Absolutely Continuous Spectrum for random Schroedinger operators on the Fibonacci and similar tree-strips
AABSOLUTELY CONTINUOUS SPECTRUM FOR RANDOMSCHR ¨ODINGER OPERATORS ON THE FIBONACCI ANDSIMILAR TREE-STRIPS
CHRISTIAN SADEL
Abstract.
We will consider cross products of finite graphs with a class oftrees that have arbitrarily but finitely long line segments, such as the Fi-bonacci tree. Such cross products are called tree-strips. We prove that forsmall disorder random Schr¨odinger operators on such tree-strips have purelyabsolutely continuous spectrum in a certain set. Introduction
It will be most convenient to describe the trees considered in this work by asubstitution rule given by a substitution matrix ( S pq ) Lp,q =0 ∈ Z ( L +1) × ( L +1)+ withpositive integer entries. The trees are constructed starting from a root and thematrix gives the rule how to substitute a vertex by its children going to the nextgeneration. Here, ’generation’ describes the graph distance from the root, the’children’ of a vertex are all connected neighbors whose graph distance to the rootis increased by one, the other neighbor will be called ’parent’.The precise substitution rule is the following: Each vertex x of the tree has alabel l ( x ) ∈ { , . . . , L } , one starts with the root with a certain label, then eachvertex of label p has exactly S pq children of label q . The (isomorphy class of the)tree is then determined by the matrix S and the label of the root.We will consider the trees for the substitution matrices S K,L = [ S K,Lp,q ] Lp,q =0 with S K,L , = K , S K,Lp,p +1 = 1, S K,LL, = 1, and all other entries 0, where K ≥ L ≥ S K,L = K · · · , e.g. for L = 1 , S K, = (cid:18) K
11 0 (cid:19) . (1.1)We denote the tree with root label p ∈ { , . . . , L } and substitution matrix S K,L by T K,Lp and then define the forest T K,L to be the disjoint, disconnected union of the T K,Lp , i.e. T K,L = (cid:83) Lp =0 T K,Lp . Another way to think of the tree T K,L is to startwith the rooted Bethe lattice where each vertex has K + 1 children (i.e. the root Mathematics Subject Classification.
Primary 82B44, Secondary 47B80, 60H25.
Key words and phrases. random Schr¨odinger operators, Anderson model, Fibonacci tree, ex-tended states, absolutely continuous spectrum.This research was supported by NSERC Discovery grant 92997-2010 RGPIN and by the PeopleProgramme (Marie Curie Actions) of the EU 7th Framework Programme FP7/2007-2013, REAgrant 291734. a r X i v : . [ m a t h - ph ] S e p CHRISTIAN SADEL has K + 1 neighbors and any other vertex has K + 2 neighbors) and then for eachvertex one takes one of the K + 1 forward edges (going to the next generation) andputs L additional vertices on them (cf. Figure 1). Figure 1.
The tree T , , K = 2 , L = 3. The open circles arevertices of label 0, the full filled circles are vertices of label 1, eachvertex of label 1 is followed by one vertex of label 2 and one oflabel 3. These labels are indicated by different shadings of thecircles.The Fibonacci trees are the trees associated to the substitution matrix S , =( ), i.e. each vertex x of the tree has either the label l ( x ) = 0 or l ( x ) = 1, eachvertex with label 0 has a child with label 0 and one with label 1 and each child withlabel 1 has one child with label 0. So each vertex of label 1, except for possibly theroot, has 2 neighbors (one parent and 1 child), and each vertex of label 0, exceptfor possibly the root, has 3 neighbors (one parent and 2 children). (see Figure 2).These trees are called Fibonacci trees for the following reason. Let n ( T , j )denote the number of vertices in the n -th generation of the tree T , j , the root beingthe first generation. Moreover, let f n denote the n -th Fibonacci number startingwith f = 1 , f = 1. Then, one has n ( T , ) = f n and n ( T , ) = f n +1 .By d ( x, y ) we denote the graph distance of x, y ∈ T K,L , where d ( x, y ) = ∞ if x and y are elements of different connected components T K,Lp .A tree-strip is the cross product of a tree with a finite set I = { , . . . , m } . On (cid:96) ( T K,L × I ) ∼ = (cid:96) ( T K,L , C m ) = { u : T K,L → C m , (cid:80) x ∈ T K,L (cid:107) u ( x ) (cid:107) < ∞} whichis also canonically equivalent to (cid:76) Lp =0 (cid:96) ( T K,Lp , C m ) and (cid:96) ( T K,L ) ⊗ C m , we definethe random operators( H λ u )( x ) = (cid:88) y : d ( x,y )=1 u ( y ) + Au ( x ) + λV ( x ) u ( x ) . (1.2) .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 3 Figure 2.
Fibonacci tree T , . Vertices of label 1 are filled circlesand vertices of label 0 are non-filled circles. The trees T ,L looksimilar where each filled circle has to be replaced by a chain (line-segment) of L vertices.Here, A ∈ Sym( m ) represents the ’free vertical operator’ and the matrices V ( x ) ∈ Sym( m ) for x ∈ T K,L are independent identically distributed random variables,distributed according to some probability measure ν on Sym( m ) and scaled by thecoupling constant λ . Sym( m ) denotes the set of real symmetric m × m matrices.These operators might be either thought of to model one particle on the product T K,L × I or to model one particle on T K,L with internal degrees of freedom andrandom hopping between these internal degrees, described by A and V ( x ). Clearly, H λ = (cid:76) Lp =0 H ( p ) λ , where H ( p ) λ is the restriction of H λ to (cid:96) ( T K,Lp , C m ) and can beseen as random Schr¨odinger operator on the tree strip T K,Lp × I .If I = G is a finite graph, then T K,L × G can be interpreted as the productgraph where ( x, k ) , ( y, j ) ∈ T K,L × G are connected by an edge, if either x = y and k, j ∈ G are connected by an edge, or k = j and x, y ∈ T K,L are connectedby an edge. If A is chosen to be the adjacency matrix of G and V ( x ) diagonalwith i.i.d. entries, then H is the adjacency operator on this product graph and H λ corresponds to the Anderson model on this product graph.For the Anderson model on Z d or R d in any dimension d , Anderson localizationis proved at spectral edges and for high disorder [FS, FMSS, DLS, SW, CKM,DK, Kl2, AM, Aiz, Wa, Klo]. It is also known to hold for one dimensional [GMP,KuS, CKM] and quasi-one dimensional models like strips [Lac, KLS] and finitedimensional trees [Breu], except if a built in symmetry prevents localization ase.g. in [SS]. In dimensions d ≥ d = 2 one expects localization. These conjectures remain open problems. Theexistence of a.c. spectrum has only been proved for the Anderson model on trees andother tree-like graphs of infinite dimension with exponentially growing boundary[Kl3, Kl4, Kl6, ASW, FHS, FHS2, Hal, AW, KLW, KLW2, FHH, KS, Sad, Sha].This work adds some more examples to this list. It appears that the hyperbolicnature of such graphs leads to conservation of a.c. spectrum and ballistic dynamicalbehavior [Kl5, KS2, AW2] and these results should hold for much more generalhyperbolic graphs. Therefore, it may be worth it to further generalize the resultsand identify the technical problems occurring in this process. Also, a recent reviewemphasized the importance of trees of finite cone type [KLW3] and the trees T K,Lp belong to this class. If a tree has an assigned root 0 then the n -th generation is the CHRISTIAN SADEL set of vertices x with graph distance d ( x,
0) = n . The cone of descendants of x isthen defined as the set of vertices y , where the shortest path to the root goes through x , i.e. the set of y such that d (0 , y ) = d (0 , x ) + d ( x, y ). The phrase ’finite cone type’refers to the fact, that there are only finitely many different (isomorphy classes of)cones of descendants. Using the isomorphy class of the cone as label, each tree offinite cone type can be associated to a substitution matrix, and each tree associatedto a substitution matrix is a tree of finite cone type. For the case considered here,the trees T K,Lp for p ∈ { , . . . , L } are exactly the different isomorphy classes of conesof descendants.In my previous work [Sad] I already considered random Schr¨odinger operatorson tree-strips of finite cone type. However, none of these trees T K,Lp were coveredthere. One of the main assumptions needed in [Sad] was that every vertex hasat least 2 children which played a significant role at various places. This meansthe trees could not have any line segment, that is a vertex or chain of vertices notbeing the root which has only two neighbors, one parent and one child. The trees T K,Lp have line segments of length L . In some sense these are the simplest treeswith that property. The main argument in [Sad] is adapted from [Kl3, KS] anduses a fixed point equation and the Implicit Function Theorem performed in someBanach spaces that are associated to supersymmetric functions. As we will see,for the tree-strips considered here this technique still works, but there are quite afew technical subtleties that are pointed out in this work. The Implicit FunctionTheorem has to be applied in a slightly different Banach space ( H ∞ × H L insteadof H L +1 ∞ ). In general, H p , 1 ≤ p ≤ ∞ is in some sense the intersection of asupersymmetric L and L ∞ space. For the set up of the fixed point equation in[Sad] it was important to always have a product of at least two such functions whichthen is a L and L function, so that the Fourier transform is mapping them backto a L and L ∞ function. The line segments in T K,Lp lead to the fact that we donot have such products of functions so that a Fourier transform is just giving an L but not necessarily an L ∞ function. The change of this Banach spaces also leadsto adjustments in the inductive Proposition 6.3 and the final arguments giving acontinuous extension of the Green’s matrix to real energies which is given by certainintegrals (cf. (6.6) and (6.7)). For this it is important that the terms inside theintegral extend continuously in a supersymmetric L , something that can still beachieved here. In former works [KS, Sad] these terms even extended in H , which isnot anymore the case here. For the analysis of the Frechet derivative, compactnessof a certain operator is needed which also demands some additional work in this case(cf. Proposition C.1). It relies on the identities mentioned in Appendix B. Some ofthe different used arguments need a stronger assumption on the distribution of thematrix valued potential V ( x ), namely it has to be compactly supported. Anothernew aspect in this work is the use of the identity given in Proposition A.1 to obtainthat a certain Frechet derivative is invertible. This method would not work for allthe trees considered in [Sad] where an extremely technical described set of energieswhere the Frechet derivative is not invertible, had to be removed.For the Fibonacci tree one can explicitly calculate the spectrum for the adjacencyoperator (cf. Proposition 1.1), therefore the main theorem is less technical for thiscase.Moreover, in [Sad, Theorem 1.2] some set of energies had to be excluded to getthe almost sure a.c. spectrum. This set was given by a very technical condition .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 5 which was shown to remove a nowhere dense set in a certain case. In Lemma 5.2we show that this condition is never satisfied for the trees considered here, hencewe do not have to remove certain energies. The argument is based on an identitysatisfied by the Green’s functions and shown in Appendix A.Considering (1.2), let us remark that there is an orthogonal matrix O ∈ O( m )such that O (cid:62) AO is diagonal. Then ( ⊗ O ) is unitary and one obtains the equivalentfamily of operators[( ⊗ O ) ∗ H λ ( ⊗ O )] u ( x ) = (cid:88) y : d ( x,y )=1 u ( y ) + O (cid:62) AOu ( x ) + λO (cid:62) V ( x ) Ou ( x ) . Hence, without loss of generality, we can assume that A is a diagonal matrix andwe will do so in the proofs. In particular, the non-random operator H is unitarilyequivalent to a direct sum of shifted adjacency operators on T K,L , H = ∆ ⊗ + ⊗ A ∼ = (cid:76) mj =1 ∆ + a j on (cid:96) ( T K,L ) ⊗ C m where the a j are the eigenvalues of A and∆ describes the adjacency operator on (cid:96) ( T K,L ) given by(∆ v )( x ) = (cid:88) y : d ( x,y )=1 v ( y ) , v ∈ (cid:96) ( T K,L ) . (1.3)Our interest lies in the spectral type of H λ . In order to state the main theoremswe have to consider the adjacency operator first. From now on we have a fixed K, L and will omit these indices in many future defined quantities. The root of T K,Lp will be called 0 ( p ) . For x ∈ T K,L we let | x (cid:105) ∈ (cid:96) ( T K,L ) denote the elementgiven by | x (cid:105) ( y ) = δ x,y = (cid:40) x = y x (cid:54) = y . For some operator H , (cid:104) x | H | y (cid:105) denotes thescalar product between | x (cid:105) and H | y (cid:105) , where we use the physics convention that thescalar product is anti-linear in the first and linear in the second component. For p ∈ { , . . . , L } and Im( z ) > ( p ) z := (cid:104) ( p ) | (∆ − z ) − | ( p ) (cid:105) . (1.4)For E ∈ R we further define Γ ( p ) E by the limitΓ ( p ) E := lim η ↓ Γ ( p ) E + iη if the limit exists in C . (1.5)Let us define the following sets of energies E ∈ R , I K,Lp := { E ∈ R : Γ ( p ) E exists, and Im(Γ ( p ) E ) > } . (1.6)Furthermore, for d ∈ N let ∆ d denote the d × d adjacency matrix for the finite linewith d vertices and d − d = ∈ Mat( d × d, R ) with the convention ∆ = 0 . (1.7)We set E L := L (cid:91) d =1 σ (∆ d ) and I K,L := I K,L \ E L (1.8) CHRISTIAN SADEL where σ (∆ d ) denotes the set of eigenvalues of the matrix ∆ d and therefore, E L is afinite set. Proposition 1.1.
We have: (i)
For all p ∈ { , . . . , L } , I K,Lp = I K,L . I K,L and I K,L are non-empty unionsof finitely many open intervals and for all p , Γ ( p ) E depends analytically on E ∈ I K,L . (ii) The absolutely continuous spectrum of the adjacency operator ∆ on (cid:96) ( T K,L ) is given by the closure σ ac (∆) = I K,L = I K,L . (iii) For any fixed L and E there is a K such that for K > K the closure of I K,L includes the interval [ − E , E ] , i.e. [ − E , E ] ⊂ I K,L for all
K > K = K ( L, E ) . (1.9)(iv) For the Fibonacci trees ( K = L = 1 ) we have I , = (cid:18) − √ , (cid:19) ∪ (cid:18) , √ (cid:19) . (1.10)Letting a ≤ a ≤ . . . ≤ a m be the eigenvalues of A , one obtains that the a.c.spectrum of H is given by the union of the bands (cid:83) mj =1 ( I K,L + a j ). However,as in previous work using this method [KS, Sad] we have to restrict ourselves tothe intersections of such bands and we need to assume that this intersection is notempty. Therefore, define I K,LA := m (cid:92) j =1 I K,L + a j . (1.11)For the regular tree-strip, the technique with resonances does not have this weak-ness (cf. [Sha]) and in fact gives existence of a.c. spectrum in a set that correspondsto the (cid:96) spectrum of the free operator (intersected with the real line ). However,[Sha] needs a full random matrix potential V ( x ) as e.g. from the GOE ensem-ble and therefore does not handle the Anderson model on tree-strips. Extendingthis method to a non regular tree or tree-strip such as the Fibonacci tree wouldbe interesting in order to confirm that the (cid:96) spectrum of the adjacency operatordetermines the mobility edge for small λ . Besides we will also need to assume thatthe random potential is almost surely bounded: Assumptions.
The following assumptions turn out to be crucial for the results. (V)
The distribution ν of V ( x ) is compactly supported in Sym( m ) . (A) Assume that the eigenvalues of A are such that the set I K,LA is not emptyin which case it is a union of finitely many open intervals.By Proposition 1.1 (iii) for any fixed L and A there is a K such that for K > K , I K,LA is not empty, so the condition is fulfilled.In the Fibonacci case K = L = 1 this assumption reduces to a max − a min < √ , The (cid:96) spectrum of the adjacency operator is in general not a subset of the real line .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 7 where a max is the biggest and a min the smallest eigenvalue of A , and then I , A = (cid:18) − √ a max , √ a min (cid:19) \ { a , a , . . . , a m } . (1.12)In order to consider the spectrum of H λ we introduce the matrix-valued spectralmeasures at the vertices of the forest T K,L . For x ∈ T K,L , j ∈ I = { , . . . , m } let | x, j (cid:105) denote the element in (cid:96) ( T , C m ) satisfying | x, j (cid:105) ( y ) = δ x,y e j where e j is the j -th canonical basis vector in C m . Similar as before, (cid:104) x, j | H | y, k (cid:105) denotes the scalarproduct between | x, j (cid:105) and H | y, k (cid:105) with the convention that the scalar product islinear in the second and anti-linear in the first component. Then, for x ∈ T K,L wedefine the random, positive matrix valued measure µ x on R by (cid:90) f ( E ) dµ x ( E ) = [ (cid:104) x, j | f ( H λ ) | x, k (cid:105) ] j,k ∈I (1.13)for all compactly supported, continuous functions f on R . Theorem 1.2.
Let the assumptions (A) and (V) be satisfied. Moreover, for p (cid:54) = 0 let ( p )0 be the first vertex (smallest distance to root) with label , i.e. ( p )0 is theunique vertex with d (0 ( p ) , ( p )0 ) = L − p + 1 . Then, there is an open neighborhood U of { } × I K,LA in R such that the following holds: (i) The spectrum of H λ is almost surely purely absolutely continuous in U λ = { E : ( λ, E ) ∈ U } . (ii) For every x ∈ T K,L and any x ∈ T K,Lp with p = 1 , . . . , L and d ( x, ( p ) ) > L − p ,the density of the absolutely continuous average spectral measure E ( µ x ) in U λ depends continuously on ( λ, E ) ∈ U . The condition d ( x, ( p ) ) > L − p meansthat either x = 1 ( p )0 or that x is a descendant of ( p )0 in T K,Lp , p (cid:54) = 0 . (iii) The density of E ( µ (0) ) and E ( µ ( p )0 ) for p = 1 , . . . , L are positive definite in U λ . This implies that H λ and also all the parts H ( p ) λ on (cid:96) ( T K,Lp × I ) havespectrum in U λ with positive probability. Remark 1.3.
Except for the removal of E L , the set I K,LA here corresponds to I A,S = I A,S
K,L in [Sad] . The handling of the line segments requires to exclude thefinitely many energies in E L . In [Sad] we also needed to remove some more energiesdefined by a very technical condition to get some smaller set ˆ I A,S . This set occurredbecause the main argument is based on the Implicit Function Theorem and one needsa certain Frechet derivative to be invertible. Here, this Frechet derivative will alwaysbe invertible for all E ∈ I K,LA by Lemma 5.2 which relies on the identity given inProposition 2.1. This identity holds more general for the Green’s functions on treesof finite cone type as shown in Appendix A. Using this identity and essentially thesame line of arguments as in Lemma 5.2, one can actually show that I A,S = ˆ I A,S asdefined in [Sad] for any substitution matrix S ∈ Z k × k + that has at least one positiveentry on the diagonal and where the determinants of all diagonal minors of − S arenon-positive. Here a diagonal minor of a square matrix is a matrix obtained bydeleting finitely many rows and the same columns, i.e. one deletes the i , . . . , i l row and column to get a ( k − l ) × ( k − l ) matrix. The × diagonal minors areexactly the diagonal elements. For × substitution matrices this condition simplymeans that there is one positive diagonal element and the determinant is negative. CHRISTIAN SADEL
The important objects we work with are the matrix Green’s functions given by G [ x ] λ ( z ) := (cid:2) (cid:104) x, j | ( H − z ) − | x, k (cid:105) (cid:3) j,k ∈I ∈ C m × m (1.14)for Im( z ) >
0. The most important ingredient to obtain Theorem 1.2 is the follow-ing.
Theorem 1.4.
Under assumptions (V) and (A) there exists an open neighborhood U of { } × I K,LA in R such that for all vertices x ∈ T K,L and all x ∈ T K,Lp with d (0 ( p ) , x ) > L − p the functions ( λ, E, η ) (cid:55)→ E (cid:16) G [ x ] λ ( E + iη ) (cid:17) , ( λ, E, η ) (cid:55)→ E (cid:18)(cid:12)(cid:12)(cid:12) G [ x ] λ ( E + iη ) (cid:12)(cid:12)(cid:12) (cid:19) , defined for η > , have continuous extensions to U × [0 , ∞ ) . Let us show now that Theorem 1.4 implies Theorem 1.2.
Proof of Theorem 1.2.
Part (ii) follows immediately as E ( G [ x ] λ ( z )) is the Stieltjestransform of E ( µ x ) and hence E ( µ x )( dE ) = ∗ − lim η ↓ π Im E ( G λ,E + iη ) dE . Here ∗ − lim denotes a limit in the weak ∗ topology on bounded measures. For part (iii)we remark that for λ = 0 one can explicitly calculate the matrix Green’s functionsand see that for energies E ∈ I K,LA the limit of the imaginary parts are positivedefinite (cf. Remark 2.2). By continuity around λ = 0 this remains true for apossibly smaller neighborhood U ⊃ { } × I K,LA .To get (i) note that for any compact interval [ a, b ] ⊂ U λ one has by Fatou’slemma E (cid:16) lim inf Im( z ) ↓ (cid:90) ba Tr (cid:0)(cid:12)(cid:12) G [ x ] λ ( z ) (cid:12)(cid:12) (cid:1) dE (cid:17) ≤ lim inf Im( z ) ↓ (cid:90) ba E (cid:16) Tr (cid:0)(cid:12)(cid:12) G [ x ] λ ( z ) (cid:12)(cid:12) (cid:1)(cid:17) dE < ∞ Thus, lim inf
Im( z ) ↓ (cid:90) ba Tr (cid:18)(cid:12)(cid:12)(cid:12) G [ x ] λ ( z ) (cid:12)(cid:12)(cid:12) (cid:19) dE < ∞ with probability one . (1.15)As G [ x ] λ ( z ) is the Stieltjes transform of µ x this implies that, almost surely, µ x isabsolutely continuous with respect to the Lebesgue measure in ( a, b ) ⊂ U λ and thedensity is a positive matrix valued L function for all x ∈ T K,L and all x ∈ T K,Lp with d (0 ( p ) , x ) > L − p . (cf. [Kl6, Theorem 4.1] and [Kel, Theorem 2.6]). This givesthe almost sure pure a.c. spectrum on any interval ( a, b ) with closure in U λ as thecyclic spaces of the vectors | x, j (cid:105) for those x span (cid:96) ( T K,L × I ). Writing U λ as acountable union of such closed compact intervals [ a, b ] one realizes by taking theintersection of the corresponding sets in the probability space that with probabilityone the spectrum of H λ is purely absolutely continuous in U λ . (cid:3) The paper is structured as follows. In Section 2 the Green’s functions Γ ( p ) z areconsidered in more detail and Proposition 1.1 is proved. In Section 3 we consider therecursion of the forward Green’s matrices that lead to a fixed point equation. Then,in Section 4, we introduce the important Banach spaces in which this fixed pointequation has to be analyzed and in Section 5 we investigate the Frechet derivative. .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 9 Finally, in Section 6 we conclude to obtain Theorem 1.4. Appendices A, B and Cstate some general facts that are used along the way.2.
The unperturbed Green’s functions
The main goal of this section is to prove Proposition 1.1. Some of the statementsare consequences of more general considerations as done in [Kel, KLW].Recall that we denoted the Green’s functions at the roots 0 p by Γ ( p ) z for Im( z ) > z = E ∈ I K,Lp it also denotes the limit for Im( z ) ↓ ν ( p ) by (cid:90) f ( x ) dν ( p ) ( x ) = (cid:104) ( p ) | f (∆) | ( p ) (cid:105) . (2.1)As z (cid:55)→ Γ ( p ) z is the Stieltjes transform of the measure ν ( p ) the absolutely continuouspart is the closure of I K,Lp . Now for any vertex x ∈ T K,Lp we can take the path tothe root and cut off the trees connected to this path, which are all equivalent to oneof the T K,Lq . Then, using an induction argument (induction over distance to root)as done in [Kel, Prop. 2.9] or alternatively using [FLSSS, Lemma 2.2] one finds σ ac (∆) = L (cid:91) p =0 I K,Lp . (2.2)Note that when cutting the connections to the root in T K,Lp then for 0 < p < L we get T K,Lp +1 , for p = L we get T K,L and for p = 0 we get the union of K times T K,L and one T K,L . Therefore, the standard recursion relations for the Green’s functionsthat can be obtained from the resolvent identity (cf. [ASW, FHS, Kl3, KLW, Kel])are given byΓ (0) z = − z + K Γ (0) z + Γ (1) z , Γ ( p ) z = − z + Γ ( p +1) z , Γ ( L ) z = − z + Γ (0) z , (2.3)where p = 1 , . . . , L −
1. By an hyperbolic contraction argument as in [KLW, Kel]one obtains that for Im( z ) >
0, these equations combined with the restrictionIm(Γ ( p ) z ) > , p = 0 , , . . . , L determine the Green’s functions uniquely.The right hand sides of the last two equations in (2.3) can be seen as M¨obiusactions (cid:0) − z (cid:1) · Γ = − z . Defining the polynomials c ( z ) = 0 , c ( z ) = 1 anditeratively c k +1 ( z ) = zc k ( z ) − c k − ( z ) one obtains by induction (cid:18) − z (cid:19) k = (cid:18) − c k − ( z ) − c k ( z ) c k ( z ) c k +1 ( z ) (cid:19) , and (2.4) c k ( z ) = c + ( z ) k − c − ( z ) k c + ( z ) − c − ( z ) where c ± ( z ) = 12 (cid:16) z ± (cid:112) z − (cid:17) . (2.5)Here one can choose any of the two roots for √ z −
4, changing the root switches c + ( z ) and c − ( z ) but does not change c k ( z ). As the determinant of these matricesare always one, we also find c k ( z ) − c k − ( z ) c k +1 ( z ) = 1 . (2.6)It can be seen from the recursion relations defining c k ( z ) that c k ( z ) is a polynomialof degree k − k is odd and odd if k is even. Now, (2.3) and (2.4) lead toΓ (0) z (cid:20) z + K Γ (0) z + (cid:18) − c L − ( z ) − c L ( z ) c L ( z ) c L +1 ( z ) (cid:19) · Γ (0) z (cid:21) + 1 = 0 (2.7)which using zc L − c L − = c L +1 can be rewritten as Kc L ( z )(Γ (0) z ) + ( K + 1) c L +1 ( z ) (Γ (0) z ) + zc L +1 ( z ) Γ (0) z + c L +1 ( z ) = 0 (2.8)Using the uniqueness of solutions in the upper half plane we see that I K,L is givenby the set of real energies E such that the cubic equation with real coefficients Kc L ( E ) x + ( K + 1) c L +1 ( E ) x + Ec L +1 ( E ) x + c L +1 ( E ) = 0 (2.9)has a solution x with a positive imaginary part and Γ (0) E is that solution.With this characterization we have now established everything to prove Propo-sition 1.1. Proof of Proposition 1.1.
Using equations (2.3) in the limits Im( z ) → I K,L ⊂ I K,L ⊂ · · · ⊂ I K,LL ⊂ I K,L , hence I K,Lp = I K,L . (2.10)For the inclusion I K,L ⊂ I K,L note that the first equation of (2.3) gives Γ (0) z = K (cid:18) − Γ (1) z − z + (cid:113) (Γ (1) z + z ) − K (cid:19) where we have to take the square root withthe positive imaginary part (the other one will have a negative imaginary part).Hence for E ∈ I K,L we have a limit for Γ (0) E + iη for η ↓ (1) E ) > (0) E ) > E ∈ I K,L .By the characterization of I K,L as in (2.9) one sees that I K,L is in fact the setof energies E where the discriminant D = D ( E ) of (2.9) is negative, D = c L +1 (cid:16) K ( K + 1) Ec L c L +1 + ( K + 1) E c L +1 − KE c L c L +1 − K c L − K + 1) c L +1 (cid:17) . (2.11)If c L +1 ( E ) = 0 then the term inside the parenthesis for D is negative coming fromthe term − K c L ( E ), using (2.6), c L ( E ) (cid:54) = 0. By induction one also gets that c L +1 ( E ) is the characteristic polynomial det( E − ∆ L ) for the matrix ∆ L as in(1.7), so it has real roots and in a neighborhood of these roots, D <
0. Therefore, I K,L is not empty. Moreover, as D ( E ) is a polynomial in E , I K,L and I K,L \ E L are unions of finitely many open intervals. As there is a solution formula for cubicequations we also find that Γ (0) E and by (2.3) all Γ ( p ) E are analytic in E ∈ I K,L . Thisfinishes part (i).As I K,L has only finitely many points removed from I K,L , we have I K,L = I K,L = L (cid:91) p =0 I K,Lp = σ ac (∆) (2.12)giving part (ii).To get part (iii) note that we find an open neighborhood O of these zeros, suchthat for fixed L , all E ∈ O and all K > K ( K + 1) Ec L c L +1 + ( K + 1) E c L +1 − KE c L c L +1 − K c L < . (2.13) .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 11 In particular, if E ∈ O and c L +1 ( E ) (cid:54) = 0 then for all K we find D ( E ) <
0. In thecompact set [ − E , E ] \ O , c L +1 ( E ) attains a minimum value bigger than zero and c L and c L +1 are bounded. The highest power in K appearing inside the parenthesison the right hand side of (2.11) is the negative term − K +1) c L +1 . Therefore, wefind K = K ( L, E ), such that D ( E ) < E ∈ [ − E , E ] \ O and K > K .In this case we obtain [ − E , E ] ⊂ I K,L finishing part (iii).For part (iv) note that for K = L = 1 using c ( E ) = 1 , c ( E ) = E we find D = 4 E − E = E (4 E −
27) and E = { } giving I , = I , = (cid:32) − √
32 ; 3 √ (cid:33) \ { } . (2.14) (cid:3) For our further analysis we will also need the following property.
Proposition 2.1.
For E ∈ I K,L one has K (cid:12)(cid:12)(cid:12) Γ (0) E (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Γ (0) E Γ (1) E · · · Γ ( L ) E (cid:12)(cid:12)(cid:12) = 1 . (2.15) Proof.
Equation (2.15) can be rewritten asdet − (cid:12)(cid:12)(cid:12) Γ (0) E (cid:12)(cid:12)(cid:12) . . . (cid:12)(cid:12)(cid:12) Γ ( L ) E (cid:12)(cid:12)(cid:12) S K,L = 0 (2.16)which is an identity shown for the more general case of trees of finite cone type inProposition A.1 in Appendix A. (cid:3)
Remark 2.2. As I K,LA = (cid:84) mj =1 ( I K,L + a j ) we get for E ∈ I K,LA that E − a j ∈ I K,L for all eigenvalues a j of A . In particular Γ ( p ) E − a j exists and Im(Γ ( p ) E − a j ) > . Recursion for matrix Green’s functions
A key identity for the analysis of Schr¨odinger operators on trees and tree-stripsare the well known identities like (2.3) for forward matrix Green’s functions thatcan be obtained from the resolvent identity. Let T x be some tree that has a vertex x and let N ( x ) be the set of neighboring points in T x . If H λ is given as in (1.2) on (cid:96) ( T x × I ) with I = { , . . . , m } then the Green’s matrix ( G x,λ,z ) jk = (cid:104) x, j | ( H λ − z ) − | x, k (cid:105) satisfies G x,λ,z = − (cid:88) y ∈N ( x ) G ( y | x ) λ,z + z − A − λV ( x ) − . (3.1)Here and in many equations below the upper index ( y | x ) indicates that we lookat the vertex y and remove the branch of the tree that is emanating from y andgoing through x . This means we let T ( y | x ) x denote the tree with vertex y wherethe branch going from y to x is removed (i.e. the tree of vertices x (cid:48) satisfying d ( x, x (cid:48) ) = d ( x, y ) + d ( y, x (cid:48) ) ). Furthermore, H ( y | x ) λ is the operator H λ restricted to (cid:96) ( T ( y | x ) x × I ) with Dirichlet boundary conditions and ( G ( y | x ) λ,z ) jk = (cid:104) y, j | ( H ( y | x ) λ − z ) − | y, k (cid:105) . This equation is valid in any tree, in particular it is valid in subtreeswhen certain branches were cut.Now if a tree (cid:101) T has an assigned root 0 ∈ (cid:101) T and we set T x := (cid:101) T ( x | to be thetree where we disconnect the branch at x going through the root then for y ∈ T x one finds T ( y | x ) x = T y . The corresponding Green’s matrices are G x,λ,z = G ( x | λ,z and G ( y | x ) λ,z = G ( y | λ,z for y (cid:54) = x, y ∈ T x and they only depend on the matrix potential onthe branches at x or y , respectively, that go away from the root. Therefore we callthem forward matrix Green’s functions. Then, (3.1) becomes G ( x | λ,z = − (cid:88) y ∈N ( x ) G ( y | λ,z + z − A − λV ( x ) − (3.2)where N ( x ) is the set of neighbors of x in the tree T x = (cid:101) T ( x | , i.e. the set offorward neighbors (or children).In the case studied in this paper we consider the trees ( T K,Lp ) x = ( T K,Lp ) ( x | ( p ) ) for x ∈ T K,Lp . Now if l ( x ) is the label of x , then, ( T K,Lp ) x ∼ = T K,Lp and the distribu-tion of G ( x | ( p ) ) λ,z (for x ∈ T K,Lp ) only depends on λ, z and the label l ( x ). Moreover,the different G ( y | ( p ) ) λ,z occurring on the right hand side are independent and theyare independent of V ( x ). Therefore one might want to work with some averagedquantities. However, the occurring inverse on the right hand side of (3.2) preventsone from getting something useful by just applying the expectation to this equation.The key idea is now to represent the operation G (cid:55)→ − G − as a linear operatorin some function space. More precisely, as in [KS, KS2, Sad] we associate to sym-metric m × m matrices G with positive imaginary part the following functions: LetSym + ( m ) denote the real, positive semi-definite matrices, i.e. M ∈ Sym + ( m ) ⇔ M ∈ Sym( m ) , M ≥
0, and for Re( G ) ∈ Sym( m ) , Im( G ) ∈ Sym + ( m ) we define thebounded functions ζ G : Sym + ( m ) → C , ζ G ( M ) = e i Tr( GM ) (3.3) ξ G : (Sym + ( m )) → C , ξ G ( M + , M − ) = ζ G ( M + ) ζ G ( M − ) = e i Tr( GM + ) − i Tr( GM − ) (3.4)Note that if Im( G ) > ζ G , ξ G and all its derivatives are exponentially de-caying. A supersymmetric Fourier transform T and T = T ⊗ T as defined in[KS, KS2, Sad] gives for Im( G ) > T ζ G = ζ − G − and T ξ G = ξ − G − . For theconvenience of the reader these operators and all important Banach spaces as usedin previous works will be defined precisely in the next section.Using these linear operators, (3.2) can be rewritten as ζ G ( x | λ,z = T (cid:16) ζ z − A − λV ( x ) (cid:89) y ∈N ( x ) ζ G ( y | λ,z (cid:17) , ξ G ( x | λ,z = T (cid:16) ξ z − A − λV ( x ) (cid:89) y ∈N ( x ) ξ G ( y | λ,z (cid:17) . (3.5)For p ∈ { , , . . . , L } we take some x ∈ T K,Lq with label l ( x ) = p let 0 = 0 ( q ) anddefine ζ ( p ) λ,z := E ( ζ G ( x | ) λ,z ) , ξ ( p ) λ,z := E ( ζ G ( x | λ,z ) . (3.6) .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 13 Then, taking expectations in (3.5) gives for Im( z ) >
0, 1 ≤ p ≤ L − ζ (0) λ,z = T B λ,z (cid:16) ( ζ (0) λ,z ) K ζ (1) λ,z (cid:17) , ζ ( p ) λ,z = T B λ,z ζ ( p +1) λ,z , ζ ( L ) λ,z = T B λ,z ζ (0) λ,z (3.7) ξ (0) λ,z = T B λ,z (cid:16) ( ξ (0) λ,z ) K ξ (1) λ,z (cid:17) , ξ ( p ) λ,z = T B λ,z ξ ( p +1) λ,z , ξ ( L ) λ,z = T B λ,z ξ (0) λ,z (3.8)where B λ,z or B λ,z are the multiplication operators defined by B λ,z f ( M ) = E ( ζ z − A − λV ( x ) ( M )) f ( M ) (3.9) B λ,z g ( M + , M − ) = E ( ξ z − A − λV ( x ) ( M + , M − )) g ( M + , M − ) . (3.10)Recall that we assume without loss of generality that A = diag( a , . . . , a m ) isdiagonal. Therefore, in the free case λ = 0 one obtains ζ ( p )0 ,z = ζ A ( p ) z , ξ ( p )0 ,z = ξ A ( p ) z , where A ( p ) z := diag(Γ ( p ) z − a , Γ ( p ) z − a , . . . , Γ ( p ) z − a m ) . (3.11)For E ∈ I K,LA the point-wise limits ζ ( p )0 ,E := lim η ↓ ζ ( p )0 ,E + iη = ζ A ( p ) E , ξ ( p )0 ,E := lim η ↓ ξ ( p )0 ,E + iη = ξ A ( p ) E (3.12)exist for p ∈ { , . . . , L } , where A ( p ) E = lim η ↓ A ( p ) E + iη , [ A ( p ) E ] jk = δ jk Γ ( p ) E − a j (3.13)are diagonal m × m matrices with strictly positive imaginary part. The impor-tant point is to understand equations (3.7) and (3.8) as fixed point equations inappropriate Banach spaces.4. The proper Banach spaces
Let us first briefly introduce the important Banach spaces as in [KS, KS2, Sad]and for the readers convenience we will list all important notation and definitionsfrom previous works in the next definition. In particular we will also give the precisedefinitions of the operators T and T mentioned above. All proofs and arguments areomitted as [Sad, Section 3] uses the exact same notations and has the statementsin more detail. We will also skip to mention the connection to supersymmetryand how these spaces naturally evolve from this formalism. For a supersymmetricbackground see [Sad, Appendix B] or [KS]. As above we set I = { , . . . , m } . Definition 4.1. (a) P ( I ) = { a : a ⊂ I} denotes the set of all subsets of I (b) P denotes the set of pairs (¯ a, a ) of subsets of I with the same cardinality, P := { (¯ a, a ) : ¯ a, a ⊂ I , | ¯ a | = | a |} (c) n = n ( m ) will be the smallest integer such that n ≥ m (d) With ¯ a = (¯ a , . . . , ¯ a n ) , a = ( a , . . . , a n ) ∈ ( P ( I )) n define P n := { (¯ a , a ) ∈ ( P ( I )) n × ( P ( I )) n : (¯ a l , a l ) ∈ P} . (4.1) We also let | a | := | a | + | a | + . . . + | a n | and a c := ( a c , . . . , a cn ) with a cj = I \ a j . (e) For functions on (Sym + ( m )) let ∂ j,k denote the derivative with respect to the j, k - entry of M , ∂ j,k f ( M ) = ∂∂M j,k f ( M ) (by symmetry, ∂ j,k = ∂ k,j ) and let ˜ ∂ j,k = ∂ j,k for j (cid:54) = k and ˜ ∂ j,j = ∂ j,j . (f) For (¯ a, a ) ∈ P with a = { k , . . . , k c } , k < k . . . < k c ; ¯ a = { ¯ k , . . . , ¯ k c } , ¯ k < ¯ k < . . . < ¯ k c , define ∂ ¯ a,a := ˜ ∂ ¯ k ,k · · · ˜ ∂ ¯ k ,k c ... . . . ... ˜ ∂ ¯ k c ,k · · · ˜ ∂ ¯ k c ,k c , D ¯ a,a := det( ∂ ¯ a,a ) (4.2) with the convention that D ∅ , ∅ is the identity operator. (g) For (¯ a , a ) ∈ P n , ¯ a = (¯ a , . . . , ¯ a ) , a = ( a , . . . , a n ) , let D ¯ a , a := (cid:81) n(cid:96) =1 D ¯ a (cid:96) ,a (cid:96) .There is a function sgn(¯ a , a , ¯ b , b , ¯ b (cid:48) , b (cid:48) ) ∈ {− , , } such that D ¯ a , a ( f g ) = (cid:88) (¯ b , b ) , (¯ b (cid:48) , b (cid:48) ) ∈P n sgn(¯ a , a , ¯ b , b , ¯ b (cid:48) , b (cid:48) )( D ¯ b , b g ) ( D ¯ b (cid:48) , b (cid:48) f ) . (4.3)(h) For f ∈ C ∞ (Sym + ( m )) , g ∈ C ∞ ((Sym + ( m )) ) and p ≥ we introduce thenorms ||| f ||| p and |||| g |||| p by ||| f ||| p := (cid:88) (¯ a , a ) ∈P n | a | (cid:20)(cid:90) R m × n (cid:12)(cid:12) D ¯ a , a f ( ϕϕ (cid:62) ) (cid:12)(cid:12) p d mn ϕ (cid:21) /p (4.4) where ϕ denotes a m × n matrix, ϕ ∈ R m × n , and |||| g |||| p := (cid:88) (¯ a , a ) , (¯ b , b ) ∈P n | a | + | b | (4.5) (cid:34)(cid:90) ( R m × n ) (cid:12)(cid:12)(cid:12) D (+)¯ a , a D ( − )¯ b , b g ( ϕ + ϕ (cid:62) + , ϕ − ϕ (cid:62)− ) (cid:12)(cid:12)(cid:12) p d mn ϕ + d mn ϕ − (cid:35) /p . (4.6) Here, D ( ± )¯ a , a denotes the operator D ¯ a , a with respect to the entry M ± = ϕ ± ϕ (cid:62)± .Note that the map R m × n (cid:51) ϕ (cid:55)→ ϕϕ (cid:62) ∈ Sym + ( m ) is surjective as n ≥ m .We also define the corresponding norms ||| f ||| ∞ and |||| g |||| ∞ using the limit p → ∞ which are given by the sums of the corresponding suprema. (i) Let
Sym C ( m ) denote the complex symmetric, m × m matrices and for B ∈ Sym C ( m ) with strictly positive imaginary part (i.e., Im B > ), let PE ( B ) denote the vector space spanned by functions of the form f ( M ) = p ( M ) ζ B ( M ) ,where p ( M ) is a polynomial in the entries of M ∈ Sym + ( m ) . Clearly, due tothe exponential decay in ζ B one has ||| f ||| p < ∞ for all p ∈ [1 , ∞ ] . (j) Define PE ( m ) as the smallest vector space containing all vector spaces PE ( B ) for all B ∈ Sym C ( m ) with Im( B ) > . (k) For ≤ p ≤ ∞ , let L p be the completion of PE ( m ) with respect to the norm ||| f ||| p . We furthermore set H = L , define ||| f ||| ,p := ||| f ||| + ||| f ||| p and let H p bethe completion of PE ( m ) w.r.t. to ||| · ||| ,p . (l) For ≤ p ≤ ∞ let (cid:98) L p be the completion of PE ( m ) ⊗ PE ( m ) w.r.t. |||| g |||| p ,set K = (cid:98) L and |||| g |||| ,p := |||| g |||| + |||| g |||| p , and let K p be the completion of PE ( m ) ⊗ PE ( m ) w.r.t. |||| g |||| ,p . In fact, K = H ⊗ H as Hilbert space tensorproduct. (m)
Let I ∈ P ( I ) n be given by I = ( I , I , . . . , I ) . (n) The supersymmetric Fourier transform T is given by T f ( ϕ (cid:48) ϕ (cid:48)(cid:62) ) := ( − mn π mn (cid:90) e ± i Tr( ϕ (cid:48) ϕ (cid:62) ) D I , I f ( ϕϕ (cid:62) ) d mn ϕ . (4.7) .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 15 Note that the sign of ± does not matter by the symmetry ϕ → − ϕ . T = T ⊗ T is given by T g ( ϕ (cid:48) + ϕ (cid:48)(cid:62) + , ϕ (cid:48)− ϕ (cid:48)(cid:62)− ) (4.8)= 1 π mn (cid:90) e ± i Tr( ϕ (cid:48) + ϕ (cid:62) + ± ϕ (cid:48)− ϕ (cid:62)− ) D + I , I D − I , I g ( ϕ + ϕ (cid:62) + , ϕ − ϕ (cid:62)− ) d mn ϕ + d mn ϕ − where D ± I , I denotes the operator D I , I with respect to the entry M ± = ϕ ± ϕ (cid:62)± . Roughly speaking, L p , (cid:98) L p are the supersymmetric analogs of L p spaces, H p and K p resemble L ∩ L p and the operators T and T have the role of the Fouriertransform. One might think that H p = L ∩ L p as a set which is equivalent tosaying that PE ( m ) is dense w.r.t. the ||| · ||| ,p norm in L ∩ L p . Unfortunately,we were not able to prove this. The spaces L p , (cid:98) L p have not been used in formerpapers, but some technical difficulties in this paper requires the use of L ∞ and L in Section 6 for proving Theorem 1.4.One finds (cf. [Sad, eq. (B.22)], [KS, eq. (2.37)]) D ¯ a , a ( T f ) = mn | a | sgn( a , ¯ a ) F ( D a c , ¯ a c f ) for all (¯ a , a ) ∈ P n , (4.9)where F f ( ϕϕ (cid:62) ) denotes the Fourier transform w.r.t. to ϕ in R m × n ∼ = R mn andsgn( a , ¯ a ) ∈ {− , } is some sign.Using the fact that the Fourier transform F f ( ϕϕ (cid:62) ) maps PE ( m ) to PE ( m ) andcontinuously L ( d ϕ ) to L ∞ ( d ϕ ) and L ( d ϕ ) to L ( d ϕ ) we find the following (cf.[Sad, Lemma 2.6, Lemma 3.3]). Lemma 4.2. (i) H is a Hilbert space and T extends to a unitary operator on H . It also definesa bounded linear operator from H to H ∞ and from L to L ∞ . (ii) K is a Hilbert space and T is unitary on K . It also defines a bounded linearoperator from K to K ∞ and (cid:98) L to (cid:98) L ∞ . (iii) For any B ∈ Sym C ( m ) , Im( B ) > , p ∈ [1 , ∞ ] we find that PE ( B ) is densein H and H p . (iv) For any
B, C ∈ Sym C ( m ) , Im( B ) > , Im( C ) > , p ∈ [1 , ∞ ] we find that PE ( B ) ⊗ PE ( C ) is dense in K and K p . From (4.9) and [Sad, equ. (B.26)] one obtains T = id , T = id , T ζ G = ζ − G − , T ξ G = ξ − G − . (4.10)We need to get back to G and | G | from the functions ζ G and ξ G . Therefore wedefine D := (cid:16) ( − mn + j + k π mn D n − I , I D I\{ k } , I\{ j } (cid:17) j,k ∈I (4.11)which is a m × m matrix of differential operators. Then for G ∈ Sym C ( m ) , Im( G ) > G − = i (cid:90) D ζ G ( ϕϕ (cid:62) ) d mn ϕ , G = − i (cid:90) D T ζ G ( ϕϕ (cid:62) ) d mn ϕ . (4.12)As ξ G ( M + , M − ) = ζ G ( M + ) ζ G ( M − ), Fubini leads to | G | = G ∗ G = (cid:90) D ( − ) D (+) T ξ G ( ϕ + ϕ (cid:62) + , ϕ − ϕ (cid:62)− ) d mn ϕ + d mn ϕ − (4.13)where D ( ± ) is the operator D with respect to the entries M ± = ϕ ± ϕ (cid:62)± . Equation (4.12) can either be obtained using a super symmetric formalism or byrealizing that D I , I ζ G = ( i ) m det( G ) ζ G and ( − j + k D I\{ k } , I\{ j } ζ G = ( i ) m − G j,k ,where G j,k is the cofactor of the j, k element of G . Moreover, using the Gaussianintegral identity as in (B.1), one has (cid:90) ζ G ( ϕϕ (cid:62) ) d mn ϕ = (2 π ) mn ( − i ) mn (det( G )) n . (4.14)Combining all these facts with Kramer’s rule, ( G − ) jk = G j,k det( G ) , one obtains thefirst equation in (4.12). The second one follows from the first one and T ζ G = ζ − G − .Now we can turn back to the analysis of the recursion equations (3.7) and (3.8).For simplified notations, let us introduce (cid:126)ζ λ,z = ζ (0) λ,z ... ζ ( L ) λ,z and (cid:126)ξ λ,z = ξ (0) λ,z ... ξ ( L ) λ,z . (4.15)Using H¨older’s inequality, Dominated Convergence and the exponential decay ofthe functions ζ G , ξ G for Im G > (cid:126)ζ ,E and (cid:126)ξ ,E for E ∈ I K,LA , oneobtains the following completely analogue to [Sad, Proposition 4.1 and 4.2].
Proposition 4.3.
We have: (i)
For η = Im z ≥ the operator B λ,z is a bounded operator on H and H . Themap F : ( λ, E, η, f , f , . . . , f L ) (cid:55)→ T B λ,E + iη (( f K f ) , f , f , . . . , f L , f ) (4.16) is a continuous map from R × R × [0 , ∞ ) × H ∞ × H L to H ∞ × H L . Here T B λ,z applied to a vector of functions means that we apply it to every function,
T B λ,z (cid:16) (cid:101) f , . . . , (cid:101) f L (cid:17) = (cid:16) T B λ,z (cid:101) f , . . . , T B λ,z (cid:101) f L (cid:17) .Analogously, for η = Im z ≥ the operator B λ,z is a bounded operator on K and K and the map Q : ( λ, E, η, g , g , . . . , g L ) (cid:55)→ T B λ,E + iη (( g K g ) , g , . . . , g L , g ) (4.17) is a continuous map from R × R × [0 , ∞ ) × K ∞ × K L to K ∞ × K L . (ii) (cid:126)ζ λ,z ∈ H L +1 ∞ ⊂ H ∞ × H L , (cid:126)ξ λ,z ∈ K L +1 ∞ ⊂ K ∞ × K L for all λ ∈ R and z = E + iη with η > . The maps ( λ, E, η ) → (cid:126)ζ λ,E + iη and ( λ, E, η ) → (cid:126)ξ λ,E + iη are continuous from R × R × (0 , ∞ ) to H ∞ ×H L and to K ∞ ×K L , respectively. (iii) If E ∈ I K,LA , then (cid:126)ζ ,E ∈ ( PE ( m )) L +1 ⊂ H ∞ × H L , (cid:126)ξ ,E ∈ K ∞ × K L and lim η ↓ (cid:126)ζ ,E + iη = (cid:126)ζ ,E in H ∞ × H L , lim η ↓ (cid:126)ξ ,E + iη = (cid:126)ξ ,E in K ∞ × K L . (4.18)(iv) The equalities (3.7) and (3.8) can be rewritten as fixed point equations in
H × H ∞ and K × K ∞ , respectively, (cid:126)ζ λ,z = F ( λ, E, η, (cid:126)ζ λ,z ) , (cid:126)ξ λ,z = Q ( λ, E, η, (cid:126)ξ λ,z ) , (4.19) valid for all λ ∈ R and z = E + iη with η > , and also valid for λ = 0 and z = E with E ∈ I K,LA . .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 17 Remark 4.4.
One of the differences between this work and [Sad] is that for fixed ( λ, E, η ) the maps F and Q are operators on H ∞ × H L and K ∞ × K L , respectively,but not on H L +1 ∞ or K L +1 ∞ . We can not use the space H L +1 ∞ as for η = 0 and f p ∈ H ∞ we can not show that B λ,E f p is in H so that T would map it back to H ∞ . We can only say that one lands in H after applying T . This in turn comesfrom the fact that there is only one factor and not a product of more than onefunction after B λ,z in the last L entries which resembles the fact that vertices oflabel , . . . , L do have only one child in T K,L . Recall that in [Sad] every vertex hadto have at least two children.The other difference is that we do not need the smaller spaces H (0) ∞ , K (0) ∞ introducedin [Sad] to avoid some further assumption (but one could work in the spaces H (0) ∞ × ( H (0) ) L , K (0) ∞ × ( K (0) ) L if one wanted to). Spectrum of Frechet derivatives
It is now time to introduce some more notations to properly describe the spec-trum of the Frechet derivatives. These notations were also used in [Sad]. By∆( m, Z + ) we denote the set of upper triangular matrices with non-negative integerentries. For J ∈ ∆( m, Z + ) and E ∈ I K,LA we define θ ( p ) J,E := (cid:89) j,k ∈{ ,...,m } j ≤ k (cid:104) Γ ( p ) E − a j Γ ( q ) E − a k (cid:105) J jk ∈ C , p = 0 , , . . . , L , and (5.1) θ J,E := diag( θ (0) J,E , . . . , θ ( L ) J,E ) = θ (0) J,E . . . θ ( L ) J,E (5.2)With the help of these matrices we will express the spectrum of the importantFrechet derivatives. The following lemma corresponds to [Sad, Lemma 5.1 and 5.2].
Lemma 5.1.
We have: (i)
The map F as in (4.16) is continuous and Frechet-differentiable w.r.t. (cid:126)f =( f , f , . . . , f L ) ∈ H ∞ × H L . For (cid:126)f ∈ H L +1 ∞ the Frechet derivative F (cid:126)f extendsnaturally to a bounded operator on H L +1 which we will also denote as F (cid:126)f .Similarly, the map Q is Frechet-differentiable w.r.t. (cid:126)g ∈ K ∞ × K L . Thederivative Q (cid:126)g is a bounded linear operator on K ∞ × K L and for (cid:126)g ∈ K L +1 ∞ itextends naturally to a bounded operator on K L . (ii) For E ∈ I K,LA let C E = F (cid:126)f (0 , E, , (cid:126)ζ ,E ) and C E = Q (cid:126)g (0 , E, , (cid:126)ξ ,E ) . Then C L +1) E is a compact operator on H ∞ ×H L and H L +1 and C L +1) E is a compactoperator on K ∞ × K L and K L +1 . (iii) The spectrum of C E as an operator on the Hilbert space H L +1 is given by theeigenvalues of the matrices θ J,E S K,L for J ∈ ∆( m, Z + ) and the accumulationpoint . Thus, denoting the spectrum of C E on H L +1 by σ H ( C E ) one obtains σ H ( C E ) = (cid:91) J ∈ ∆( m, Z + ) σ ( θ J,E S K,L ) ∪ { } . (5.3) Similarly, denoting the spectrum of C E on K L +1 by σ K ( C E ) one finds σ K ( C E ) = (cid:91) J,J (cid:48) ∈ ∆( m, Z + ) σ ( θ J,E θ ∗ J (cid:48) ,E S K,L ) ∪ { } . (5.4) Here θ J,E are the matrices as defined in (5.2) . (iv) The spectra of C E and C E as operators on H ∞ ×H L and K ∞ ×K L , respectively,(denoted by σ H ∞ ×H L ( C E ) and σ K ∞ ×K L ( C E ) ), are the same as their spectraas operators on H L +1 and K L +1 , respectively, σ H ∞ ×H L ( C E ) = σ H ( C E ) , σ K ∞ ×K L ( C E ) = σ K ( C E ) . (5.5) Proof.
For the proof we will mostly just consider the function F and operator C E .The corresponding statements for Q and C E are proved analogously.(i) The derivative F (cid:126)f can be written as a ( L + 1) × ( L + 1) matrix of operatorsand we obtain formally F (cid:126)f = KT B λ,z M ( f K − f ) T B λ,z M ( f K )0 0 T B λ,z ... . . .0
T B λ,z
T B λ,z · · · (5.6)where M ( f ) denotes the multiplication operator that multiplies a function with f . For f , g ∈ H ∞ and f , g ∈ H one finds B λ,z f K − f g , B λ,z f K g ∈ H andby Lemma 4.2 F (cid:126)f defines a bounded linear operator on H ∞ × H L . Thus, F isFrechet-differentiable. Similarly, if f , f ∈ H ∞ , then F (cid:126)f defines also a boundedlinear operator on H L +1 .To get (ii) note that (cid:126)ζ ,E ∈ ( PE ( m )) , and B ,E = M ( e i Tr(( E − A ) M ) ). Forsimplicity, let us simply write B for B ,E , M for M (( ζ (0)0 ,E ) K − ζ (1)0 ,E ) and M for M (( ζ (0)0 ,E ) K ), then C E = KT B (cid:18) M (cid:19) + T B M ... 10 . . . 11 0 · · · . (5.7)Now taking C L +1 E , each of the occurring non-zero terms has at least one factor M or M in it. Therefore, each term occurring in C L +1) E has at least two terms M j , j = 0 , T B in between. Considering the structure of C E , theseterms in between M j ’s either come from ( C E ) ( C E ) j = T BM T BM j , j = 0 , C E ) ( C E ) ( C E ) · · · ( C E ) L − ,L ( C E ) L, ( C E ) ,j = T BM ( T B ) L +1 M j for j =0 ,
1. Hence, each appearing term in C L +1) E includes either a term M i T BM j or M i ( T B ) L +1 M j . We claim that we can use Proposition C.1 (ii) to obtain that theseoperators are compact. As B = M ( ζ E − A ) this means that we need to check thatthe L matrices A , . . . , A L are invertible, where A j is a mj × mj matrix with a .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 19 tri-diagonal block structure of m × m blocks given by A j = A − E . . . . . .. . . . . . A − E , where A = A − E . (5.8)For E ∈ I K,LA one finds for each eigenvalue a j of A that E − a j ∈ I K,L . In particular, E − a j (cid:54)∈ E L as defined in (1.8) which yields invertibility of A , . . . , A L as can beeasily seen when assuming that A is diagonal. Similar considerations can be donefor the operator C E using Proposition C.1 (iii).Part (iii) is the exact same calculation as in [Sad]. For the reader’s conveniencelet us point out the main ideas. Define the ( L + 1) × ( L + 1) matrix ζ ,E =diag( ζ (0)0 ,E , . . . , ζ ( L )0 ,E ), let D ∈ Sym( m ) and (cid:126)v ∈ R L +1 and start with the identity C E ( ζ tD ζ ,E (cid:126)v ) = ζ B ζ B ... ζ BL S K,L (cid:126)v (5.9)where for small t , B = − (cid:16) E − A + tD + KA (0) E + A (1) E (cid:17) − = A (0) E + ∞ (cid:88) k =1 t k ( A (0) E D ) k A (0) E (5.10) B p = − (cid:16) E − A + tD + A ( p +1) E (cid:17) − = A ( p ) E + ∞ (cid:88) k =1 t k ( A ( p ) E D ) k A ( p ) E (5.11)with 1 ≤ p ≤ L and the convention A ( L +1) E = A (0) E . Here we use ζ A ζ B = ζ A + B and T ζ A = ζ − A − as well as the recursion equations satisfied by the free Green’sfunction for λ = 0. This gives C E ( ζ tD ζ ,E (cid:126)v ) = ζ ,E ∞ (cid:89) k =1 diag (cid:16)(cid:2) ζ t k ( A ( p ) E D ) k A ( p ) E (cid:3) p =0 ,...,L (cid:17) S K,L (cid:126)v . (5.12)A further expansion of the exponential functions ζ tD and ζ t k ... in powers of t andvarying the matrix D leads to C E ([Tr( DM )] k ζ ,E ( M ) (cid:126)v ) = (5.13) ζ ,E ( M )diag (cid:16)(cid:2) (Tr( A ( p ) E DA ( p ) E M )) k (cid:3) p =0 ,...,L (cid:17) S K,L (cid:126)v + P k,D,E ( M ) (cid:126)v where P k,D,E ( M ) is a matrix of polynomials of degree less than k . Mapping themap P ( M ) = [Tr( DM )] k to ˆ P ( M ) = [Tr( A ( p ) E DA ( p ) E M )] k defines a linear map onthe set of homogeneous polynomials in entries of M of order k . Using the diagonalstructure of A ( p ) E one realizes that these linear maps (for all k ) are also representedby P J ( M ) (cid:55)→ θ ( p ) J,E P J ( M ) for J ∈ ∆( m, Z + ), where P J ( M ) = (cid:81) j,k ( M jk ) J jk . Thus, C E ( P J ζ ,E (cid:126)v ) = P J ζ ,E θ J,E
S (cid:126)v + ζ ,E p J,E (cid:126)v (5.14)where p J,E is a matrix of polynomials of degree less than the one of P J .Let f J,k = P J ζ E, (cid:126)e k with ( (cid:126)e k ) k =0 ,...,L being the standard basis in R L +1 . Usingthe functions f J,k ordered in some way with increasing degree of the polynomial P J , the operator C E can be represented as an infinite block triangular matrix, where the ( L + 1) × ( L + 1) blocks along the diagonal are given by the matrices θ J,E S K,L . Using the fact that C L +1) E is compact and the density of the span ofthese functions in H L +1 , [Sad, Proposition A.1] immediately implies (5.3).For the operator C E we start with a similar calculation as (5.9), replacing ζ tD by ζ tD ⊗ ζ tD (cid:48) ( M + , M − ) = ζ tD ( M + ) ζ tD (cid:48) ( M − ) and ζ ,E by ξ ,E ( M + , M − ) = ζ ,E ( M + ) ζ ,E ( M − ), then one obtains similar to (5.13) C E ([Tr( DM + − D (cid:48) M − )] k ξ ,E ( M + , M − ) (cid:126)v ) = (5.15) ξ ,E diag (cid:16) [( g ( p ) ( M + , M − )) k ] p =0 ,...,L (cid:17) S K,L (cid:126)v + (cid:101) P k,D,E ( M + , M − ) (cid:126)v where g ( p ) ( M + , M − ) = Tr( A ( p ) E DA ( p ) E M + − A ( p ) E D (cid:48) A ( p ) E M − ), and (cid:101) P k,D,E ( M + , M − )is a matrix of polynomial of degree less than k in the entries of M + and M − . Using P J ⊗ P J (cid:48) ( M + , M − ) = P J ( M + ) P J (cid:48) ( M − ) this leads to C E (cid:0) P J ⊗ P J (cid:48) ξ ,E (cid:126)v (cid:1) = P J ⊗ P J (cid:48) ξ ,E θ J,E θ ∗ J (cid:48) ,E S K,L (cid:126)v + ξ ,E (cid:101) p J,J (cid:48) ,E (cid:126)v (5.16)where (cid:101) p J,J (cid:48) ,E ( M + , M − ) is a matrix of polynomials of degree less than the one of P J ( M + ) P J (cid:48) ( M − ) = P J ⊗ P J (cid:48) ( M + , M − ). Now we follow the same line of argumentsas for the spectrum of C E .For (iv) note that σ H ∞ ×H L ( C E ) ⊂ σ H ( C E ) by compactness of C L +1) E in H ∞ ×H L ⊂ H L +1 . Equality follows as one finds eigenfunctions corresponding to theeigenvalues of θ J,E S K,L in H L +1 ∞ by considering the finite dimensional subspaces V c spanned by f J,k with (cid:107) J (cid:107) ≤ c (where (cid:107) J (cid:107) = (cid:80) j,k | J j,k | ) that are left invariantby C E . (cid:3) The following result will ensure that we can use the Implicit Function Theorem.
Lemma 5.2.
For E ∈ I K,LA and any
J, J (cid:48) ∈ ∆( m, Z + ) we find det( − θ J,E θ ∗ J (cid:48) ,E S K,L ) (cid:54) = 0 . (5.17) This means, the matrices θ J,E θ ∗ J (cid:48) ,E S do not have an eigenvalue 1. In particular,noting θ ,E = , this implies / ∈ σ H ( C E ) and / ∈ σ K ( C E ) . (5.18) Proof.
For
J, J (cid:48) ∈ ∆( m, Z + ) and E ∈ I K,LA define f ( J, J (cid:48) , E ) = det( − θ J,E θ ∗ J (cid:48) ,E S K,L ) = 1 − Kθ (0) J,E (cid:16) θ (0) J (cid:48) ,E (cid:17) ∗ − L (cid:89) p =0 θ ( p ) J,E (cid:16) θ ( p ) J (cid:48) ,E (cid:17) ∗ . For J = J (cid:48) = 0 we have f ( , , E ) = det( − S K,L ) = − K (cid:54) = 0. Now let (cid:107) J (cid:107) denote the norm given by the sum of the absolute values of all entries of J . Next,we consider the case (cid:107) J (cid:107) + (cid:107) J (cid:48) (cid:107) = 1, i.e. one of these matrices is zero and theother has one entry. Both cases are completely analogous so let us just consider J (cid:48) = 0, (cid:107) J (cid:107) = 1. Then by (5.1) one has θ ( q ) J,E = Γ ( q ) E − b Γ ( q ) E − b (5.19) .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 21 for some b , b ∈ { a , . . . , a m } . Using (2.15) in Proposition 2.1 and the Cauchy-Schwarz inequality we find K (cid:12)(cid:12)(cid:12) θ (0) J,E (cid:12)(cid:12)(cid:12) + L (cid:89) p =0 (cid:12)(cid:12)(cid:12) θ ( p ) J,E (cid:12)(cid:12)(cid:12) = K (cid:12)(cid:12)(cid:12) Γ (0) E − b (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Γ (0) E − b (cid:12)(cid:12)(cid:12) + (cid:32) L (cid:89) p =0 (cid:12)(cid:12)(cid:12) Γ ( p ) E − b (cid:12)(cid:12)(cid:12)(cid:33) (cid:32) L (cid:89) p =0 (cid:12)(cid:12)(cid:12) Γ ( p ) E − b (cid:12)(cid:12)(cid:12)(cid:33) ≤ (cid:89) k =1 (cid:118)(cid:117)(cid:117)(cid:116) K (cid:12)(cid:12)(cid:12) Γ (0) E − b k (cid:12)(cid:12)(cid:12) + L (cid:89) p =0 (cid:12)(cid:12)(cid:12) Γ ( p ) E − b k (cid:12)(cid:12)(cid:12) = 1 . (5.20)Since θ (0) J,E is the product of two factors with positive imaginary part, it can not be apositive real number, hence | f ( J, , E ) | > − | θ (0) J,E | − (cid:81) Lp =0 | θ ( p ) J,E | ≥
0, so f ( J, , E )can not be zero in this case.Finally, consider (cid:107) J (cid:107) + (cid:107) J (cid:48) (cid:107) ≥
2. We may assume without loss of generalitythat J (cid:54) = . Then θ ( p ) J,E (cid:16) θ ( p ) J (cid:48) ,E (cid:17) ∗ = Γ ( p ) E − b Γ ( p ) E − b · X ( p ) (5.21)where X ( p ) itself is a product of an even number of factors (at least 2) Γ ( p ) E − b or com-plex conjugates. By (2.15) and the fact that for E ∈ I K,LA and b ∈ { a , . . . , a m } noneof the imaginary parts of Γ ( p ) E − b can be zero, we find | Γ (0) E − b | < (cid:81) Lp =0 | Γ ( p ) E − b | < | X (0) | < (cid:81) Lp =0 | X ( p ) | <
1. Using this and Cauchy-Schwartz as in(5.19) we find K (cid:12)(cid:12)(cid:12) θ (0) J,E (cid:16) θ (0) J (cid:48) ,E (cid:17) ∗ (cid:12)(cid:12)(cid:12) + L (cid:89) p =0 (cid:12)(cid:12)(cid:12) θ ( p ) J,E (cid:16) θ ( p ) J (cid:48) ,E (cid:17) ∗ (cid:12)(cid:12)(cid:12) < | f ( J, J (cid:48) , E ) | >
0. Hence, in any case, f ( J, J (cid:48) , E ) will notbe zero. (cid:3) Conclusions
The most important ingredient for the proof of Theorem 1.4 is the following.
Proposition 6.1.
There exists an open set U ⊂ R with { } × I K,LA ⊂ U , suchthat the maps ( λ, E, η ) ∈ U × (0 , ∞ ) (cid:55)→ (cid:126)ζ λ,E + iη ∈ H ∞ × H L (6.1)( λ, E, η ) ∈ U × (0 , ∞ ) (cid:55)→ (cid:126)ξ λ,E + iη ∈ K ∞ × K L (6.2) have continuous extensions to maps from U × [0 , ∞ ) to H ∞ × H L and K ∞ × K L ,respectively, that satisfy (4.19) .Proof. By Lemma 5.1 and Lemma 5.2 we can use the Implicit Function Theoremon Banach Spaces as stated in [Kl6, Appendix B] for the functions ˆ F ( λ, E, η, (cid:126)f ) = F ( λ, E, η, (cid:126)f ) − (cid:126)f and ˆ Q ( λ, E, η, (cid:126)g ) = Q ( λ, E, η, (cid:126)g ) − (cid:126)g at the points (0 , E, , (cid:126)ζ ,E )and (0 , E, , (cid:126)ξ ,E ) with E ∈ I K,LA . Uniqueness of the continuous implicit functionand the continuity properties of (cid:126)ζ λ,E + iη and (cid:126)ξ λ,E + iη as stated in Proposition 4.3give the continuous extensions. (cid:3) For η = Im( z ) > ζ ( x | y ) λ,z = E ζ G ( x | y ) λ,z and ξ ( x | y ) λ,z = E ξ G ( x | y ) λ,z where as in Section 3 the upper index ( x | y ) for x, y ∈ T K,Lp indicates that we consider the Green’s function at x for the operator H ( x | y ) λ whichis the restriction of H λ to the subtree ( T K,Lp ) ( x | y ) that is obtained by removing thebranch at x going through y . If x is a child or descendant of y then ζ ( x | y ) λ,z = ζ ( l ( x )) λ,z and ξ ( x | y ) λ,z = ξ ( l ( x )) λ,z , where l ( x ) is the label of x . But if y is a descendant of x then weget different quantities. In the following arguments it will often be used implicitlythat B λ,z and B λ,z is a strongly continuous family of operators on any space L r , H r and (cid:98) L r , K r , respectively, for r ∈ [1 , ∞ ) which follows from the Leibniz rule (4.3),boundedness and Dominated Convergence. Proposition 6.2.
There is an open set U ⊂ R , { } × I K,LA ⊂ U , such that forall p ∈ { , . . . , L } , x ∈ T K,Lp with d (0 ( p ) , x ) ≤ L − p and y being the unique child of x , one has that the maps ( λ, E, η, M ) (cid:55)→ ζ ( x | y ) λ,E + iη ( M ) and ( λ, E, η, M + , M − ) (cid:55)→ ξ ( x | y ) λ,E + iη ( M + , M − ) (6.3) extend continuously as maps from U × [0 , ∞ ) × Sym + ( m ) and U × [0 , ∞ ) × (Sym + ( m )) to C , respectively. Moreover, ||| ζ ( x | y ) λ,z ||| ∞ and |||| ξ ( x | y ) λ,z |||| ∞ are uniformlybounded on compact subsets of U × [0 , ∞ ) .Proof. First note that d (0 ( p ) , x ) ≤ L − p means that x is in the starting line segmentof T K,Lp and hence ( T K,Lp ) ( x | y ) is a finite line with d (0 ( p ) , x ) edges and j = d (0 ( p ) , x )+1 ≤ L vertices. In fact, H ( x | y )0 − E is given by the matrix A j as in (5.8) and using E ∈ I K,LA which implies E − a j (cid:54)∈ E L for any eigenvalue of a j one obtains that ζ ( x | y )0 ,E and ξ ( x | y )0 ,E exist. Using assumption (V), the boundedness of the distribution of V ( x ), one obtains existence of ζ ( x | y ) λ,E and ξ ( x | y ) λ,E for ( λ, E ) in an open neighborhoodof { } × I K,LA . Point wise continuity of the maps follows immediately. Note thatthe infinity norm of ζ G and ξ G are bounded by 1 and the derivatives appearing inthe ||| · ||| ∞ and |||| · |||| ∞ norms lead to multiplication by determinants of minors of G .Therefore, using assumption (V) again we obtain the uniform bounds of the ||| · ||| ∞ and |||| · |||| ∞ norm on compact subsets of U × [0 , ∞ ). (cid:3) Proposition 6.3.
Let U = U ∩ U with U and U as in Propositions 6.1 and 6.2.Clearly, U ⊂ R is open and { } × I K,LA ⊂ U . For all x ∈ T K,L and all x ∈ T K,Lp with d ( x, ( p ) ) > L − p and all children y of x there is r ( x, y ) ∈ { , ∞} such thatthe maps ( λ, E, η ) ∈ R × R × (0 , ∞ ) (cid:55)→ ζ ( x | y ) λ,E + iη ∈ L r ( x,y ) , (6.4)( λ, E, η ) ∈ R × R × (0 , ∞ ) (cid:55)→ ξ ( x | y ) λ,E + iη ∈ (cid:98) L r ( x,y ) , (6.5) have continuous extensions to maps from ( λ, E, η ) ∈ U × [0 , ∞ ) to L r ( x,y ) and (cid:98) L r ( x,y ) , respectively. Moreover, for l ( x ) = 0 and l ( y ) = 1 as well as for l ( x ) (cid:54) = 0 wehave r ( x, y ) = 2 and hence L r ( x,y ) = L = H . For l ( x ) = 0 = l ( y ) both, r ( x, y ) = 2 and r ( x, y ) = ∞ are possible. Here, L r and (cid:98) L r denote the spaces as defined inDefinition 4.1. For such continuous extensions of maps from ( λ, E + iη ) that extend as functionsfrom U × [0 , ∞ ) to L r we will use the notion that such a family of functions extendscontinuously in L r . .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 23 Proof.
All arguments will implicitly use some specific version of the Green’s matrixrecursion (3.1) in the form as in (3.7). We will also implicitly use Proposition 6.1and H¨older’s inequalities.Note that H ∞ ⊂ L ∩ L ∞ , thus ζ (0) λ,z extends continuously in L and L ∞ , where ζ ( p ) λ,z extends continuously in L = H . The proof will be done by induction over thedistance from the root. For the start on T K,L we have to consider the root 0 (0) andon T K,Lp we have to consider the vertex 1 ( p )0 as defined in Theorem 1.2 which is theclosest vertex to the root of label 0 and characterized by d (0 ( p ) , ( p )0 ) = L + 1 − p .Let y be a child of 0 (0) , then using the general recursion relation (3.1) in the formas (3.5) and taking expectations leads to ζ (0 (0) | y ) λ,z = T B λ,z (cid:0) ( ζ (0) λ,z ) K − ζ ( l ( y )) λ,z (cid:1) which by Proposition 6.1 gives the continuous extension in H = L (even H ∞ if K ≥ r (0 (0) , y ) = 2. Similar, letting x be the parent of 1 ( p )0 and y a child,then ζ (1 ( p )0 | y ) = T B λ,z (cid:0) ζ ( x | ( p )0 ) λ,z ( ζ (0) λ,z ) K − ζ ( l ( y )) λ,z (cid:1) Using Propositions 6.1 and 6.2 and Dominated Convergence one obtains that theproduct after the operators
T B λ,z on the right hand side extend continuously in L and hence the whole term does too. In particular, r (1 ( p )0 , y ) = 2.For the induction step, let x be a descendant of 0 (0) or 1 ( p )0 for some p ∈{ , . . . , L } . Let x be the parent and y some child of x . We have several cases:Case 1: l ( x ) = q (cid:54) = 1, then l ( x ) = q − l ( y ) = q + 1 or l ( y ) = 0 if q = L . Wehave by induction assumption that r ( x , x ) = 2 and so ζ ( x | x ) λ,z extends continuouslyin L . Hence, ζ ( x | y ) λ,z = T B λ,z ζ ( x | x ) λ,z does as well and r ( x, y ) = 2.Case 2: l ( x ) = 0 and l ( y ) = 1, then ζ ( x | y ) λ,z = T B λ,z (cid:0) ( ζ (0) λ,z ) K ζ ( x | x ) λ,z (cid:1) . By inductionassumption ζ ( x | x ) λ,z either extends in L or L ∞ . As K ≥ L in either case, so r ( x, y ) = 2.Case 3: l ( x ) = 0 and l ( y ) = 0, then ζ ( x | y ) λ,z = T B λ,z (cid:0) ( ζ (0) λ,z ) K − ζ (1) λ,z ζ ( x | x ) λ,z (cid:1) . If ζ ( x | x ) λ,z extends continuously in L , then the product after T B λ,z extends continuously in L and hence ζ ( x | y ) λ,z extends continuously in L ∞ . If ζ ( x | x ) λ,z extends continuously in L ∞ then we obtain a continuous extension of ζ ( x | y ) λ,z in L .All arguments for the functions ξ ( x | y ) λ,z are completely analogue. (cid:3) Proof of Theorem 1.4.
Using (4.12), the recursion relation (3.1) and T = id oneobtains E (cid:0) G [ x ] λ ( z ) (cid:1) = − i (cid:90) D T E ζ G [ x ] λ ( z ) ( ϕϕ (cid:62) ) d mn ϕ = − i (cid:90) D B λ,z (cid:89) y : d ( x,y )=1 ζ ( y | x ) λ,z ( ϕϕ (cid:62) ) d mn ϕ (6.6) and similarly, based on (4.13) one obtains E (cid:18)(cid:12)(cid:12)(cid:12) G [ x ] λ ( z ) (cid:12)(cid:12)(cid:12) (cid:19) = (cid:90) D ( − ) D (+) B λ,z (cid:89) y : d ( x,y )=1 ξ ( y | x ) λ,z ( ϕ + ϕ (cid:62) + , ϕ − ϕ (cid:62)− ) d mn ϕ + d mn ϕ − , (6.7)where D is defined by (4.11), D ( ± ) represent the matrix-operator D acting withrespect to M ± = ϕ ± ϕ (cid:62)± and D ( − ) D (+) has to be understood as a matrix product.Using Propositions 6.1 and 6.3 one obtains for x ∈ T K,L or d ( x, ( p ) ) > L − p thatthe products of the ζ ’s on the right hand side of (6.6) have 2 factors that extendcontinuously in L and if l ( x ) = 0 some additional bunch of factors that extendcontinuously in L ∞ . Therefore, the product extends continuously in L . Hence,when applying D , each entry of the matrix extends continuously in L ( d mn ϕ ).Therefore, the map ( λ, E, η ) (cid:55)→ E ( G [ x ] λ ( z )) ∈ Sym C ( m ) extends continuously toa map from U × [0 , ∞ ) to Sym C ( m ) with U as in Proposition 6.3. By similararguments the same is true for ( λ, E, η ) (cid:55)→ E ( | G [ x ] λ ( z ) | ) ∈ Sym + ( m ). This provesTheorem 1.4. (cid:3) Appendix A. An identity for the unperturbed Green’s functions ontrees of finite cone type
Recall that associated to an s × s substitution matrix S ∈ Mat( s, Z + ) with non-negative integer entries are the following s rooted trees of finite cone type, denotedby T r , r = 1 , . . . , s . Each vertex has a label, the root of the tree T r has label r ,any vertex of label p has S pq children of label q . Denoting by ∆ the adjacencyoperator on the forest (cid:83) r T ( r ) and by 0 ( r ) ∈ T r the root of the tree T r , we definefor Im( z ) > ( r ) z := (cid:104) ( r ) | (∆ − z ) − | ( r ) (cid:105) . We define the setΣ = { E ∈ R : Γ ( r ) E := lim η ↓ Γ ( r ) E + iη exists for all r and Im(Γ ( q ) E ) > q } Proposition A.1.
Let Γ E = diag(Γ (1) E , . . . , Γ ( s ) E ) denote the diagonal s × s matrixwith the Green’s functions along the diagonal. Then one has for E ∈ Σdet( − | Γ E | S ) = 0 . (A.1) Proof.
The recursion relation for the Green’s functions is given byΓ ( p ) E = − (cid:32) E + s (cid:88) q =1 S pq Γ ( q ) E (cid:33) − . Multiplying by (Γ ( p ) E ) ∗ and some algebra leads to (cid:12)(cid:12)(cid:12) Γ ( p ) E (cid:12)(cid:12)(cid:12) s (cid:88) q =1 S pq Γ ( q ) E = − E (cid:12)(cid:12)(cid:12) Γ ( p ) E (cid:12)(cid:12)(cid:12) − (cid:16) Γ ( p ) E (cid:17) ∗ . Taking imaginary parts gives (cid:12)(cid:12)(cid:12) Γ ( p ) E (cid:12)(cid:12)(cid:12) s (cid:88) q =1 S pq Im(Γ ( q ) E ) = Im(Γ ( p ) E ) .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 25 Defining the vector (cid:126) Γ E = (Γ (1) E , . . . , Γ ( s ) E ) (cid:62) these equations can be read as | Γ E | S Im( (cid:126) Γ E ) = Im( (cid:126) Γ E )and for E ∈ Σ, Im( (cid:126)
Γ) is not the zero vector. Hence, | Γ E | S has an eigenvalue 1which proves (A.1). (cid:3) Appendix B. Gaussian integrals and the Fourier transform
The following identities are used at various parts in the article.
Lemma B.1.
Let D be an invertible, symmetric k × k matrix with positive definitereal part, i.e. D = D (cid:62) , Re( D ) > . Then, for any complex vector v ∈ C k one hasthe Gaussian integral (cid:90) R k e − ( x + v ) · D ( x + v ) d k x = (2 π ) k/ (cid:112) det( D ) . (B.1) Some care needs to be taken to select the correct branch of (cid:112) det( D ) . If D = A + iB where A > is the real part, then we write D = √ A (1 + iA − / BA − / ) √ A where √ A has the same eigenspaces as A and the corresponding eigenvalues are the posi-tive square roots of the eigenvalues of A . Furthermore, A − / BA − / is diagonal-izable by a real orthogonal matrix. This diagonalizes iA − / BA − / as well andthe eigenvalues have all real part . Hence, we may define √ iA − / BA − / bytaking the same eigenspaces and the principal branch of the square roots of the eigen-values. Then (B.1) is correct with (cid:112) det( D ) = det( √ A ) det( √ iA − / BA − / ) .Proof. In one dimension one has the well known integral formula (cid:90) ∞−∞ e − z ( x + c ) dx = √ π √ z (B.2)for Re( z ) >
0, where the square root is the principal branch and c is any fixedcomplex number. Now if D = A + iB , then use a basis change y = O √ Ax , where O is a real orthogonal matrix such that OA − / BA − / O (cid:62) is diagonal. This leads toa Gaussian integral with a diagonal matrix and then (B.1) follows from (B.2). (cid:3) For functions f ( x ) on R k and a k × k matrix D we define M ( D ) and C ( D ) tobe the multiplication and convolution operator by e ix · Dx , i.e.( M ( D ) f )( x ) = e ix · Dx f ( x ) , ( C ( D ) f )( x ) = (cid:90) e i ( x − y ) · D ( x − y ) f ( y ) d k y . For D invertible we also define S ( D ) by ( S ( D ) f )( x ) = f ( Dx ) which is a change ofvariables and defines a bounded operator on any L p space. Lemma B.2.
Let F denote the Fourier transform on R k , and let D be a symmetric,invertible k × k matrix with positive semi-definite imaginary part Im( D ) ≥ . (i) Then as a map from L ( R k ) ∩ L ( R k ) to L ( R k ) one has F ∗ M ( D ) F = C ( − D − )(2 π ) k/ (cid:112) det( − iD ) (B.3) where (cid:112) det( − iD ) is selected as in Lemma B.1 (note that Re( − iD ) ≥ ). (ii) If D is real, i.e. Im( D ) = 0 , this can be re-written as F ∗ M ( D ) F = 1 (cid:112) det( − iD ) M ( − D − ) S ( D − ) F M ( − D − ) (B.4) Equation (B.4) is valid in operator sense on L ( R k ) . (iii) For a real invertible, symmetric matrix D define D := − D − and iterativelydefine D j := − ( D + D j − ) − as long as the inverses exist, i.e. D = − ( D − D − ) − , D = − ( D − ( D − D − ) − ) − , and so on. Assume that the first L matrices D , . . . , D L , exist. Then, one has as operators on L ( R k ) (cid:0) FM ( D ) (cid:1) L +1 = L (cid:89) j =1 M ( D j ) S ( D j ) (cid:113) det( iD − j ) F M ( D + D L ) (B.5) Note that all D j are invertible and therefore these are indeed bounded opera-tors.Proof. First assume Im( D ) >
0, then for f ∈ L ( R k ) and any y ∈ R k , the map( x, w ) (cid:55)→ e − iy · x e ix · Dx e ix · w f ( w ) is in L ( R k ) and one finds(2 π ) k F ∗ M ( D ) F f )( y ) = (cid:90) e − iy · x e ix · Dx e ix · w f ( w ) d k w d k x = (cid:90) (cid:20)(cid:90) e [ x + D − ( w − y )] · iD [ x + D − ( w − y )] d k x (cid:21) e − i ( y − w ) · D − ( y − w ) f ( w ) d k w = (2 π ) k/ (cid:112) det( − iD ) C ( − D − ) f ( y ) . (B.6)Now, if Im( D ) is only positive semi-definite, we approach D by D + i(cid:15) and let f ∈ L ( R k ) ∩ L ( R k ). Then as (cid:15) ↓
0, the right hand side converges point wise (forfixed y ). As the L norm is uniformly bounded by (cid:107) f (cid:107) , Dominated Convergenceshows convergence in L ( R k ). As the operators M ( e ix · ( D + i(cid:15) ) x ) converge for (cid:15) ↓ L ( R k ). For part (ii) and (B.4) note that (cid:90) e − i ( y − w ) · D − ( y − w ) f ( w ) d k w = e − i y · D − y (cid:90) e iD − y · w e − i w · D − w f ( w ) d k w . As the left hand side and right hand side of (B.4) are compositions of boundedoperators on L ( R k ) the validity for functions in the dense subset L ( R k ) ∩ L ( R k )implies the validity on L ( R k ).For (iii) note that (B.4) also implies FM ( D ) F = M ( − D − ) S ( − D − ) (cid:112) det( − iD ) FM ( − D − ) = M ( D ) S ( D ) (cid:113) det( iD − ) FM ( D ) . Now iteration and using M ( A ) M ( B ) = M ( A + B ) yields (B.5). (cid:3) Remark B.3.
For part (iii) one can reformulate the condition that the iterativelydefined k × k matrices D j exist for j = 1 , . . . , L . Note, D − = − D + D − isthe Schur complement w.r.t. the first upper block of the block matrix (cid:0) − D − D (cid:1) .Inductively, one obtains that D j is the inverse of the Schur complement of theupper left k × k block of a jk × jk matrix D j , This matrix has a tri-diagonal block .C. SPECTRUM ON THE FIBONACCI AND SIMILAR TREE-STRIPS 27 structure given by k × k blocks with − D along the diagonal and identity matrices on the side diagonals, i.e. D j := − D . . . . . .. . . . . . − D , where D = − D . (B.7)
Now, for a matrix Y = ( U VW X ) with X being invertible one has that the invertibilityof Y and the invertibility of the Schur complement U − V X − W are equivalent.Therefore, one obtains by induction that the existence of all the matrices D , . . . , D L is equivalent to the invertibility of all the matrices D , . . . , D L . Appendix C. Compact operators involving Fourier transforms
In the analysis of the Frechet derivative it is important that s certain power is acompact operator. In this work we need a little bit more general results compared toprevious work as [Sad] to prove that. As above, for functions f ( M ) , g ( M + , M − ) and h ( x ), M ( f ( M )) , M ( g ( M + , M − )) and M ( h ( x )) will denote the corresponding mul-tiplication operators. Recall ζ B ( M ) = e i Tr( BM ) , ξ B ( M + , M − ) = e i Tr( BM + − BM − ) .For a real, symmetric k × k matrix D define the jk × jk matrix D j = D j ( D ) asa tri-diagonal k × k block matrix with − D along the diagonal and the unit matrix along the side diagonal, as in (B.7). We denote the set of real, symmetric k × k matrices D , where D , D . . . , D L are invertible, by S ( k, L ). Proposition C.1. (i)
Let h , h be exponentially decaying, continuous functions on R k , let F denotethe Fourier transform on R k and let D ∈ S ( k, L ) . Then, M ( h ) FM ( h ) and M ( h ) (cid:16) FM ( e ix · Dx ) (cid:17) L +1 M ( h ) (C.1) are compact operators from L ( R k ) to L p ( R k ) for any p ∈ [1 , ∞ ] . (ii) For f , f ∈ PE ( m ) and B ∈ S ( m, L ) the operators M ( f ) T M ( f ) , and M ( f ) ( T M ( ζ B )) L +1 M ( f ) (C.2) are compact operators from H p to H q for any p, q ∈ [1 , ∞ ] . (Note that thecase where one Banach space is H is included as H = H as a set, only thenorm differs technically by a factor ). (iii) For g , g ∈ PE ( m ) ⊗ PE ( m ) , B ∈ S ( m, L ) , the operators M ( g ) T M ( g ) , and M ( g ) ( T M ( ξ B )) L +1 M ( g ) (C.3) are compact operators from K p to K q for any p, q ∈ [1 , ∞ ] .Proof. For (i) let us first assume that h and h are compactly supported andconsider M ( h ) FM ( h ). Let K be the compact support of h . There exists aconstant C K such that for all x ∈ K and all y ∈ R k we have | e ix · y − e ix · y (cid:48) | ≤ C K | y − y (cid:48) | . Therefore | ( FM ( h ) f )( y ) − ( FM ( h ) f )( y (cid:48) ) | ≤ (2 π ) − k/ C K | y − y (cid:48) | (cid:107) h f (cid:107) ≤ (2 π ) − k/ C K (cid:107) h (cid:107) (cid:107) f (cid:107) | y − y (cid:48) | and hence M ( h ) FM ( h ) maps a L bounded sequence of functions into a sequenceof equi-continuous functions, supported on the compact support of h . By thetheorem of Arzela Ascoli we obtain a convergent subsequence in L ∞ and hence inany L p norm.If h and h are continuous and exponentially decaying, then we can approachthem in (cid:107) · (cid:107) ∞ norm by compactly supported continuous functions h ,n , h ,n . Then M ( h ,n ) FM ( h ,n ) approaches M ( h ) FM ( h ) in L → L p operator norm for any p ∈ [1 , ∞ ] (here, consider M ( h ) as map from L to L , F as map from L to L ∞ and M ( h ) as map from L ∞ to L p ).For the second operator in (C.1) note that by Remark B.3 for D ∈ S ( k, L )Lemma B.2 (iii) applies. By (B.5) one finds after commuting the multiplicationand shift operators on the left hand side that M ( h ) (cid:0) FM ( e ix · Dx ) (cid:1) L +1 FM ( h ) = S A M (ˆ h ) FM (ˆ h ) where ˆ h and ˆ h are exponentially decaying functions and A isthe product of the D j as in B.2 (iii). Therefore, by the previous statement, thisdefines a compact operator from L to any L p , p ∈ [1 , ∞ ].Using (4.9) and the Leibniz-rule (4.3) the statements (ii) and (iii) immediatelyfollow from (i). For the connection of the matrix B in (ii) with D as used in(i), note that the operator T involves a Fourier transform on R m × n ∼ = R mn , so k = 2 mn , and combining the column vectors of ϕ ∈ R m × n to one large vector ϕ ∈ R mn vector, Tr( ϕϕ (cid:62) B ) = Tr( ϕ (cid:62) B ϕ ) can be written as ϕ · D ϕ where D is a2 mn × mn matrix which is block-diagonal with the repeated m × m block B alongthe diagonal, D = (cid:16) B · B (cid:17) . Then B ∈ S ( m, L ) implies D ∈ S (2 mn, L ) so we canuse part (i). Starting with an H bounded sequence one can subsequently constructa subsequence converging in all L and L p norms involved in the definition of H p .Similar considerations can be made to obtain part (iii). (cid:3) References [Aiz] M. Aizenman,
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Mathematics Department, University of British Columbia, Vancouver, BC, V6T 1Z2,Canada; and Institute of Science and Technology Austria, 3400 Klosterneuburg
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