,l ( J ) = C ′ N,L (( S lγ − Z l ) J ) = O N → + ∞ ,γ (1) L − s by Corollary 9.5. Using (9.16)and k ( S rγ − Z r ) J k ≤ kS rγ J k + kZ r J k , we get by iteration kS rγ J k ν ≤ (cid:0) − L − c ( D, β ) (cid:1) r/ k J k ν + r − X l =0 C ′ N,L ,l ( J ) / + 2 r − X l =0 kZ l J k ν = (cid:0) − L − c ( D, β ) (cid:1) r/ k J k ν + (cid:0) rL − s + r η (cid:1) O N → + ∞ ,γ (1)for any J ⊥ in ℓ ( ν ) satisfying (Hol) .Decomposing P tj =1 c j S jγ J = P ⌊ t/ ⌋ r =1 c r S rγ J + P ⌊ t/ ⌋ r =0 c r +10 S γ S rγ J we thus get(10.6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ γ (B ) N n M n M X r =1 ⌊ nM − rt ⌋− X a =0 D J ∗ K ′ γ , e i θ t X j =1 c j S jγ P ⊥ ,ν e − i θ S atu γ T K ′ γ E ν γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ t (cid:16) L c ( D, β ) + t L − s + t η (cid:17) O N → + ∞ ,γ (1) . Recall that t = M α , α < . We now choose α = 3 / s = 4 and let L = M / .Then collecting (10.2), (10.4), (10.5), (10.6), we see that whether G N is in case (i) or (ii),the variance is bounded by quantities vanishing in the limit N −→ ∞ followed by η ↓ M −→ ∞ . This concludes the proof of Theorem 2.1. Appendix A. Further properties of Green’s function on quantum trees
This appendix is devoted to a corollary of Lemma 3.1, which can be thought of as akind of “recursive relation” for the imaginary parts of Green’s functions. Its origin in thecombinatorial case is the recursive relation of the spherical function of regular trees. Wealso discuss the proof of Lemma 3.2 afterwards.
Corollary A.1.
Define Ψ γ,v ( w ) = Im G γ ( v, w ) for γ ∈ C \ R . For any b ∈ B , we have (A.1)Ψ γ,o b ( t b ) − ζ γ (ˆ b )Ψ γ,t b ( t b ) − ζ γ ( b )Ψ γ,o b ( o b ) + ζ γ (ˆ b ) ζ γ ( b )Ψ γ,t b ( o b ) = − ζ γ (ˆ b ) ζ γ ( b ) Im S γ ( L b ) , while, if k ≥ , for any ( b , . . . , b k ) ∈ B k ( T ) , we have (A.2) Ψ γ,o b ( t b k ) − ζ γ (ˆ b )Ψ γ,t b ( t b k ) − ζ γ ( b k )Ψ γ,o b ( o b k ) + ζ γ (ˆ b ) ζ γ ( b k )Ψ γ,t b ( o b k ) = 0 . Finally, we have (A.3) Im (cid:18) G γ ( t b , t b ) S γ ( L b ) (cid:19) − ζ γ ( b ) S γ ( L b ) Im (cid:18) G γ ( o b , t b ) S γ ( L b ) (cid:19) − ζ γ ( b ) S γ ( L b ) Im (cid:18) G γ ( o b , t b ) S γ ( L b ) (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12) ζ γ ( b ) S γ ( L b ) (cid:12)(cid:12)(cid:12)(cid:12) Im G γ ( o b , o b ) = Im (cid:18) ζ γ ( b ) S γ ( L b ) (cid:19) ≤ Im R + γ ( o b ) . Proof.
Since G γ ( v , v k ) = G γ ( v , v k − ) ζ γ ( b k ), taking the imaginary parts (and remember-ing that Im( zz ′ ) = z Im z ′ + z ′ Im z ) yields(A.4) Im G γ ( o b , t b k ) − ζ γ ( b k ) Im G γ ( o b , o b k ) = Im ζ γ ( b k ) · G γ ( o b , o b k ) . In particular,(A.5) Ψ γ,o b ( t b ) − ζ γ ( b )Ψ γ,o b ( o b ) = Im ζ γ ( b ) G γ ( o b , o b ) . Next, we have G γ ( t b , t b ) = G γ ( o b ,t b ) ζ γ (ˆ b ) and ζ γ (ˆ b ) = ζ γ ( b ) + S γ ( L b ) G γ ( o b ,o b ) by (3.7). Hence,(A.6) G γ ( t b , t b ) = ζ γ ( b ) G γ ( o b , t b ) + S γ ( L b ) ζ γ ( b ) UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 49 and thusΨ γ,t b ( t b ) = Im ζ γ ( b ) Re G γ ( o b , t b ) + Re ζ γ ( b )Ψ γ,o b ( t b ) + Im ( ζ γ ( b ) S γ ( L b )) . It follows that(A.7) Ψ γ,t b ( t b ) − ζ γ ( b )Ψ γ,o b ( t b ) = Im ζ γ ( b ) G γ ( o b , t b ) + Im ( ζ γ ( b ) S γ ( L b )) . Using (A.6) and Im( zz ′ ) = z Im z ′ + z ′ Im z again, we deduce that ζ γ (ˆ b )[Ψ γ,t b ( t b ) − ζ γ ( b )Ψ γ,t b ( o b )] = Im ζ γ ( b ) ζ γ (ˆ b ) G γ ( o b , t b ) + ζ γ (ˆ b ) Im ( ζ γ ( b ) S γ ( L b ))= Im ζ γ ( b ) h G γ ( o b , o b ) − S γ ( L b ) ζ γ (ˆ b ) i + ζ γ (ˆ b ) Im ( ζ γ ( b ) S γ ( L b ))= Im ζ γ ( b ) G γ ( o b , o b ) + ζ γ (ˆ b ) ζ γ ( b ) Im S γ ( L b ) . Recalling (A.5), this proves (A.1).Now let k ≥
2. If we apply (A.4) to ( b , . . . , b k ), we obtainIm G γ ( o b , t b k ) − ζ γ ( b k ) Im G γ ( o b , o b k ) = Im ζ γ ( b k ) · G γ ( o b , o b k ) . Multiplying this equation by ζ γ (ˆ b ) and using (3.10), we get ζ γ (ˆ b ) Im G γ ( o b , t b k ) − ζ γ (ˆ b ) ζ γ ( b k ) Im G γ ( o b , o b k ) = Im ζ γ ( b k ) · G γ ( o b , o b k ) . But, by (A.4), the RHS is equal to Im G γ ( o b , t b k ) − ζ γ ( b k ) Im G γ ( o b , o b k ), so (A.2) follows.Finally, let us prove (A.3). By (A.6), G γ ( t b , t b ) S γ ( L b ) = ζ γ ( b ) S γ ( L b ) G γ ( o b , t b ) S γ ( L b ) + ζ γ ( b ) S γ ( L b ) . To show the equality in (A.3), we must therefore show thatIm (cid:18) ζ γ ( b ) S γ ( L b ) G γ ( o b , t b ) S γ ( L b ) (cid:19) = 2 Re (cid:18) ζ γ ( b ) S γ ( L b ) (cid:19) Im (cid:18) G γ ( o b , t b ) S γ ( L b ) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) ζ γ ( b ) S γ ( L b ) (cid:12)(cid:12)(cid:12)(cid:12) Im G γ ( o b , o b ) . We have (cid:12)(cid:12)(cid:12) ζ γ ( b ) S γ ( L b ) (cid:12)(cid:12)(cid:12) G γ ( o b , o b ) = ζ γ ( b ) S γ ( L b ) G γ ( o b ,t b ) S γ ( L b ) , so thatIm (cid:18) ζ γ ( b ) S γ ( L b ) G γ ( o b , t b ) S γ ( L b ) (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12) ζ γ ( b ) S γ ( L b ) (cid:12)(cid:12)(cid:12)(cid:12) Im G γ ( o b , o b )= Im (cid:18) ζ γ ( b ) S γ ( L b ) G γ ( o b , t b ) S γ ( L b ) (cid:19) +Im (cid:18) ζ γ ( b ) S γ ( L b ) G γ ( o b , t b ) S γ ( L b ) (cid:19) = 2 Re (cid:18) ζ γ ( b ) S γ ( L b ) (cid:19) Im (cid:18) G γ ( o b , t b ) S γ ( L b ) (cid:19) , and the equality in (A.3) follows. For the inequality in (A.3), we use that Im R + γ ( o b ) =Im ζ γ ( b ) − C γ ( L b ) S γ ( L b ) ≥ Im ζ γ ( b ) S γ ( L b ) by Remark A.2 below. (cid:3) Proof of Lemma 3.2.
We only prove the “current” relations, see [6, Section 2] for theremaining parts. We will also use that ζ γ ( b ) = V + γ ; o ( t b ) V + γ ; o ( o b ) , as follows from [6, Lemma 2.1].Since V + γ ; o satisfies the δ -conditions, we have P b + ∈N + b R + γ ( o b + ) = R + γ ( t b ) + α t b , so P b + ∈N + b Im R + γ ( o b + ) = Im R + γ ( t b ). Similarly, as P w − ∈N − w U ′ w − ( L w − ) + α w U w (0) = U ′ w (0),we get P b − ∈N − b Im R − γ ( t b − ) = Im R − γ ( o b ).Suppose Hf = γf and let J γ ( x b ) = Im[ f ( x b ) f ′ ( x b )]. Then J ′ γ ( x b ) = Im[ | f ′ ( x b ) | + f ( x b )[ W ( x b ) f ( x b ) − γf ( x b )] = − Im γ | f ( x b ) | . Therefore,(A.8) J γ ( t b ) = J γ ( o b ) − Im γ Z L b | f b ( x b ) | d x b . But for f b ( x b ) = ξ γ + ( x b ) = V + γ ; ob ( x b ) V + γ ; ob ( o b ) , we have J ( t b ) = | ζ γ ( b ) | Im R + γ ( t b ) and J ( o b ) =Im R + γ ( o b ), so (3.13) follows. Equation (3.14) also follows by taking f b ( x b ) = ξ γ − ( x b ) = U − γ ; v ( x b ) U − γ ; v ( t b ) . To deduce (3.15), express U − γ ; v ( x b ) using the data at t b , which reads U − γ ; v ( x b ) = U − γ ; v ( t b )[ S ′ γ ( L b ) C γ ( x b ) − C ′ γ ( L b ) S γ ( x b )] + ( U − γ ; v ) ′ ( t b )[ − S γ ( L b ) C γ ( x b ) + C γ ( L b ) S γ ( x b )]. Re-calling (2.6) completes the proof. (cid:3) Remark A.2.
It also follows that Im( S ′ γ ( L b ) S γ ( L b ) ) ≤ S ′ γ ( L b ) S γ ( L b ) ) = | S γ ( L b ) | Im[ S γ ( L b ) S ′ γ ( L b )] ≤ | S γ ( L b ) | Im[ S γ (0) S ′ γ (0)] = 0. If thepotentials are symmetric, then S ′ γ ( L b ) = C γ ( L b ), so we also get Im( C γ ( L b ) S γ ( L b ) ) ≤ Appendix B. Proofs of the reduction
B.1.
Reduction to a discrete variance.
The aim of this subsection is to prove Propo-sition 5.3. In the course of the proof, we will omit the subscripts N from f N and K N tolighten notations. Proof of (1).
Let ψ j be an eigenfunction on G N corresponding to λ j . On the edge b , we have ψ j,b ( x b ) = ψ j ( o b ) C λ j ( x b ) + ψ ′ j ( o b ) S λ j ( x b ). Evaluating at x b = L b , we get ψ j,b ( x b ) = [ C λ j ( x b ) − C λj ( L b ) S λj ( L b ) S λ j ( x b )] ψ j ( o b ) + S λj ( x b ) S λj ( L b ) ψ j ( t b ). From (2.6), we get(B.1) ψ j,b ( x b ) = S λ j ( L b − x b ) S λ j ( L b ) ψ j ( o b ) + S λ j ( x b ) S λ j ( L b ) ψ j ( t b ) . Since 2 h ψ j , f ψ j i = P b R L b f b ( x b ) | ψ j,b ( x b ) | d x b , it easily follows that(B.2) 2 h ψ j , f ψ j i = h ˚ ψ j , ( K f,j + J f,j + M (1) f,j + M (2) f,j ) G ˚ ψ j i , where ˚ ψ j ( v ) = ψ j ( v ), K f,j , J f,j ∈ H and M f,j ∈ H are defined by K f,j ( v ) = X b : o b = v S λ j ( L b ) Z L b f b ( x b ) S λ j ( L b − x b ) d x b ,J f,j ( v ) = X b : t b = v S λ j ( L b ) Z L b f b ( x b ) S λ j ( x b ) d x b ,M (1) f,j ( b ) = 1 S λ j ( L b ) Z L b f b ( x b ) S λ j ( L b − x b ) S λ j ( x b ) d x b = M (2) f,j (ˆ b ) . Let us write(B.3) 2 [ f ] γ j := D ˚ ψ j , (cid:16)h h K f,j i γ j + h J f,j i γ j + h M (1) f,j i γ j + h M (2) f,j i γ j i (cid:17) G ˚ ψ j E = ( h K f,j i γ j + h J f,j i γ j + 2 h M f,j i γ j ) k ˚ ψ j k , where h M (1) f,j i γ j = h M (2) f,j i γ j =: h M f,j i γ j because Ψ γ,t b ( o b ) = Ψ γ,o b ( t b ).From (B.2) and (B.3), we have(B.4)1 N N ( I ) X λ j ∈ I |h ψ j , f ψ j i − [ f ] γ j | ≤ Var I η (cid:0)(cid:0) K f,λ (cid:1) G − h K f,λ i γ (cid:1) +Var I η (cid:0)(cid:0) J f,λ (cid:1) G − h J f,λ i γ (cid:1) + Var I η (cid:16)(cid:0) M (1) f,λ (cid:1) G − h M (1) f,λ i γ (cid:17) + Var I η (cid:16)(cid:0) M (2) f,λ (cid:1) G − h M (2) f,λ i γ (cid:17) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 51
We now need to replace [ f ] γ by h f i γ . By (B.3),(B.5) [ f ] γ j = (cid:13)(cid:13)(cid:13) ˚ ψ ( N ) j (cid:13)(cid:13)(cid:13) P v ∈ V N Im ˜ g γ j N ( v, v ) X b ∈ B Z L b f b ( x b )Ψ γ j b ( x b ) d x b , where(B.6) Ψ γ j b ( x b ) =Im ˜ g γ j N ( o b , o b ) S λ j ( L b − x b ) + Im ˜ g γ j N ( t b , t b ) S λ j ( x b ) + 2 Im ˜ g γ j N ( o b , t b ) S λ j ( L b − x b ) S λ j ( x b ) S λ j ( L b ) . Let us also introduce the quantity(B.7) f γ = f γN = R G N f ( x )Ψ γ ( x )d x R G N Ψ γ ( x )d x . Then(B.8) 1 N N ( I ) X λ j ∈ I |h ψ j , f ψ j i − h f i γ j | ≤ N N ( I ) X λ j ∈ I |h ψ j , f ψ j i − [ f ] γ j | + 1 N N ( I ) X λ j ∈ I (cid:12)(cid:12)(cid:12) [ f ] γ j − f γ j (cid:12)(cid:12)(cid:12) + 1 N N ( I ) X λ j ∈ I (cid:12)(cid:12)(cid:12) f γ j − h f i γ j (cid:12)(cid:12)(cid:12) . We already controlled the first term. Let us turn to the second. Noting that[ f ] γ j = [ f γ j ˆ ] γ j we have 1 N N ( I ) X λ j ∈ I | [ f ] γ j − f γ j | = 1 N N ( I ) X λ j ∈ I | [ f γ j ˆ ] γ j − h ψ j , f γ j ˆ ψ j i| , which can be bounded by discrete variances as in (B.4). For example K f,j ( v ) is nowreplaced by K f γj ˆ ,j ( v ). By definition we see that K f γj ˆ ,j ( v ) = f γ j K ˆ ,j ( v ).To deal with the last term, we use [6, eq.(A.12)], which shows that(B.9) ˜ g z ( x b , x b ) = 1 S z ( L b ) (cid:0) [ S z ( L b ) C z ( x b ) − C z ( L b ) S z ( x b )]˜ g z ( o b , x b )+ S z ( x b )[˜ g z ( t b , x b ) + S z ( L b ) C z ( x b ) − C z ( L b ) S z ( x b )] (cid:1) = S z ( L b − x b )˜ g z ( o b , x b ) + S z ( x b )˜ g z ( t b , x b ) + S z ( x b ) S z ( L b − x b ) S z ( L b )= S z ( L b − x b )˜ g z ( o b , o b ) + 2 S z ( L b − x b ) S z ( x b )˜ g z ( o b , t b ) + S z ( x b )˜ g z ( t b , t b ) S z ( L b )+ S z ( x b ) S z ( L b − x b ) S z ( L b ) , where we used (2.6) in the second equality. Recalling (B.6), we deduce that(B.10) | Ψ γ j b ( x b ) − Im ˜ g γ j ( x b , x b ) | ≤ Cη ( | ˜ g γ j ( o b , o b ) | + 2 | ˜ g γ j ( o b , t b ) | + | ˜ g γ j ( t b , t b ) | + 1) , where we used that the function S z is Lipschitz continuous on I + i[0 , C is uniform in ( z, L, W ) ∈ ( I + i[0 , × Lip M [m , M] and depends on C Dir . Comparing (2.9)and (B.7) we get (B.11) | f γ − h f i γ | ≤ | R G N f ( x )(Im ˜ g γ ( x, x ) − Ψ γ ( x ))d x | R G N Im ˜ g ( x, x )d x + (cid:12)(cid:12)(cid:12) Z G N f ( x )Ψ γ ( x )d x (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) R G N Im ˜ g γ ( x, x )d x − R G N Ψ γ ( x )d x (cid:12)(cid:12)(cid:12) ≤ Cη P b ∈ B N ( | ˜ g γ ( o b , o b ) | + 2 | ˜ g γ ( o b , t b ) | + | ˜ g γ ( t b , t b ) | + 1) R G N Im ˜ g γ ( x, x )d x (cid:16) R G N | Ψ γ ( x ) | d x | R G N Ψ γ ( x )d x | (cid:17) , where we used that | f | ≤
1. It follows from Proposition C.4 thatlim sup N →∞ sup λ ∈ I | f γ − h f i γ |≤ C ′ η E P [ L b ( G γ ( o b , o b ) + 2 | G γ ( o b , t b ) | + | G γ ( t b , t b ) | + 1)] E P (Im G z ) (cid:16) E P ( | Ψ γ | ) | E P (Ψ γ ) | (cid:17) . Note that, by (B.10), | E P (Ψ γ ) | ≥ E P (Im ˜ g γ ( x b , x b )) − Cη E P ( | ˜ g γ j ( o b , o b ) | + 2 | ˜ g γ j ( o b , t b ) | + | ˜ g γ j ( t b , t b ) | + 1) . Therefore, thanks to Proposition C.7 lim sup N →∞ sup λ ∈ I | f γ − h f i γ | vanishes as η ↓ η ↓ lim sup N →∞ N N ( I ) X λ j ∈ I (cid:12)(cid:12) f γ j − h f i γ j (cid:12)(cid:12) = 0 . Recalling (B.8), this concludes the proof of (1), up to verifying that the operators arein (Hol) . For K f,λ ( v ), the bounds | K f,λ ( v ) | ≤ P o b = v S λ ( L b ) R L b S λ ( L b − x b ) d x b and | K f,λ ( v ) − K f,λ ′ ( v ) | ≤ P o b = v R L b | S λ ( L b − x b ) S λ ( L b ) − S λ ′ ( L b − x b ) S λ ′ ( L b ) | d x b easily imply this. In facthere we can simply bound uniformly | K f,λ ( v ) | ≤ C D, M , Dir and | K f,λ ( v ) − K f,λ ′ ( v ) | ≤ C ′ | λ − λ ′ | . The operators J f,λ , M f,λ are similar. Finally | K f γ ˆ ,λ ( v ) | ≤ C R G N | Ψ γ ( x ) | d x | R G N Ψ γ ( x ) d x | and | K f γ ,λ ( v ) − K f γ ′ ,λ ′ ( v ) | ≤ | K f γ ,λ ( v ) − K f γ ,λ ′ ( v ) | + | K f γ ,λ ′ ( v ) − K f γ ′ ,λ ′ ( v ) | . The first isbounded by C ′ | λ − λ ′ | R G N | Ψ γ ( x ) | d x | R G N Ψ γ ( x ) d x | , the second by C | f γ − f γ ′ | ≤ C R G N | Ψ γ − Ψ γ ′ || R G N Ψ γ | (1+ R G N | Ψ γ || R G N Ψ γ ′ | ),by arguing as in (B.11). Using Corollary C.8 and Proposition C.9 concludes the proof. Proof of (2).
We will suppose that k ≥
2. The case k = 1 is very similar to the proofof (1). Let K ∈ K k . We have2 h ψ j , K ψ j i = X b ∈ B Z L b ψ j ( x b )( K ψ j )( x b ) d x b = X ( b ; b k ) ∈ B k Z L b Z L bk K b ,b k ( x b , y b k ) (cid:20) S λ j ( L b − x b ) S λ j ( L b ) ψ j ( o b )+ S λ j ( x b ) S λ j ( L b ) ψ j ( t b ) (cid:21) · (cid:20) S λ j ( L b k − y b k ) S λ j ( L b k ) ψ j ( o b k ) + S λ j ( y b k ) S λ j ( L b k ) ψ j ( t b k ) (cid:21) d x b d y b k . (B.12)In analogy to the previous step, we define the operators J γ K ∈ H k , M γ K , , M γ K , ∈ H k − and P γ K ∈ H k − by(B.13) J γ K ( b ; b k ) = Z L b Z L bk K b ,b k ( r b , s b k ) S Re γ ( L b − r b ) S Re γ ( s b k ) S Re γ ( L b ) S Re γ ( L b k ) d r b d s b k , UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 53 M γ K , ( b ; b k − )= X b k ∈N + bk − Z L b Z L bk K b ,b k ( r b , s b k ) S Re γ ( L b − r b ) S Re γ ( L b k − s b k ) S Re γ ( L b ) S Re γ ( L b k ) d r b d s b k M γ K , ( b ; b k ) = X b ∈N − b Z L b Z L bk K b ,b k ( r b , s b k ) S Re γ ( r b ) S Re γ ( s b k ) S Re γ ( L b ) S Re γ ( L b k ) d r b d s b k P γ K ( b ; b k − ) = X b k ∈N + bk − b ∈N − b Z L b Z L bk K b ,b k ( r b , s b k ) S Re γ ( r b ) S Re γ ( L b k − s b k ) S Re γ ( L b ) S Re γ ( L b k ) d r b d s b k . Then expanding (B.12) gives2 h ψ j , K ψ j i = X ( b ; b k ) ∈ B k ψ j ( o b ) J γ K ( b ; b k ) ψ j ( o b k )+ X ( b ; b k − ) ∈ B k − ψ j ( o b ) M γ K , ( b ; b k − ) ψ j ( o b k − )+ X ( b ; b k ) ∈ B k − ψ j ( o b ) M γ K , ( b ; b k ) ψ j ( o b k ) + X ( b ; b k − ) ∈ B k − ψ j ( o b ) P γ K ( b ; b k − ) ψ j ( o b k − ) . In other words, setting M γ K = M γ K , + M γ K , , we have(B.14) 2 h ψ j , K ψ j i = h ˚ ψ j , (cid:0) J λ j K + M λ j K + P λ j K (cid:1) G ˚ ψ j i . We define 2 [ K ] γ j := (cid:2) h J γ j K i γ j + h M γ j K i γ j + h P γ j K i γ j (cid:3) · k ˚ ψ j k = h ˚ ψ j , (cid:0) [ h J γ j K i γ j + h M γ j K i γ j + h P γ j K i γ j ] (cid:1) G ˚ ψ j i , so that(B.15) 1 N N ( I ) X λ j ∈ I |h ψ j , K ψ j i − [ K ] γ j |≤ Var Iη (cid:0)(cid:0) J γ K (cid:1) G − h J γ K i γ (cid:1) + Var Iη (cid:0)(cid:0) M γ K (cid:1) G − h M γ K i γ (cid:1) + Var Iη (cid:0)(cid:0) P γ K (cid:1) G − h P γ K i γ (cid:1) . Recalling the definition (5.3) and the fact that Ψ γ,v ( w ) = Im ˜ g γN ( v, w ), we note that[ K ] γ j = (cid:13)(cid:13)(cid:13) ˚ ψ ( N ) j (cid:13)(cid:13)(cid:13) P v ∈ V N Im ˜ g γ j N ( v, v ) X ( b ; b k ) ∈ B k Z L b Z L bk K b ,b k ( x b , y b k )Ψ γ j ( x b , y b k ) d x b d y b k , whereΨ γ j ( x b , y b k ) = S λ j ( L b − x b ) S λ j ( y b k ) S λ j ( L b ) S λ j ( L b k ) Im ˜ g γ j ( o b , t b k )+ S λ j ( L b − x b ) S λ j ( L b k − y b k ) S λ j ( L b ) S λ j ( L b k ) Im ˜ g γ j ( o b , o b k )+ S λ j ( x b ) S λ j ( y b k ) S λ j ( L b ) S λ j ( L b k ) Im ˜ g γ j ( t b , t b k ) + S λ j ( x b ) S λ j ( L b k − y b k ) S λ j ( L b ) S λ j ( L b k ) Im ˜ g γ j ( t b , o b k ) . Just as in the proof of (1), we introduce K γ := P ( b ; b k ) ∈ B k R L b R L bk K ( x b , y b k )Ψ( x b , y b k )d x b d y b k R G N Ψ( x )d x . We have(B.16) 1 N N ( I ) X λ j ∈ I |h ψ j , K ψ j i − hKi γ j | ≤ N N ( I ) X λ j ∈ I (cid:12)(cid:12) h ψ j , K ψ j i − [ K ] γ j (cid:12)(cid:12) + 1 N N ( I ) X λ j ∈ I (cid:12)(cid:12) [ K ] γ j − K γ j (cid:12)(cid:12) + 1 N N ( I ) X λ j ∈ I (cid:12)(cid:12) K γ j − hKi γ j (cid:12)(cid:12) . The first term is bounded by (B.15). The second term is estimated as in the proof of(1), noting that [ K ] γ j = [ K γ j ˆ ] γ j .To deal with the last term in (B.16), we use [6, eq.(A.11)] and argue as in the proof of(1), to see that | Ψ γ j ( x b , y b k ) − Im ˜ g γ j ( x b , y b k ) |≤ Cη ( | ˜ g γ j ( o b , o b k ) | + | ˜ g γ j ( o b , t b k ) | + | ˜ g γ j ( t b , o b k ) | + | ˜ g γ j ( t b , t b k ) | ) . As in (1), we deduce that N N ( I ) P λ j ∈ I (cid:12)(cid:12) K γ j − hKi γ j (cid:12)(cid:12) →
0, concluding the proof of thebound. The proof of property (Hol) also goes as before.B.2.
Reduction to non-backtracking variances.
The aim of this subsection is toprove Proposition 5.4. Recall definition (4.1) of f j , f ∗ j . Proof of (1).
Since ψ j ( o b ) ψ j ( t b ) ∈ R then ψ j ( t b ) ψ j ( o b ) = ψ j ( o b ) ψ j ( t b ). Hence, f ∗ j ( b ) f j ( b ) = (1 + ζ γ (ˆ b ) ζ γ ( b )) ψ j ( o b ) ψ j ( t b ) − ζ γ ( b ) | ψ j ( o b ) | − ζ γ (ˆ b ) | ψ j ( t b ) | S λ ( L b ) . Thus, f ∗ j ( b b ) f j ( b b ) = (1 + ζ γ ( b ) ζ γ (ˆ b )) ψ j ( o b ) ψ j ( t b ) − ζ γ (ˆ b ) | ψ j ( t b ) | − ζ γ ( b ) | ψ j ( o b ) | S λ ( L b ) . Letting ( L J )( b ) = ζ γ (ˆ b ) ζ γ ( b ) ( T J )( b ) and ( L J )(ˆ b ) = ζ γ ( b ) ζ γ (ˆ b ) ( T J )( b ), we get( L J )( b ) f ∗ j ( b ) f j ( b ) − ( L J )(ˆ b ) f ∗ j ( b b ) f j ( b b )= ( T J )( b ) S λ ( L b ) (cid:26) | ψ ( t b ) | ·
2i Im (cid:18) ζ γ (ˆ b )1 + ζ γ (ˆ b ) ζ γ ( b ) (cid:19) − | ψ ( o b ) | ·
2i Im (cid:18) ζ γ ( b )1 + ζ γ ( b ) ζ γ (ˆ b ) (cid:19)(cid:27) . Let
T J ( b ) = β b J ( t b ) with β b to be chosen later. Then, h f ∗ j , [( L − L ) J ] B f j i = X b ∈ B ( L J )( b ) f ∗ j ( b ) f j ( b ) − X b ∈ B ( L J )( b b ) f ∗ j ( b b ) f j ( b b )= 2i X b ∈ B β b J ( t b ) S λ ( L b ) (cid:26) | ψ ( t b ) | · Im (cid:18) ζ γ (ˆ b )1 + ζ γ (ˆ b ) ζ γ ( b ) (cid:19) − | ψ ( o b ) | · Im (cid:18) ζ γ ( b )1 + ζ γ ( b ) ζ γ (ˆ b ) (cid:19)(cid:27) = 2i X b ∈ B | ψ ( o b ) | S λ ( L b ) Im (cid:18) ζ γ ( b )1 + ζ γ ( b ) ζ γ (ˆ b ) (cid:19) (cid:8) β b b J ( o b ) − β b J ( t b ) (cid:9) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 55
But by (3.7), ζ γ ( b )1+ ζ γ ( b ) ζ γ (ˆ b ) = ζγ ( b ) + ζ γ (ˆ b ) =
12 Re ζ γ (ˆ b )+ Sγ ( Lb )˜ gγ ( tb,tb ) , soIm (cid:18) ζ γ ( b )1 + ζ γ ( b ) ζ γ (ˆ b ) (cid:19) = − Im( S γ ( L b )˜ g γ ( t b ,t b ) ) | ζ γ (ˆ b ) + S γ ( L b )˜ g γ ( t b ,t b ) | = N γ ( t b ) Re S γ ( L b ) | ˜ g γ ( t b ,t b ) | − Re ˜ g γ ( t b ,t b ) Im S γ ( L b ) | ˜ g γ ( t b ,t b ) | | ζ γ (ˆ b ) + S γ ( L b )˜ g γ ( t b ,t b ) | = | ζ γ ( b ) | | ζ γ ( b ) ζ γ (ˆ b ) | · N γ ( t b ) Re S γ ( L b ) | ˜ g γ ( t b , t b ) | (cid:18) − Re ˜ g γ ( t b , t b ) Im S γ ( L b ) N γ ( t b ) Re S γ ( L b ) (cid:19) . Choose β b b = S λ ( L b ) | ˜ g γ ( t b ,t b ) | | ζ γ ( b ) ζ γ (ˆ b ) | Re S γ ( L b ) | ζ γ ( b ) | N γ ( t b ) , so by (3.7) β b = S λ ( L b ) | ˜ g γ ( t b ,t b ) | | ζ γ ( b ) ζ γ (ˆ b ) | Re S γ ( L b ) | ζ γ ( b ) | N γ ( o b ) = N γ ( t b ) N γ ( o b ) β b b . We thus get h f ∗ j , [( L − L ) J ] B f j i = 2i X b | ψ ( o b ) | (cid:16) − Im S γ ( L b ) Re ˜ g γ ( t b , t b )Re S γ ( L b ) N γ ( t b ) (cid:17)n J ( o b ) − N γ ( t b ) N γ ( o b ) J ( t b ) o = 2i (cid:16) X o b | ψ ( o b ) | J ( o b ) d ( o b ) − X o b | ψ ( o b ) | N γ ( o b ) X t b ∼ o b N γ ( t b ) J ( t b ) + h ˚ ψ j , ( E γ j J ) G ˚ ψ j i (cid:17) = 2i {h ˚ ψ, [( I − P γ ) dJ ] G ˚ ψ i + h ˚ ψ, ( E γ j J ) G ˚ ψ j i} , since P γ = dN γ P N γ d and ( E γ J )( o b ) = P t b ∼ o b [ Im S γ ( L b ) Re ˜ g γ ( t b ,t b )Re S γ ( L b ) ( J ( t b ) N γ ( o b ) − J ( o b ) N γ ( t b ) )]. Thus,Var Iη [( I − P γ ) J ] ≤ Var I nb ,η [( L − L ) d − J ] + Var Iη ( E γ d − J ) . But P γ ( S T,γ K ) = T P Ts =1 ( T − s + 1) P sγ K = S T,γ K − K + e S T,γ K , so K = ( I − P γ ) S T,γ K + e S T,γ
K .
Hence, Var Iη ( K ) ≤ Var Iη [( I − P γ ) S T,γ K ] + Var Iη [ e S T,γ K ](B.17) ≤ Var I nb ,η [( L − L ) d − S T,γ K ] + Var Iη [ E γ d − S T,γ K ] + Var Iη [ e S T,γ K ] . Finally, L K ′ ( b ) − L K ′ ( b ) = ( T K ′ )( b ) − ( T K ′ )(ˆ b )1+ ζ γ ( b ) ζ γ (ˆ b ) = β b K ′ ( t b ) − β b b K ′ ( o b )1+ ζ γ ( b ) ζ γ (ˆ b ) , so( L − L ) K ′ ( b ) = β b b ζ γ ( b ) ζ γ (ˆ b ) (cid:16) N γ ( t b ) N γ ( o b ) K ′ ( t b ) − K ′ ( o b ) (cid:17) = ( L γ K ′ )( b )using the above relation on β b . Replacing K by J − h J i γ in (B.17) completes the proof. Proof of (2)
Define T γ : H → H and O γ : H → H by( T γ K )( b ) = S γ ( L b )1 + ζ γ (ˆ b ) ζ γ ( b ) K ( b ) , (B.18) ( O γ K )( v ) = − X b ∈ B ; t b = v ζ γ (ˆ b ) S γ ( L b ) K ( b ) − X b ∈ B ; o b = v ζ γ ( b ) S γ ( L b ) K ( b ) . For k ≥
2, define T γk : H k → H k , O γk : H k → H k − and P γk : H k → H k − by( T γk K )( b , . . . , b k ) = S Re γ ( L b ) S Re γ ( L b k ) K ( b , . . . , b k ) , ( O γk K )( b , . . . , b k − ) = − X b ∈N − b ζ γ (ˆ b ) K ( b , . . . , b k − ) − X b k ∈N + bk − K ( b , . . . , b k ) ζ γ ( b k ) , ( P γk K )( b , . . . , b k − ) = X b ∈N − b X b k ∈N + bk − ζ γ (ˆ b ) K ( b , . . . , b k ) ζ γ ( b k ) . Then we have
Lemma B.1.
For any K ∈ H satisfying (5.1) , Var Iη ( K − h K i γ ) ≤ Var I nb ,η ( T γ K ) + Var Iη ( O γ T γ K − hO γ T γ K i γ ) + O N → + ∞ ,γ ( η ) . For any K ∈ H k , k ≥ , we have Var Iη ( K − h K i γ ) ≤ Var I nb ,η ( T γk K ) + Var Iη ( O γk K − hO γk K i γ ) + Var Iη ( P γk K − hP γk K i γ ) . Before proving the lemma, we note that it implies point (2) of Proposition 5.4. Indeed,by hypothesis, J satisfies (Hol) . Applying the lemma several times if necessary, startingfrom J we obtain variances as in point (2), with operators L γp , F γp consisting of compo-sitions of L γ , S T,γ , O γk , T γ , . . . etc, all of which preserve (Hol) by Remark 5.2, since theyadd/multiply by terms in L γk . Proof of Lemma B.1.
Let K ∈ H k , k ≥
1. Then(B.19) h f ∗ j , K B f j i = X ( b ; b k ) ∈ B k K ( b ; b k ) S λ j ( L b ) S λ j ( L b k ) (cid:16) ψ j ( o b ) ψ j ( t b k ) − ζ γ j (ˆ b ) ψ j ( t b ) ψ j ( t b k ) − ψ j ( o b ) ζ γ j ( b k ) ψ j ( o b k ) + ζ γ j (ˆ b ) ψ j ( t b ) ζ γ j ( b k ) ψ j ( o b k ) (cid:17) . Assume k = 1. Define U γ : H → H by( U γ K )( b ) = 1 + ζ γ (ˆ b ) ζ γ ( b ) S γ ( L b ) K ( b ) , Since ψ j ( o b ) ψ j ( t b ) ∈ R , we have ψ j ( t b ) ψ j ( o b ) = ψ j ( o b ) ψ j ( t b ) and thus(B.20) h f ∗ j , K B f j i = h ˚ ψ j , ( U γ j K + O γ j K ) G ˚ ψ j i . We now note that(B.21) hU γ K i γ = −hO γ K i γ + O N → + ∞ ,γ ( η ) . Indeed, we have (cf. (5.3)) hU γ K i γ = 1 P v ∈ V Ψ γ,v ( v ) X b ∈ B ζ γ (ˆ b ) ζ γ ( b ) S γ ( L b ) K ( b )Ψ γ,o b ( t b ) . On the other hand, recalling (B.18), −hO γ K i γ = − P v ∈ V Ψ γ,v ( v ) X v ∈ V Ψ γ,v ( v )( O γ K )( v )= 1 P v ∈ V Ψ γ,v ( v ) X b ∈ B ζ γ (ˆ b )Ψ γ,t b ( t b ) + ζ γ ( b )Ψ γ,o b ( o b ) S γ ( L b ) K ( b )= hU γ K i γ + 1 P v ∈ V Ψ γ,v ( v ) X b ζ γ (ˆ b ) ζ γ ( b ) Im S γ ( L b ) S γ ( L b ) K ( b ) , where we used (A.1) and the fact that Ψ γ,o b ( t b ) = Ψ γ,t b ( o b ) by (3.11). Since | Im S γ ( L b ) | ≤ C I η , applying Cauchy-Schwarz and Corollary C.8 now proves (B.21).Recalling (B.20), we thus showed that for any K ∈ H ,Var Iη ( K − h K i γ ) = Var Iη ( U γ T γ K − hU γ T γ K i γ ) ≤ Var I nb ,η ( T γ K ) + Var Iη ( O γ T γ K − hO γ T γ K i γ ) + O N → + ∞ ,γ ( η ) . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 57
Now let k ≥
2. Recalling (B.19) we have h f ∗ j , ( T γ j k K ) B f j i = h ˚ ψ j , (cid:0) K + O γ j k K + P γ j k K (cid:1) G ˚ ψ j i , so that Var Iη ( K − h K i γ ) ≤ Var I nb ,η ( T γk K ) + Var Iη ( O γk K + P γk K + h K i γ ) . The result hence follows if we show that(B.22) h K i γ = −hO γk K i γ − hP γk K i γ . Indeed, − hO γk K i γ − hP γk K i γ = 1 P v ∈ V Ψ γ,v ( v ) X ( b ; b k − ) ∈ B k ζ γ (ˆ b ) K ( b ; b k − )Ψ γ,o b ( t b k − )+ 1 P v ∈ V Ψ γ,v ( v ) X ( b ; b k ) ∈ B k K ( b ; b k ) ζ γ ( b k )Ψ γ,o b ( t b k − ) − P v ∈ V Ψ γ,v ( v ) X ( b ; b k ) ∈ B k ζ γ (ˆ b ) K ( b ; b k ) ζ γ ( b k )Ψ γ,o b ( t b k − ) , so (B.22) follows from (A.2). (cid:3) Appendix C. Control of averaged Green functions
Throughout the proof we have to control averages of spectral quantities on the finitegraphs ( Q N ). The averages have two forms, either the energy is fixed and the averaging ison the vertices/edges of the graph, or we have a double averaging over the graph and theenergy. We discuss the first case here which is technically simpler and illustrates directlyhow we use our main assumptions. The idea is always as follows: we use the Benjamini-Schramm convergence to replace the averages on the finite graph by some E P -average overthe limiting random graph ( § C.1), then use the moment condition (Green) to show theselimit averages do not explode ( § C.2). The second case is handled in Section 6 using resultsof the present Appendix.C.1.
Consequences of (BST).
A rooted quantum graph ( Q , x ) is a quantum graph Q = ( V, E, L, W, α ) with a root x = ( b , x ) ∈ G . Let Q D, m , M ∗ be the set of (isomorphismclasses of) rooted quantum graphs [ Q , x ] satisfying (Data) . The set Q D, m , M ∗ is endowedwith a metric inspecting the local structure of the quantum graph, see [5]. Convergenceof the sequence ( Q N ) in the sense of Benjamini-Schramm means that the probabilitymeasures ν Q N := 12 L ( Q N ) X b ∈ B Z L b δ [ Q N , ( b ,x )] d x = 1 L ( Q N ) Z G N δ [ Q N , x ] d x on Q D, m , M ∗ converge weakly to a probability measure P on Q D, m , M ∗ . Such convergencemeans that if we take an r -ball B ( x , r ) uniformly at random in G N (this is encoded by ν Q N ), it will resemble a P -random ball in Q D, m , M ∗ . In formulas, the weak convergence ν Q N w −→ P means that for any continuous F : Q D, m , M ∗ −→ C , we havelim N →∞ L ( Q N ) Z G N F ([ Q N , x ]) d x = Z Q D, m , M ∗ F ([ Q , x ]) d P ([ Q , x ]) =: E P [ F ] . Assumption (BST) says that the limiting law P is supported on the subset T D, m , M ∗ ofquantum trees. In [5] we proved that, for any fixed z ∈ C + , the important function G z : Q D, m , M ∗ −→ C given by G z : [ Q , x ] G z ( x , x ) is continuous. So by weak convergence,(C.1) lim N →∞ L ( Q N ) Z G N g zN ( x , x ) d x = Z T D, m , M ∗ G z ( x , x ) d P ([ Q , x ]) = E P ( G z ) . This was used to prove that the empirical spectral measures of Q N converge vaguelyto some averaged spectral measure E P ( µ x ). In particular, [5, Theorem 3.11, Lemma 7.3]imply that for any bounded interval,(C.2) lim sup N → + ∞ N N ( I ) | V N | ≤ C I . More importantly, if P ( H Q has spectrum in I ) >
0, then by [5, Lemma 3.12], thereexists C ′ I > N → + ∞ N N ( I ) | V N | ≥ C ′ I . Remark C.1.
The bound (C.3) holds in particular on the interval I where (Green) holds. In fact, we have R I E P (Im R + λ +i0 ( o b ) − s + Im R + λ +i0 ( o ˆ b ) − s )d λ < ∞ by Fatou’s lemmaand (Green) . Here the integral is more precisely on I ∩ S , where S is a set of fullLebesgue measure on which all limits R + λ +i0 ( o b ) exist for P -a.e. [ Q , ( b, x b )]. The existenceof S follows from the Herglotz property of R + z ( o b ), see e.g. Remark C.3. It follows that R I E P (Im R + λ +i0 ( o b ) + Im R + λ +i0 ( o ˆ b ))d λ >
0. Using [6, Lemma A.2], if follows that thereis an L function f b = φ − λ ; b such that R I E P (Im h f b , G λ +i0 f b i + Im h f ˆ b , G λ +i0 f ˆ b i ) d λ > E P ( µ H Q f b ( I ) + µ H Q f ˆ b ( I )) >
0, where µ H Q f is the spectral measure of H Q at f and weused the standard inversion formula of Borel transform along with Fatou’s lemma and thespectral theorem h f, G z f i = R R t − z d µ H Q f ( t ). It follows that µ H Q f b ( I ) + µ H Q f ˆ b ( I ) > H Q has spectrum in I with positive probability.We are now interested in finding the limits of more general spectral quantities than(C.1). We first make the observation that, by definition of universal covers and the localmetric on Q D, m , M ∗ , we have d ([ e Q , f x ] , [ e Q , f x ]) ≤ d ([ Q , x ] , [ Q , x ]). It follows that themap Q D, m , M ∗ ∋ [ Q , x ] ˜ g z ( x , x ) is also continuous. So by weak convergence,(C.4) lim N →∞ L ( Q N ) Z G N ˜ g zN ( x , x ) d x = Z T D, m , M ∗ G z ( x , x ) d P ([ Q , x ]) = E P ( G z ) . Here we just have G z on the RHS because the universal cover of a tree is the tree itself.We will often need to control combinatorial averages like N P v ∈ V N g zN ( v, v ), for z ∈ C + .For this, we consider the map φ z : [ Q , ( b, x b )] G z ( o b ,o b ) L b d ( o b ) + G z ( o ˆ b ,o ˆ b ) L b d ( o ˆ b ) . Note that this mapis constant over the edge b . Here and in what follows, we duplicate all terms (for instancewe use both o b and o ˆ b ) just to ensure the map is well-defined, since [ Q , ( b, x b )] can also berepresented as [ Q , (ˆ b, L b − x b )].By arguments similar (and simpler) to [5, Section 6], we see that φ z is continuous on Q D, m , M ∗ . In fact, consider f z ( x ) := G z ( o b , x ) for x ∈ b . This solves Hf z = zf z , soit can be expressed in some basis of solutions E zb and E z ˆ b . We choose them to satisfy E zb (0) = 1 and ( E zb ) ′ (0) = − i √ z . This yields f z ( x ) = a ( b ) E zb ( x ) + a (ˆ b ) E z ˆ b ( L b − x ) for somecoefficients a ( b ) which depend on f z only through its initial values f z (0), ( f z ) ′ (0). Inparticular G z ( o b , o b ) = a ( b ) + a (ˆ b ) E z ˆ b ( L b ). Consequently φ z [ Q , ( b, x b )] = a ( b )+ a (ˆ b ) E z ˆ b ( L b ) L b d ( o b ) + a (ˆ b )+ a ( b ) E zb ( L b ) L b d ( o ˆ b ) . Now the arguments of [5, Section 4,6] show that φ z is indeed continuous.For later use (Proposition C.4), we mention that the arguments of [5] imply more generally UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 59 that it is continuous in ( z, [ Q , ( b, x )]). For now fix z . Since ν Q N w −→ P , this implies that L ( Q N ) R G N φ z [ Q N , x ] d x −→ E P ( φ z ). But an easy expansion gives R G N φ z [ Q N , x ] d x = P v ∈ V N g zN ( v, v ). Similarly N = P b ∈ B N R L b ( d ( o b ) L b + d ( o ˆ b ) L b ) d x b . Hence,(C.5) 1 N X v ∈ V N g zN ( v, v ) −→ E P ( d ( o b ) L b + d ( o ˆ b ) L b ) E P (cid:16) G z ( o b , o b ) d ( o b ) L b + G z ( o ˆ b , o ˆ b ) d ( o ˆ b ) L b (cid:17) . The same limit holds for ˜ g zN ( v, v ). In a similar way, the map [ Q , ( b, x b )] G z ( o b ,t b ) L b + G z ( o ˆ b ,t ˆ b ) L b can be expressed as a ( b ) E zb ( L b )+ a (ˆ b ) L b + a (ˆ b ) E z ˆ b ( L b )+ a ( b ) L b , showing that it is continuousand we deduce as before that N P b ∈ B N g zN ( o b , t b ) −→ E P ( d ( ob ) Lb + d ( o ˆ b ) Lb ) E P ( G z ( o b ,t b ) L b ), using(3.11).Recalling that ζ z ( b ) = G z ( o b ,t b ) G z ( o b ,o b ) , we immediately deduce the continuity of [ Q , ( b, x b )] ζ z ( b ) L b + ζ z (ˆ b ) L b , thus obtaining N P b ∈ B N ζ z ( b ) −→ E P ( d ( ob ) Lb + d ( o ˆ b ) Lb ) E P (cid:0) ζ z ( b )+ ζ z (ˆ b ) L b (cid:1) .It follows from [24, p.10] that [ Q , ( b, x b )] ( C z ( L b ) , S z ( L b )) is continuous. Indeed, thisis just saying that ( C z ( L b ) , S z ( L b )) depends continuously on ( W b , L b ). So using (3.4), wededuce that N P b ∈ B N R + z ( o b ) −→ E P ( d ( ob ) Lb + d ( o ˆ b ) Lb ) E P (cid:0) R + z ( o b )+ R + z ( o ˆ b ) L b (cid:1) .Similarly considering [ Q , ( b, x b )] L b ( ζ z ( b ) P b + ∈N + b ζ z ( b + ) + ζ z (ˆ b ) P ˆ b + ∈N +ˆ b ζ z (ˆ b + )), N P ( b ,b ) ∈ B ζ z ( b ) ζ z ( b ) −→ c d,L E P ( L b ( ζ z ( b ) P b + ∈N + b ζ z ( b + ) + ζ z (ˆ b ) P ˆ b + ∈N +ˆ b ζ z (ˆ b + )).Note that N +ˆ b = { b b ′ : b ′ ∈ N − b } .Throughout the paper, we need continuity of more general functionals, depending onseveral bonds. This is why we introduce the following class. Definition C.2. If k ∈ N , we define L γk ⊂ H k to be the set of all F γ ( b , . . . , b k ) that aresums and products of the following quantities, their inverses and complex conjugates: • S γ ( L b i ), S Re γ ( L b i ), R L bi S nγ ( x b i ) S mγ ( L b i − x b i )d x i , R L bi S n Re γ ( x b i ) S m Re γ ( L b i − x b i )d x i for m, n ∈ N ∪ { } and i ∈ { , . . . , k } . • ζ γ ( b i ), ζ γ ( ˆ b i ), | ζ γ ( ˆ b i ) ζ γ ( b i ) | , R + γ ( o b i ), R − γ ( t b i ), Im R + γ ( o b i ), Im R − γ ( t b i ). • ˜ g γN ( v, w ), g γN ( v, w ), Im g γN ( v, w ), v, w ∈ { o b i , t b i } ≤ i ≤ k . • R L bi Im ˜ g ( x b i , x b i )d x b i , R L bi g ( x bi ,x bi ) d x b i for i ∈ { , . . . , k } . Remark C.3.
Consider F γ ∈ L γk +1 , γ ∈ C + . We note that there is a Lebesgue-null set A ⊂ R such that, for each λ / ∈ A , there exists a set Ω λ ⊂ Q D, m , M ∗ with P (Ω λ ) = 1 havingthe following property: if [ Q , ( b , x )] ∈ Ω λ , ( b , . . . , b k ) ∈ B b k , then lim η ↓ F λ +i η ( b , . . . , b k )exists and is finite.Indeed, if F γ ∈ L γk +1 , then for every fixed [ Q , ( b , x )] and every ( b , . . . , b k ) ∈ B b k , themap F λ +i η ( b , . . . , b k ) is a product of Herglotz functions, their complex conjugate, andfunctions having limits on R . We deduce that the limit lim η ↓ F λ +i η ( b , . . . , b k ) exists and isfinite for all ( b , . . . , b k ) ∈ B b k and all λ ∈ R \ A [ Q ,b ] for some Borel set A [ Q ,b ] of measurezero (see e.g. [28, Corollary 3.29]). By Fubini’s theorem, the set C = { ([ Q , ( b , x )] , λ ) ∈ Q D, m , M ∗ × R : λ ∈ A [ Q ,b ] } has ( P ⊗ Leb)-measure zero, and hence, for almost all λ ∈ R , the set O λ = { [ Q , ( b , x )] ∈ Q D, m , M ∗ : ([ Q , ( b , x )] , λ ) ∈ C} has P -measure zero. Taking Ω λ = O cλ proves our claim. Proposition C.4.
Let Q N be a sequence of quantum graphs satisfying (Data) andconverging to P in the sense of Benjamini-Schramm. Let Ξ ⊂ C + be compact and let F γ ∈ L γk +1 . Then uniformly in γ ∈ Ξ , we have (C.6) 1 N X ( b ,...,b k ) ∈ B k +1 F γ ( b , . . . , b k ) −→ N →∞ c d,L E P (cid:20) X ( b ,...,b k ) ∈ B b k F γ ( b , . . . , b k ) + X ( b − k ,...,b − ) ∈ B k,b F γ (ˆ b , . . . , ˆ b − k ) (cid:21) , where c d,L = E P ( d ( ob ) Lb + d ( o ˆ b ) Lb ) .Proof. Let ˜ F γ ( b ) = P ( b ; b k ) ∈ B b k F γ ( b ; b k ). The previous arguments show that ˜ F γ ( b )is continuous on Q D,m,M ∗ for fixed γ , so (C.6) holds. For the Green’s functions, e.g. g γN ( o b , t b p ), one just expands f γ ( x ) = G γ ( o b , x ) for x ∈ b p using the basis of E γb p , E γ ˆ b p asbefore. It remains to justify uniformity in γ .Let us consider the limit in (C.5), the general case is similar. We first observe that φ : ( γ, [ Q , ( b, x b )]) G γ ( o b ,o b ) L b d ( o b ) + G γ ( o ˆ b ,o ˆ b ) L b d ( o ˆ b ) is uniformly continuous on Ξ × Q D, m , M ∗ . In fact,since φ does not depend on the value of x ∈ b , it is enough to study uniform continuity onthe compact set Ξ × Q D, m , M ,δ ∗ , where Q D, m , M ,δ ∗ := { ( Q , ( b, x b )) ∈ Q D, m , M ∗ | x b ∈ [ δ, L b − δ ] } (see [5, Lemma 3.6]). We are thus reduced to checking continuity in ( γ, [ Q , ( b, x b )]), whichholds as pointed out before (C.5).Using uniform continuity on Ξ × Q D, m , M ∗ , it easily follows that the sequence F N ( γ ) = L ( Q N ) R G N φ ( γ, [ Q N , x ]) d x is uniformly equicontinuous. The uniform convergence of F N ( γ ) follows. (cid:3) C.2.
Consequences of (Green).
Let I ⊂ I as in (Non-Dirichlet) and z ∈ C + . If all Q N satisfy (Data) , we may fix c , c , c > z ∈ I + i[0 , b ∈ ∪ N B N ,(C.7) c ≤ | S z ( L b ) | ≤ c and | C z ( L b ) | ≤ c . Using [6, Corollary 2.5] along with Proposition C.4, we see that (Green) implies thatfor any s > λ ∈ I ,η ∈ (0 , E P ( | G z ( o b , o b ) | s + | G z ( o ˆ b , o ˆ b ) | s ) < ∞ , sup λ ∈ I ,η ∈ (0 , E P ( | ˆ ζ z ( b ) | s + | ˆ ζ z (ˆ b ) | s ) < ∞ where z := λ + i η and sup λ ∈ I ,η ∈ (0 , E P ( | ˆ R + z ( o b ) | s + | ˆ R + z ( o ˆ b ) | s ) < ∞ . Using (3.9) then (3.7), we also deduce thatsup λ ∈ I ,η ∈ (0 , E P ( | G z ( o b , o b ) | − s + | G z ( o ˆ b , o ˆ b ) | − s ) < ∞ , sup λ ∈ I ,η ∈ (0 , E P ( | ˆ ζ z ( b ) | − s + | ˆ ζ z (ˆ b ) | − s ) < ∞ . Next, (3.8) and the Herglotz property imply that Im G z ( o b , o b ) ≥ Im R + z ( o b ) | G z ( o b , o b ) | ,so we deduce that sup λ ∈ I ,η ∈ (0 , E P ( | Im G z ( o b , o b ) | − s + | Im G z ( o ˆ b , o ˆ b ) | − s ) < + ∞ . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 61
We have thus obtained bounds on the powers of all the quantities listed in Defini-tion C.2, except on the negative powers of | ζ γ (ˆ b ) ζ γ ( b ) | and the positive powers of R L b g ( x b ,x b ) d x b .These are dealt with using Lemmas C.5 and C.6 below, which imply that for all s > λ ∈ I ,η ∈ (0 , E P h | ζ γ (ˆ b ) ζ γ ( b ) | − s i < + ∞ . and sup λ ∈ I ,η ∈ (0 , E P "(cid:18)Z L b g γ ( x b , x b ) d x b (cid:19) s < + ∞ . Lemma C.5.
We have | ζ γ (ˆ b ) ζ γ ( b ) | ≤ max (cid:18) | G γ ( o b , o b ) G γ ( t b , t b ) || G γ ( o b , t b ) | , | G γ ( o b , o b ) G γ ( t b , t b ) | Im G γ ( o b , o b ) Im G γ ( t b , t b ) (cid:19) . To deduce (C.8), note that | G γ ( o b ,o b ) G γ ( t b ,t b ) || G γ ( o b ,t b ) | = | ζ γ ( b ) ζ γ (ˆ b ) | . Proof.
We have ζ γ (ˆ b ) ζ γ ( b ) = | G γ ( o b ,t b ) | G γ ( o b ,o b ) G γ ( t b ,t b ) , so ζ γ (ˆ b ) ζ γ ( b ) = G γ ( o b ,o b ) G γ ( t b ,t b ) G γ ( o b ,o b ) G γ ( t b ,t b )+ | G γ ( o b ,t b ) | .Let us write G γ ( o b , o b ) = x + i y , G γ ( t b , t b ) = x ′ − i y ′ , so that G γ ( o b , o b ) G γ ( t b , t b ) + | G γ ( o b , t b ) | = | G γ ( o b , t b ) | + xx ′ + yy ′ + i( yx ′ − y ′ x ) . We know by Lemma 3.2 that y, y ′ ≥ • If xx ′ ≥ − yy ′ , we have | G γ ( o b , o b ) G γ ( t b , t b ) + | G γ ( o b , t b ) | | ≥ | G γ ( o b , t b ) | . • If xx ′ ≤ − yy ′ , then ( yx ′ − y ′ x ) ≥ yy ′ ) , so | yx ′ − y ′ x | ≥ √ yy ′ . Hence, | G γ ( o b , o b ) G γ ( t b , t b ) + | G γ ( o b , t b ) | | ≥ Im G γ ( o b , o b ) Im G γ ( t b , t b ) . (cid:3) Lemma C.6.
Let Ξ ⊂ C + be a bounded set. Suppose that Q satisfies (Data) . We mayfind a constant C ′ Ξ such that for all b ∈ B and all γ ∈ Ξ , we have (C.9) Z L b g γ ( x b , x b ) d x b ≤ C ′ Ξ (1 + | R + γ ( o b ) | ) | ζ γ ( b ) | (Im R + γ ( t b )) + C ′ Ξ (1 + | R − γ ( t b ) | ) | ζ γ (ˆ b ) | (Im R − γ ( o b )) . Proof.
Let b ∈ B , x ∈ b . Recalling the notations and results of Section 3.1, we have − ˜ g ( x, x ) = R + γ ( x )+ R − γ ( x ) , so that by the Herglotz property,(C.10) 1Im ˜ g γ ( x, x ) = | R + γ ( x ) + R − γ ( x ) | Im( R + γ ( x ) + R − γ ( x )) ≤ | R + γ ( x ) | Im R + γ ( x ) + 2 | R − γ ( x ) | Im R − γ ( x ) . Since V + γ ; o b ( t b ) = ζ γ ( b ), arguing as for (A.8) with f ( x ) = V + γ ; o b ( x ), we have(C.11) | ζ γ ( b ) | Im R + γ ( t b ) ≤ | V + γ ; o b ( x ) | Im R + γ ( x ) . Noting that V + γ ; o b ( y ) = C γ ( y )+ R + γ ( o b ) S γ ( y ), we deduce that for any compact set Ξ ⊂ C ,there exists C Ξ > (Data) and on Ξ such that for all z ∈ Ξ, all N ∈ N and all b ∈ B N , we have(C.12) | V + γ ; o b ( y ) | + | V + ′ γ ; o b ( y ) | ≤ C Ξ (1 + | R + γ ( o b ) | ) . In particular, using (C.11), we get(C.13) Im R + γ ( x ) ≥ | ζ γ ( b ) | Im R + γ ( t b ) C (1 + | R + γ ( o b ) | ) . The negative powers of R L b g ( x b ,x b ) d x b are easily bounded using Jensen’s inequality( L b R L b g ( x b ,x b ) d x b ) − ≤ L b R L b Im ˜ g ( x b , x b ) d x b . On the other hand, we have, using (C.12), | V + γ ; o b ( x ) | Im R + γ ( x ) = Im( V + γ ; o b ( x ) V + ′ γ ; o b ( x )) ≤ | V + γ ; o b ( x ) V + ′ γ ; o b ( x ) | ≤ C Ξ (1+ | R + γ ( o b ) | ) | V + γ ; o b ( x ) | , so, by (C.11), we get | V + γ ; o b ( x ) | ≥ | ζ γ ( b ) | Im R + γ ( t b ) C Ξ (1 + | R + γ ( o b ) | ) . All in all, we have | R + γ ( x ) | = | ( V + γ ; o b ) ′ ( x ) || V + γ ; o b ( x ) | ≤ C (1 + | R + γ ( o b ) | ) | ζ γ ( b ) | Im R + γ ( t b ) . The same proof gives a similar bound for | R − γ ( x ) | and a bound similar to (C.13) forIm R − γ ( x ), exchanging the roles of o b and t b . Therefore, combining this with (C.10) and(C.13), we obtain (C.9). (cid:3) The previous considerations imply that the limits (C.6) are controlled:
Proposition C.7.
Suppose (BST) , (Data) , (Non-Dirichlet) and (Green) are satis-fied, let I ⊂ I be compact, and F γ ∈ L γk +1 . Then sup λ ∈ I,η ∈ (0 ,η Dir ) E P (cid:20) X ( b ,...,b k ) ∈ B b k F γ ( b , . . . , b k ) + X ( b − k ,...,b − ) ∈ B k,b F γ (ˆ b , . . . , ˆ b − k ) (cid:21) < + ∞ . Proof.
By definition F γ ( b ; b k ) is a sum or product of quantities in Definition C.2. If F γ ( b ; b k ) = Q ki =0 T γi ( b i ), then using H¨older’s inequality with P ki =0 1 p i = 1, we have (cid:12)(cid:12)(cid:12) N X ( b ; b k ) F γ ( b ; b k ) (cid:12)(cid:12)(cid:12) ≤ k Y i =0 (cid:16) N X ( b ; b k ) | T γi ( b i ) | p i (cid:17) /p i ≤ C D,k k Y i =0 (cid:16) N X b ∈ B N | T γi ( b ) | p i (cid:17) /p i . Taking the limit N −→ + ∞ , Proposition C.4 thus gives E P (cid:20) X ( b ,...,b k ) ∈ B b k ˆ F γ ( b , . . . , b k ) + X ( b − k ,...,b − ) ∈ B k,b ˆ F γ (ˆ b , . . . , ˆ b − k ) (cid:21) ≤ C D,k k Y i =0 (cid:16) E P h T γi ( b ) + T γi (ˆ b ) i p i (cid:17) /p i . We are thus reduced to the case k = 0. But we already showed above how to bound eachof the quantities T γ ( b ) = ζ γ ( b ), R + γ ( o b ),... etc. uniformly. R + γ ( t b ) is also controlled since R + γ ( t b ) = P b + ∈N + b R + γ ( o b + ) − α t b . R − γ ( t b ) = R + γ ( o b b ) + C γ ( L b ) − S ′ γ ( L b ) S γ ( L b ) is controlled as well.The case when F γ ( b ; b k ) = G γ ( o b , t b k ) is controlled as above, we just view it as thelimit of ˜ g γN ( o b , t b k ), which has a product form by (3.10). (cid:3) Combining Proposition C.4 and C.7, we deduce:
Corollary C.8.
Suppose Q N satisfy (BST) , (Data) , (Non-Dirichlet) and (Green) .Let k ≥ , and let F γ ∈ L γk +1 . Then we have sup η ∈ (0 ,η Dir ) lim sup N → + ∞ N sup η ∈ ( − η , η ) sup λ ∈ I X ( b ,...,b k ) ∈ B k | F λ +i η +i η ( b , . . . , b k ) | < + ∞ . UANTUM ERGODICITY FOR QUANTUM GRAPHS IN THE AC REGIME 63
Proposition C.9.
Suppose Q N satisfy (BST) , (Data) , (Non-Dirichlet) and (Green) .Let k ≥ , and let F γ ∈ L γk +1 . For almost all λ ∈ I , for all s > and all t, t ′ ∈ (0 , ,these operators also satisfy lim η ↓ lim sup N → + ∞ N X ( b ,...,b k ) ∈ B k +1 (cid:12)(cid:12)(cid:12) F λ + t i η ( b , . . . , b k ) − F λ + t ′ i η ( b , . . . , b k ) (cid:12)(cid:12)(cid:12) s = 0 . Proof.
As in (C.6), we have for every η > N → + ∞ N X ( b ; b k ) ∈ B k +1 (cid:12)(cid:12)(cid:12) F λ + t i η ( b ; b k ) − F λ + t ′ i η ( b ; b k ) (cid:12)(cid:12)(cid:12) s = c D,L E P (cid:20) X ( b ; b k ) ∈ B b k (cid:12)(cid:12)(cid:12) F λ + t i η ( b ; b k ) − F λ + t ′ i η ( b ; b k ) (cid:12)(cid:12)(cid:12) s + X ( b − k ; b − ) ∈ B k,b (cid:12)(cid:12)(cid:12) F λ + t i η (ˆ b ; ˆ b − k ) − F λ + t ′ i η (ˆ b ; ˆ b − k ) (cid:12)(cid:12)(cid:12) s (cid:21) . By Remark C.3, we know that for λ ∈ I ∩ A c , the integrand converges to zero as η ↓ P -a.s. Moreover, it follows from Proposition C.7 that for any p >
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Email address : [email protected] Laboratoire J.A.Dieudonn´e, UMR CNRS-UNS 7351, Universit´e Cˆote d’Azur, 06108 Nice,France
Email address : [email protected] Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.
Email address : [email protected] Department of Mathematical Sciences, Loughborough University, Leicestershire, LE113TU, United Kingdom.
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