Point Spectrum of Periodic Operators on Universal Covering Trees
PPOINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES
JESS BANKS, JORGE GARZA-VARGAS, AND SATYAKI MUKHERJEEAbstract. For any multi-graph ๐บ with edge weights and vertex potential, and its universal coveringtree ๎ , we completely characterize the point spectrum of operators ๐ด ๎ on ๎ arising as pull-backsof local, self-adjoint operators ๐ด ๐บ on ๐บ . This builds on work of Aomoto, and includes an alternativeproof of the necessary condition for point spectrum derived in [Aom91]. Our result gives a finitetime algorithm to compute the point spectrum of ๐ด ๎ from the graph ๐บ , and additionally allowsus to show that this point spectrum is contained in the spectrum of ๐ด ๐บ . Finally, we prove thattypical pull-back operators have a spectral delocalization property: the set of edge weight andvertex potential parameters of ๐ด ๐บ giving rise to ๐ด ๎ with purely absolutely continuous spectrum isopen and its complement has large codimension.
1. IntroductionConsider a finite graph
๐บ = (๐ , ๐ธ) and its universal cover ๎ = ( ๎ , ๎ฑ ) , together with a coveringmap ฮ โถ ๎ โ ๐บ . The purpose of this paper is to relate the point spectrum of certain local,periodic, self-adjoint operators on ๐ ( ๎ ) to the combinatorial structure of ๐บ .Precise definitions, notation and assumptions about the model in consideration will be discussedbelow in Section 2, but for now we give a high-level overview of the problem setting. By endowing ๐บ with edge weights and a potential on its vertex set, we obtain a natural self-adjoint operator ๐ด ๐บ on ๐ (๐ ) . This framework encompasses Schrยจodinger operators on graphs, weighted adjacencymatrices, graph Laplacians and transition matrices for random walks, and the correspondingpull-back of the weights and potential to ๎ induces an analogous periodic, self-adjoint operator ๐ด ๎ on ๐ ( ๎ ) .The class of operators ๐ด ๎ obtained in this way contains, but is richer than, the periodicSchrยจodinger operators in one dimension, which are of great relevance to spectral theory and thetheory of orthogonal polynomials. The spectra of these ๐ด ๎ are additionally crucial to the study ofrelative expanders [F +
03] and, as shown in [BC19], control in a strong sense the spectra of largerandom lifts of a fixed base graph. However, despite the many advances in functional analysis,operator algebras and operator theory, many natural questions regarding the spectral propertiesof ๐ด ๎ are unanswered and seem inaccessible with current techniques. We direct the reader to[ABS20] for a survey of both periodic Jacobi matrices and the difficulty in generalizing to themore generic case considered here.In this paper we will be concerned with the spectrum of ๐ด ๎ , which we denote by Spec ๐ด ๎ , thedensity of states ๐ (a natural and canonical measure on Spec ๐ด ๎ ), and most importantly those UC Berkeley
E-mail addresses : [email protected], [email protected],[email protected] . a r X i v : . [ m a t h . SP ] S e p OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 2 ๐ โ Spec ๐ด ๎ for which there exists a corresponding ๐ eigenvectorโin other words, the pointspectrum Spec ๐ ๐ด ๎ .Our main result is a set of necessary and sufficient condition on ๐บ (including its edge weightsand potential) for Spec ๐ ๐ด ๎ to be non-empty. This gives a finite algorithm to compute Spec ๐ ๐ด ๎ from ๐บ , and extends the work of Aomoto, who has already shown the necessary half of our resultin [Aom91]. However, our new and elementary argument is essentially different from his, andwe build on it to show, surprisingly, that Spec ๐ ๐ด ๎ โ Spec ๐ด ๐บ , and to give a lower bound forthe multiplicity of each eigenvalue of ๐ด ๐บ arising in this way. Finally, we prove that the set ofedge weights and potentials for which ๐ด ๎ has point spectrum is a closed semialgebraic set oflarge codimension, which in particular implies that the set has Lebesgue measure zero. This maybe regarded as a spectral delocalization result of the kind long-studied in mathematical physics[Ana18]; see [AISW20] for recent and analogous work in the context of quantum graphs. Inparticular, our result implies that even when Spec ๐ ๐ด ๎ has an isolated point, there are arbitrarilysmall perturbations of ๐ด ๎ with no point spectrum at all. In view of the Kato-Rellich theoremon stability of the discrete point spectrum, this is a strong manifestation of the fact that theeigenspaces of ๐ด ๎ are infinite-dimensional. Related Work.
The operators ๐ด ๎ defined here have been studied by several authors with differentmotivations and levels of generality, and are variously referred to as operators of nearest-neighbortype [Aom91], connected, local, pull-back operators [AFH15] or periodic Jacobi operators [ABS20];we will use the latter. When ๐บ is an unweighted ๐ -regular graph (making ๐ด ๐บ is its adjacencymatrix), classical work of Kesten in the context of Cayley graphs [Kes59], and McKay in thecontext of random graphs [McK81], proved that Spec ๐ด ๎ = [โ2โ๐ โ 1, 2โ๐ โ 1] and that ๐ followswhat is now called the Kesten-Mackay distribution with parameter ๐ . When ๐บ is an unweighted (๐, ๐) -bireguar bipartite graph with ๐ < ๐ , Godsil and Mohar showed that Spec ๐ด ๎ = {๐ โ โ โถโ๐ โ 1 โ โ๐ โ 1 โค |๐| โค โ๐ โ 1 + โ๐ โ 1} โช {0} and that ๐{0} = ๐โ๐๐+๐ [GM88]. These results implythat for adjacency matrices, when ๐บ is ๐ -regular, ๐ด ๎ has no point spectrum, while when ๐บ is (๐, ๐) -biregular and bipartite, Spec ๐ ๐ด ๎ = {0} .Subsequent work focused on the properties of Spec ๐ด ๎ and ๐ , and their relation to ๐ด ๐บ , withoutmaking any assumptions on ๐บ ; see [A +
88, Aom91, Sun92, SS92] as well the more recent [AFH15,BC19, ABS20, GVK19]. Of relevance for the current paper is a result of Avni, Breuer and Simonin [ABS20], which states that for any ๐บ , any edge weights, and any potential, the operator ๐ด ๎ has no singular continuous spectrum. As a corollary one can deduce that the continuous part of Spec ๐ด ๎ always consists of a finite union of closed non-degenerate intervals, and its singular partis the finite set of eigenvalues of Spec ๐ ๐ด ๎ . Equivalently, ๐ can be decomposed into a measure thatis absolutely continuous with respect to the Lebesgue measure on โ and a finite sum of atomicmeasures.The most noteworthy prior result regarding the point spectrum of ๐ด ๎ is the aforementionedwork of Aomoto, who in addition to deriving necessary conditions for the presence of pointspectrum of ๐ด ๎ deduced a remarkable formula relating ๐{๐} to the combinatorial structure of ๐บ for every ๐ โ Spec ๐ ๐ด ๎ . He then used these results to show that when ๐บ is a ๐ -regular graph,regardless of the edge weights and potential, ๐ด ๎ has no point spectrum. This generalizes thecase when ๐บ is a cycle, which was established by different authors in the mathematical physicsliterature; see Section 2 of [ABS20] for a discussion and survey. In a different context, Keller, Lenzand Warzel [KLW13] showed that adjacency matrices of certain trees have no point spectrum and OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 3 that this property is stable under small perturbations of the potential. For our setting, their resultsimply that if ๐บ has a loop at every vertex and ๐ด ๐บ is the adjacency matrix of ๐บ , then ๐ด ๎ has nopoint spectrum.Our results, stated in Section 3 after the preliminary material in Section 2, recover many ofthe ones above and provide a pleasant unification and generalization of the literature on pointspectra. 2. Preliminaries2.1. Graphs and Covers.
We will work in the general setting of weighted graphs with self-loopsand multi-edges. In this setup we will regard a graph as a tuple
๐บ = (๐ , ๐ธ, ๐, ๐) , consisting ofvertices, edges, edge-weights ๐ โถ ๐ธ โ โ and a potential ๐ โถ ๐ โ โ . When it is not clear fromthe context, we will write
๐ (๐บ) and
๐ธ(๐บ) to emphasize that we are referring to the set of verticesand edges of ๐บ . It will be convenient to regard ๐ธ as a set of directed edges, equipped with adirection-reversing involution ๐ โฆ ฬ๐ with no fixed points, as well as source and terminal maps ๐ , ๐ โถ ๐ธ โ ๐ so that ๐ (๐) = ๐ ( ฬ๐) for every ๐ โ ๐ธ . An edge for which ๐ (๐) = ๐ (๐) and ๐ = ฬ๐ is a self-loop , and we refer to the remainder as proper edges . We will also abuse notation and write ๐ (๐ข) and ๐ (๐ข) for the sets of directed edges whose source and terminal, respectively, are the vertex ๐ข โ ๐ .We say that a graph ๐ป covers ๐บ if there exists a covering map ๐ โถ ๐ป โ ๐บ , namely a map ofvertices and edges which is compatible with the source and terminal maps, preserves potential andedge weights, and is an isomorphism on ๐ (๐ข) and ๐ (๐ข) for each vertex ๐ข . If both are finite, then |๐ (๐บ)| necessarily divides |๐ (๐ป )| , and we call their ratio ๐ the degree of the cover; equivalentlywe say that ๐ป is an ๐ -lift of ๐บ . Each ๐ -lift ๐ป may be expressed explicitly by an assignment ofpermutations to edges ๐ โถ ๐ธ โ S ๐ , with the property that ๐ โ1๐ = ๐ ฬ๐ for each edge ๐ โ ๐ธ . Then ๐ (๐ป ) = ๐ (๐บ) ร [๐] โthroughout the paper we will use the notation [๐] = {1, ..., ๐} โand for every ๐ โ ๐ธ(๐บ) and each ๐ โ [๐] , we include an edge ฬ๐ โ ๐ธ(๐ป ) with ๐ ( ฬ๐) = (๐ (๐), ๐) and ๐ ( ฬ๐) = (๐ (๐), ๐ ๐ (๐)) .The universal cover of a connected graph ๐บ is the unique (up to isomorphism) infinite tree ๎ = ( ๎ , ๎ฑ , (cid:97) , (cid:98) ) that covers every other cover of ๐บ . It can be constructed directly in terms of non-backtracking walks on ๐บ , which are sequences of edges ๐ , ๐ , ...๐ ๐ such that, for every ๐ โ [๐ โ1] , ๐ (๐ ๐ ) = ๐ (๐ ๐ +1 ) and ๐ ๐ โ ฬ๐ ๐ +1 . If we choose a root vertex ๐ข โ ๐ , then we may set the vertex set ๎ of ๎ to be the set of non-backtracking walks on ๐บ starting at ๐ข , with directed edges ๎ฑ whenever onewalk is an immediate prefix or suffix of another, and edge weights and potential inherited fromthe final edge and vertex of the walk, respectively. Up to isomorphism ๎ is independent of theroot choice, and is manifestly a cover of ๐บ ; we will call the covering map ฮ . Finally, we note that ๎ is finite if and only if ๐บ is acyclic โthat is, if it does not contain a closed non-backtracking walk.In this case ๐บ = ๎ .Given ๐บ and a universal cover ๎ , the latter is endowed with a set of symmetries which acttransitively on ๎ by simultaneously permuting the fibres over every vertex.2.2. Jacobi Operators, Spectra, and the Density of States.
Following the convention intro-duced in [ABS20], the
Jacobi operator associated to
๐บ = (๐ , ๐ธ, ๐, ๐) acts on ๐ โ ๐ (๐ ) โ โ |๐ | Some authors additionally include so-called half-loops , which are edges ๐ with ๐(๐) = ๐ (๐) and ๐ = ฬ๐ ; see [Fri93].Our results easily extend to this case, but for simplicity we will not consider it here. OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 4 as: (๐ด ๐บ ๐)(๐ข) = ๐ ๐ข ๐(๐ข) + โ ๐โ๐(๐ข) ๐ ๐ ๐(๐ (๐)). Throughout the paper, we will assume that the edge weights satisfy a conjugate symmetrycondition ๐ ๐ = ๐ ฬ๐ and that the potential ๐ is realโthese ensure that ๐ด ๐บ is Hermitian, and we willaccordingly call such edge weights and potential Hermitian as well.When ๐ป is an ๐ -lift of ๐บ , we will always think of ๐ด ๐ป as acting on ๐ (๐ ) โ โ , regarded as the setof โ ๐ -valued functions on ๐ . A standard result characterizes the spectrum of ๐ด ๐ป . Let ๐ โถ ๐ธ โ S ๐ be the set of permutations which define ๐ป , and overload notation to write ๐ ๐ as well for theunitary operator which acts by permuting the coordinates of โ ๐ according to ๐ ๐ . Then ๐ด ๐ป acts on ๐ โ ๐ (๐ ) โ โ ๐ as (๐ด ๐ป ๐)(๐ข) = ๐ ๐ข ๐(๐ข) + โ ๐โ๐(๐ข) ๐ ๐ ๐ ๐ ๐(๐ (๐)) โ โ ๐ . By writing โ ๐ = โ ๐1 โ โ ๐0 โ โ โ โ ๐โ1 , where โ ๐1 is the span of the all-ones vector and โ ๐0 is its orthogonal complement, we can simultaneously decompose every edge permutation as ๐ ๐ = 1 โ ๐(๐ ๐ ) ; the latter unitary operator on โ ๐0 โ โ ๐โ1 is the regular representation of ๐ ๐ . Thus wemay write ๐ด ๐ป = ๐ด ๐บ โ ๐ด ๐ป/๐บ , where the second acts on ๐ โ ๐ (๐ ) โ โ ๐0 in the natural way: (๐ด ๐ป/๐บ ๐)(๐ข) = ๐ ๐ข ๐(๐ข) + โ ๐โ๐(๐ข) ๐ ๐ ๐(๐ ๐ )๐(๐ (๐)) โ โ ๐0 . In other words,
Spec ๐ด ๐ป = Spec ๐ด ๐บ โ Spec ๐ด ๐ป/๐บ , and we refer to Spec ๐ด
๐ป/๐บ as the new eigenvalues of ๐ป .Once again writing ๎ = ( ๎ , ๎ฑ , (cid:97) , (cid:98) ) for the universal cover of ๐บ , we will call the analogousoperator on ๐ ( ๎ ) the periodic Jacobi operator of ๎ . Since the edge weights (cid:97) and potential (cid:98) arerelated to those of ๐บ by (cid:97) ฬ๐ = ๐ ฮ( ฬ๐) for every ฬ๐ โ ๎ฑ and (cid:98) ฬ๐ฃ = ๐ ฮ( ฬ๐ฃ) for every ๐ฃ โ ๎ , finiteness of ๐บ , ๐ , and ๐ ensure that ๐ด ๎ belongs to the set ๎ฎ (๐ ( ๎ )) of bounded operators on ๐ ( ๎ ) , and theinherited conjugate symmetry condition (cid:97) ๐ = (cid:97) ฬ๐ guarantees that it is Hermitian. We use Spec ๐ด ๎ to denote the spectrum of the periodic Jacobi operator, that is Spec ๐ด ๎ = {๐ โ โ โถ (๐ โ ๐ด ๎ ) โ1 โ ๎ฎ (๐ ( ๎ ))} . We remind the reader that, unlike in the finite dimensional case, ๐ โ Spec ๐ด ๎ does not guaranteean ๐ eigenvector for ๐ . The subset of the spectrum with this additional propertyโthe pointspectrum โwill be our primary concern in this work. We will return to it below.For any ๐ข โ ๐ (๐บ) , the quantities โจ๐ฟ ฬ๐ข , ๐ด ๐ ๎ ๐ฟ ฬ๐ข โฉ for ๐ โ โ are real and constant over all ฬ๐ข in the fibreover ๐ข , and a routine application of the Riesz representation theorem guarantees an accompanying spectral measure ๐ ๐ข on Spec ๐ด ๎ associated to each ๐ข , satisfying โจ๐ฟ ฬ๐ข , ๐ (๐ด ๎ )๐ฟ ฬ๐ข โฉ = โซ Spec ๐ด ๎ ๐ (๐ฅ) d ๐ ๐ข (๐ฅ) โ ฬ๐ข โ ฮ โ1 (๐ข) for every bounded measurable function ๐ โถ Spec ๐ด ๎ โ โ , where ๐ (๐ด ๎ ) is defined via the Borelfunctional calculus. The density of states (DOS) of ๐ด ๎ is the unique measure obtained by averaging OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 5 these spectral measures over ๐ข โ ๐ (๐บ) : ๐ = 1|๐ (๐บ)| โ ๐ขโ๐ (๐บ) ๐ ๐ข . It is typical in the literature to work with real positive edge weights instead of Hermitian ones.The latter choice will make some of our proofs more convenient, but from the perspective of
Spec ๐ด ๎ it does not add any generality. In particular, ๐ด ๎ is gauge invariant in the following sense. Lemma 2.1.
Let
๐บ = (๐ , ๐ธ, ๐, ๐) be a graph with Hermitian edge weights and real potential, andlet ๐บ โฒ = (๐ , ๐ธ, ๐ โฒ , ๐) , where ๐ โฒ๐ = |๐ ๐ | ; write ๎ and ๎ โฒ for their respective covers. Then ๐ด ๎ = ๐ โ ๐ด ๎ โฒ ๐ ,where ๐ is a diagonal unitary operator. One may prove Lemma 2.1 by choosing a root ๐ for ๎ and letting ๐ ๐ฃ,๐ฃ be the product of edgeweight arguments along the unique shortest path connecting ๐ and ๐ฃ . An immediate corollary isthat the spectrum and density of states of ๐ด ๎ depend only on the moduli of the edge weights . In thecase ๐ โก 0 , note that the above implies that ๐ด ๎ and โ๐ด ๎ are unitarily equivalent, which has thefollowing consequence. Lemma 2.2.
Let ๐บ be a finite graph, ๎ its universal cover. If ๐ ๐ฃ = 0 for all ๐ฃ โ ๐ (๐บ) then thespectrum and density of states of ๐ด ๎ are symmetric about zero. On several occasions we will use the following well-known facts to relate the empirical spectralmeasures of finite graphs ๐บ to the densities of states of their universal covers. Lemma 2.3.
Let ๐บ be a finite graph, ๎ its universal cover, and ๐ the density of states of ๐ด ๎ . Thereexists a sequence of covers ๐บ ๐ of ๐บ whose girth diverges as ๐ goes to infinity. Moreover, for thissequence, the empirical spectral measures ๐ ๐ of ๐ด ๐บ ๐ converges weakly to ๐ . This lemma follows directly from results stated in [ABS20] whose proofs will appear in [ABKSon].Here we discuss very briefly their approach and refer the reader to Section 4 of [ABS20] for details.Begin by fixing a spanning tree ๐ of ๐บ and define ๐ = |๐ธ(๐บ) โงต ๐ธ(๐ )| . Then, if ๐บ is viewed as a1-complex, the fundamental group of ๐บ is the free group on ๐ generators, namely, F ๐ . Moreover,by contracting the lifts of ๐ in ๎ one obtains the Cayley graph of F ๐ and this allows to define anaction of F ๐ on ๎ which has the copies of ๐ as fundamental domains. Subgroups of F ๐ of finiteindex correspond to the lifts of ๐บ , and those lifts coming from normal subgroups enjoy a certaintype of symmetry. Theorem 4.3 in [ABS20] says that one can take a sequence of normal subgroupsof F ๐ of finite index, which give rise to a sequence of lifts of ๐บ , say ๐บ , ๐บ , โฆ , with the propertythat the empirical spectral distributions of ๐ด ๐บ ๐ converges weakly to ๐ . It is not hard to see fromtheir discussion that the symmetric property of the ๐บ ๐ together with the weak convergence of thespectral distributions, implies that this sequence of graphs has diverging girth.For completeness, below we provide a purely combinatorial proof of Lemma 2.3. Proof of Lemma 2.3.
By induction, it suffices to show that for every finite graph
๐ป = (๐ , ๐ธ) with girth(๐ป ) = ๐ and |๐ธ| = 2๐ , there exists a finite lift ๐ฟ of ๐ป whose girth is strictly larger (theweights and potential are irrelevant here, and we will suppress them). We will construct ๐ฟ as a ๐+1 -lift of ๐ป , with the following set of permutations ๐ โถ ๐ธ โ S ๐ +1 . Group the edges in to pairs The girth of a graph is the length of its shortest cycle.
OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 6 (๐, ฬ๐) consisting of an edge and its reversal, and order these (๐ , ฬ๐ ), ..., (๐ ๐ , ฬ๐ ๐ ) . Now let ๐ ๐ ๐ be thepermutation that maps ๐ โฆ ๐ + 2 ๐ mod 2 ๐+1 for every ๐ โ [2 ๐+1 ] , and let ๐ ฬ๐ ๐ = ๐ โ1๐ ๐ as required.Since girth ๐ฟ > girth ๐ป , we need only to show that ๐ฟ contains no cycle of length ๐ . Seekingcontradiction, assume instead that ๐ ๐ , ..., ๐ ๐ ๐ is a sequence of ๐ directed edges forming a cycle in ๐ฟ . Writing ๐ for the covering map, and ๐ (๐ ๐ ), ..., ๐ (๐ ๐ ๐ ) form a cycle in ๐ป with length ๐ , and since girth ๐ป = ๐ , they are distinct. The vertices of ๐ฟ are ๐ ร [2 ๐+1 ] , which we regard as a set of ๐+1 โlayers;โ assume ๐ (๐ ๐ ) is in the ๐ก th one. Because of how we have arranged the permutations, ๐ (๐ ๐ ๐ ) is in layer ๐ก ยฑ 2 ๐ ยฑ โฏ ยฑ 2 ๐ ๐ โ ๐ก mod 2 ๐+1 , because the ๐ , ..., ๐ ๐ are distinct and smaller than ๐ + 1 .Thus ๐ (๐ ๐ ๐ ) โ ๐ (๐ ๐ ) โa contradiction.We finally show that, given such a sequence ๐บ ๐ with diverging girth, ๐ ๐ converges weakly to ๐ .For every fixed positive integer ๐ and each vertex ๐ข of ๐บ ๐ , the quantity โจ๐ฟ ๐ข , ๐ด ๐๐บ ๐ ๐ฟ ๐ข โฉ is a weightedcount of length- ๐ closed walks in ๐บ ๐ starting and ending at ๐ข . Since the ๐บ ๐ have diverging girth,for ๐ sufficiently large the depth- ๐ neighborhood of ๐ข in ๐บ ๐ is identical to that of every ฬ๐ข โ ฮ โ1 (๐ข) ,and thus this count is eventually constant and equal to โจ๐ฟ ฬ๐ข , ๐ด ๐ ๎ ๐ฟ ฬ๐ข โฉ . Finally, as the ๐ th moments ofthe empirical spectral measures ๐ ๐ are given by normalized traces of ๐ด ๐๐บ ๐ , the method of momentsgives us weak convergence to the density of states. (cid:3) Substantially stronger versions of this result are known but will not be necessary for us; wedirect the reader for instance to the recent work of Bordenave and Collins [BC19].2.3.
Point Spectrum and the Aomoto Sets.
We will denote the point spectrum of ๐ด ๎ as Spec ๐ ๐ด ๎ = {๐ โ โ โถ Ker(๐ โ ๐ด ๎ ) โ {0}} . The following proposition collates several equivalent characterizations of
Spec ๐ ๐ด ๎ . Proposition 2.1.
Let ๐บ be a finite graph. Assume ๐บ has at least one cycle, and let ๎ be its universalcover. Then ๐ โ Spec ๐ ๐ด ๎ if and only if any of the following hold:(i) dim Ker(๐ โ ๐ด ๎ ) = โ (ii) ๐ is an atom of ๐ (iii) For some ๐ข โ ๐ (๐บ) , ๐ is an atom of ๐ ๐ข .(iv) For some ๐ข โ ๐ (๐บ) , the Cauchy transform ๐ ๐ข (๐ง) = โซ Spec ๐ด ๎ (๐ง โ ๐ฅ) โ1 d ๐ ๐ข (๐ฅ) has a pole at ๐ .(v) For some ๐ข โ ๐ (๐บ) , and every ฬ๐ข โ ฮ โ1 (๐ข) , there exists ๐ โ Ker(๐ โ ๐ด ๎ ) with ๐ ( ฬ๐ข) โ 0 .Moreover, the vertices satisfying (iii),(iv), and (v) coincide.Proof. In Section 8 of [ABS20] it is proven that ๐ โ Spec ๐ ๐ด ๎ implies (i). The intuition behind thisresult is that if ๐ โ Ker(๐ โ ๐ด ๎ ) , then precomposing ๐ with deck transformations on ๎ gives riseto many linearly independent eigenvectors of ๐ด ๎ for ๐ .The rest of the claims are standard results that are true for general self-adjoint bounded operators.The fact that ๐ โ Spec ๐ ๐ด ๎ , (ii), (iii), and (v) are equivalent follows directly from the definition of ๐ and the forthcoming Lemma 2.4. On the other hand (iii) and (iv) are clearly equivalent. (cid:3) OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 7
By way of a complicated set of coupled equations satisfied by the Cauchy transforms ๐ ๐ข , Aomotoidentified a set of vertices of ๐บ whose combinatorial structure is instrumental in understanding Spec ๐ ๐ด ๎ and will be the focus of much of this paper. Definition 2.1 (The Aomoto set) . Let ๐บ be a finite graph and assume that ๐ โ Spec ๐ ๐ด ๎ . The Aomoto set of ๐บ associated to ๐ consists of those vertices in ๐ (๐บ) that satisfy the equivalentconditions (iii-v) in Proposition 2.1. This set will be denoted by ๐ ๐ (๐บ) . -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.500.050.10.150.20.250.3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.500.0050.010.0150.020.025 ๐บ ๐ ๐ข = ๐ ๐ข = ๐ ๐ข = ๐ ๐ข ๐ ๐ข Figure 1. On the left a finite graph ๐บ , where the vertices in ๐ (๐บ) = ๐ โ1 (๐บ) arecolored in red. In the middle and on the right we show the two distinct spectralmeasures of ๐ด ๎ associated to the vertices of ๐บ . Example 2.1.
In Figure 1, ๐บ is a finite graph with ๐ ๐ โก 1 and ๐ ๐ฃ โก 0 for every ๐ โ ๐ธ(๐บ) and ๐ฃ โ ๐ (๐บ) . By the symmetries in ๐บ , the spectral measures corresponding to the vertices ๐ข , ๐ข , ๐ข and ๐ข are equal. Hence, ๐ด ๎ has only two distinct spectral measures associated to the vertices of ๐บ , sketched in Figure 1. These sketches were generated by taking a random lift of ๐บ of degree1200 and by plotting the weighted histogram for the corresponding discrete spectral measures. Asthe figures show, the spectral measures corresponding to ๐ข , ๐ข , ๐ข and ๐ข have atoms at โ1 and1, while the spectral measure corresponding to ๐ข is absolutely continuous with respect to theLebesgue measure on โ . Hence, ๐ โ1 (๐บ) = ๐ (๐บ) = {๐ข , ๐ข , ๐ข , ๐ข } . Note that the subgraph inducedby ๐ (๐บ) consists of two disconnected trees; later in Theorem 3.1 we will show that this a generalproperty of Aomoto sets.We will use repeatedly an equivalent form of Proposition 2.1 (v) above: if ๐ข โ ๐ ๐ (๐บ) , then anyeigenvector ๐ โ Ker(๐ โ ๐ด ๎ ) is identically zero on the fibre over ๐ข . We will also require a standardidentity expressing the mass assigned to ๐ โ Spec ๐ ๐ด ๎ by the spectral measure ๐ ๐ข . Lemma 2.4.
Let ๐บ be a finite graph, ๎ be its universal cover, and ๐ โ Spec ๐ ๐ด ๎ . Then if B is anyorthonormal basis for Ker(๐ โ ๐ด ๎ ) , for any ๐ข โ ๐ and ฬ๐ข โ ฮ โ1 (๐ข) , ๐ ๐ข {๐} = โ ๐โ B |๐( ฬ๐ข)| . (1)Equation (1) follows from a standard application of the Borel functional calculus, where the keyobservation is that if ๐ ๐ โถ Spec ๐ด ๎ โ โ is the indicator function of the singleton {๐} then ๐ ๐ (๐ด ๎ ) is the orthogonal projection onto ker(๐ โ ๐ด ๎ ) . This set was referred to as ๐ (1)๐ (๐บ) in [Aom91] and [ABS20]; we have dropped the superscript to lighten notation,and because we will not consider the sets ๐ (๐ผ)๐ (๐บ) for ๐ผ โ 1 which appear in that work. OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 8
3. Main ResultsOur first contribution is to strengthen a result of Aomoto [Aom91], by way of a new and moreconceptual proof. This result characterizes the induced subgraph on ๐ ๐ (๐บ) for any ๐ โ Spec ๐ ๐ด ๎ ,and relate the mass ๐{๐} to the local structure of this subgraph and neighboring vertices. Let uswrite ๐๐ ๐ (๐บ) for the set of vertices outside the Aomoto set but connected to it by an edge, cc ๐ ๐ (๐บ) for the number of connected components of the subgraph induced by ๐ ๐ (๐บ) , and define the index of ๐ as ๐ผ ๐ (๐บ) = cc ๐ ๐ (๐บ) โ |๐๐ ๐ (๐บ)|. Recall that for us a graph
๐บ = (๐ , ๐ธ, ๐, ๐) contains vertices ๐ , directed edges ๐ธ , Hermitian edgeweights ๐ โถ ๐ธ โ โ satisfying ๐ ๐ = ๐ ฬ๐ and real potential ๐ โถ ๐ โ โ . Theorem 3.1.
Let ๐บ be a finite graph, ๎ be its universal cover, and ๐ โ Spec ๐ ๐ด ๎ . Then:(i) The subgraph induced by ๐ ๐ (๐บ) is acyclic,(ii) ๐ is an eigenvalue, with multiplicity one, of the induced Jacobi operator of each connectedcomponent of this subgraph, and(iii) The density of states of ๐ด ๎ satisfies ๐{๐} = ๐ผ ๐ (๐บ)|๐ (๐บ)| (2)Assertion (i), claimed without proof in [ABS20], clarifies an ambiguity in Aomotoโs result, whichdid not rule out self-loops or multi-edges in the subgraph induced by ๐ ๐ (๐บ) ; (ii) is a new observation,and (iii) is due to Aomoto. Our new proof is combinatorial and linear algebraic, using properties ofeigenvectors of Jacobi operators on finite and infinite trees; the question of finding an alternativeto Aomotoโs original proof explaining the significance of the quantity ๐ผ ๐ (๐บ) , was posed in [ABS20,Problem 8.1]. The proofs of (i) and (ii) may be found in Section 4, and that of (iii) in Section 5.We then build on the proof of Theorem 3.1 to prove a number of novel results. First, we showthat for any graph ๐บ , the point spectrum of the periodic Jacobi operator on its unversal coveris contained in Spec ๐ด ๐บ โwith multiplicity bounded in terms of the index ๐ผ ๐ (๐บ) . In fact, we canfurther refine this result for the Jacobi operator of any cover ๐ป of ๐บ . Theorem 3.2.
Let ๐บ be a finite graph, ๐ป an ๐ -lift of ๐บ , and ๎ their common universal cover. If ๐ โ Spec ๐ ๐ด ๎ , then(i) ๐ โ Spec ๐ด ๐บ with multiplicity at least |๐ (๐บ)| โ ๐{๐} , and(ii) ๐ โ Spec ๐ด ๐ป/๐บ with multiplicity at least (๐ โ 1)|๐ (๐บ)| โ ๐{๐} ,so that the multiplicity of ๐ โ Spec ๐ด ๐ป is at least ๐|๐ (๐บ)| โ ๐{๐} . We will show at the end of Section 3.1 that these lower bounds on multiplicity need not be tight.Additionally, we prove a converse to Theorem 3.1, namely that if a graph has a set replicating thestructure of the Aomoto set for some ๐ , then its universal cover has ๐ in its point spectrum. To beprecise, let us extend the notation ๐ and cc to apply to any ๐ โ ๐ (๐บ) , and for each ๐ โ โ , let ๎ญ ๐ (๐บ) be the set of all subsets ๐ โ ๐ (๐บ) which induce an acyclic subgraph, each connected componentof which has ๐ in the spectrum of its induced Jacobi operator and such that cc (๐ ) โ |๐๐ | > 0 . Theorem 3.3.
Let ๐บ be a finite graph, and let ๎ be its universal cover. For any ๐ โ โ and ๐ โ ๎ญ ๐ (๐บ) , OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 9 (i) ๐ โ Spec ๐ด ๐บ , with multiplicity at least cc (๐ ) โ |๐๐ | .(ii) ๐ โ Spec ๐ ๐ด ๎ , and |๐ (๐บ)| โ ๐{๐} โฅ cc (๐ ) โ |๐๐ | . The lower bounds in (i) and (ii) need not be tight. For (i), this is shown at the end of Section 3.1;for (ii), when ๐ผ ๐ (๐บ) > 1 , one may choose ๐ to contain only a subset of the trees in the true Aomotoset for some ๐ โ Spec ๐ ๐ด ๎ . Furtheremore, there are cases where the inequality in (ii) is strict forelements in ๎ญ ๐ (๐บ) that are not subsets of the Aomoto set.The proofs of both Theorem 3.2 and Theorem 3.3 follow from a generalization of the latter,Theorem 6.1, which we state and verify in Section 6. The argument proceeds, roughly, by patchingtogether and extending the ๐ -eigenvectors on each component of the Aomoto set promised byTheorem 3.1 to global eigenvectors of ๐ด ๐บ and ๐ด ๐ป/๐บ . This has the interesting consequence that if ๐ โ Spec ๐ ๐ด ๎ and ๐บ , ๐บ , โฆ is a sequence of lifts of ๐บ , there is a constant fraction of |๐ (๐บ ๐ )| of ๐ -eigenvectors of ๐ด ๐บ ๐ that are localized. In contrast, under some technical assumptions, quantumergodicity results [AS19] imply that if ๐ is in the absolutely continuous part of the spectrum of ๐ด ๎ ,the proportion between the number of localized ๐ -eigenvectors of ๐ด ๐บ ๐ and |๐ (๐บ ๐ )| goes to zero.Combining Theorems 3.2 and 3.3, we find the following corollary. Note that since there arefinitely many induced subgraphs, in finite time we can find every ๐ โ โ for which ๎ญ ๐ (๐บ) isnonempty. Corollary 3.1.
Let ๐บ be a finite graph, ๎ itโs universal cover. Then for each ๐ โ โ , ๐{๐} = 1|๐ (๐บ)| max ๐โ ๎ญ ๐ (๐บ) ( cc (๐ ) โ |๐(๐ )|). (3) Moreover,
Spec ๐ ๐ด ๎ may be computed from ๐บ in finite time. Although Theorem 3.1 implies that the Aomoto set ๐ ๐ (๐บ) is an element in ๐ด ๐ (๐บ) maximizing thequantity on the right side of (3), it can happen that there is not a unique maximizer. Figure 2 givessuch an example.Figure 2. Two distinct sets of vertices (in red and blue respectively) of a graph ๐บ are shown. If ๐ โก 1 and ๐ โก 0 , it is easy to show from Corollary 3.1 that ๐ผ (๐บ) = 1 .Then both the red and the blue vertex set belong to ๎ญ (๐บ) . It will follow fromObservation 4.2 below that ๐ (๐บ) is precisely the set indicated by the red vertices.Finally, we use Theorem 3.1 and Theorem 3.3 to argue that point spectrum is rare in a certainsense. To make this precise fix ๐บ = (๐ , ๐ธ) and think of the set of possible Hermitian edge weights ๐ = (๐ ๐ ) ๐โ๐ธ and vertex potentials ๐ = (๐ ๐ฃ ) ๐ฃโ๐ as โ |๐ธ|/2 โ โ |๐ | โ โ |๐ธ|+|๐ | . Theorem 3.4.
Let
๐บ = (๐ , ๐ธ) be a finite graph with at least one cycle and ๎ be its universal cover.Assume that every vertex in ๐บ has at least ๐ min distinct neighborhs. Leaving ๐ and ๐ธ fixed, let OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 10 ๎ผ โ โ |๐ธ|+|๐ | be the set of Hermitian edge weights and potentials for which Spec ๐ ๐ด ๎ โ โ . Then, ๎ผ is asemialgebraic closed set of codimension at least max{๐ min โ 1, 1} . Remark 3.1.
Even if the bound codim( ๎ผ ) โฅ max{๐ min โ 1, 1} is tight in general, for many specificinstances a stronger bound can be obtained. We refer the reader to the discussion in Section 7 fora stronger bound that depends in a more complicated way on the combinatorial structure of ๐บ .Theorem 3.4 will be proved in Section 7 and resolves [Aom91, Question 2], which speculated thatthe existence of point spectrum was dependent on the combinatorial structure of ๐บ and not on theedge weights and potential. Results in a similar direction were obtained in [KLW12] and [KLW13].Their results are less general in the sense that they require ๐บ to have edge weights ๐ โก 1 and aloop at every vertex. However, they allow for more general potentials on the more general classof trees with finite cone type.Theorem 3.4 implies in particular that ๎ผ ๐ is an open dense set and hence that the point spectrumof ๐ด ๎ can be destroyed by adding arbitrarily small perturbations, even when ๐ด ๎ has isolatedeigenvalues. This is surprising, given a result of Kato (see [RS78, Section XII.2]) that if ๐ป is abounded self-adjoint operator and ๐ โ Spec ๐ ๐ป is isolated with dim ker(๐ โ ๐ป ) < โ , then everysufficiently small perturbation of ๐ป has non-empty point spectrum. Of course, this does notcontradict our result, since Proposition 2.1 ensures that every ๐ โ Spec ๐ ๐ด ๎ has an infinite-dimensional eigenspace. However, it is not the case that infinite-dimensional eigenspaces areunstable in general, and in many cases the phenomenon implied by Katoโs result is still present.Furthermore, it is an immediate consequence of Theorem 3.4 that ๎ผ has Lebesgue measure zero.This can interpreted as an almost sure spectral delocalization result, since it implies that under arandom absolutely continuous perturbation (with respect to the Lebesgue measure) of the edgeweights and potential of ๐บ , the spectrum of ๐ด ๎ becomes purely absolutely continuous.We conclude this section by giving some applications of the results presented above.3.1. Point Spectrum of Biregular Trees.
Let ๐บ be the complete bipartite graph ๐พ ๐,๐ for someintegers ๐ > ๐ , and denote by ๐ ๐ and ๐ ๐ the vertex components of ๐บ having ๐ and ๐ verticesrespectively. We will first analyze the case when ๐ โก 0 and ๐ is any Hermitian edge weighting. Itis easy to see that ๐ ๐ is a set satisfying the conditions of Theorem 3.3 for ๐ = 0 , and hence that ๐{0} โฅ ๐โ๐๐+๐ . Then by Theorem 3.1, ๐ (๐บ) โ โ and moreover ๐ผ (๐บ) โฅ ๐ โ ๐ . In the forthcomingObservation 4.2, we will show that that when ๐ โก 0 (regardless of the structure of ๐บ ) the set ๐ (๐บ) is an independent set in ๐บ , which together with the previous observations implies that in fact ๐ (๐บ) = ๐ ๐ . So by Theorem 3.1, when ๐ โก 0 , ๐{0} = ๐โ๐๐+๐ and the vectors in ker(๐ด ๎ ) are supportedon the fibre of ๐ ๐ , which extends a result of Godsil and Mohar [GM88].In the case of arbitrary real potential ๐ , the existence and location of eigenvalues of ๐ด ๎ dependon the particular choice of ๐ , and moreover by Theorem 3.4 we know that one may choose ๐ suchthat ๐ด ๎ has no point spectrum. This discussion resolves Problems 8.6 and 8.7 posed in [ABS20].Finally, we note in passing that in this case, when ๐ โก 0 we have ๐ผ (๐บ) = ๐ โ ๐ but the multiplicity In a previous version of this paper we only proved that ๎ผ is a closed set of Lebesgue measure 0 by showing that ๎ผ was contained in an algebraic set of codimension 1. We thank Barry Simon for pointing out this stronger versionof the theorem and suggesting a sketch of the proof. By an independent in ๐บ we mean a set ๐ โ ๐ (๐บ) which induces a subgraph with no edges.
OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 11 of zero in
Spec ๐ด ๐บ is ๐ + ๐ โ 2 , which shows that the bounds on the multiplicity given in Theorems3.2 and 3.3 may not be tight.3.2. Non-isolated Point Spectrum.
Let ๐บ be a finite graph and let ๐ = |๐ (๐บ)| . Sunadaโs gaplabeling theorem (see[ABS20, Theorem 5.1] or [GVK19, Theorem 1.8]) states that Spec ๐ด ๎ is adisjoint union of at most ๐ (possibly degenerate) closed intervals typically called bands , and that if ๐ต is one of these bands then ๐(๐ต) = ๐/๐ for some ๐ โ [๐] . If ๐ โ Spec ๐ ๐ด ๎ is isolated then {๐} is a(degenerate) band of Spec ๐ด ๎ . This is the case, for example, when ๐ = 0 , ๐บ is a bipartite biregulargraph with components of different sizes, ๐ โก 1 and ๐ โก 0 . Here we will show that it is possiblefor ๐ ๐ด ๎ to lie inside a non-degenerate band of Spec ๐ด ๎ . Our argument is similar in spiritto the one used at the end of Section 5 of [ABS20] for an unrelated purpose.Set ๐ โก 0 , noting that ๐ is symmetric about zero from Lemma 2.2, and assume that ๐ ๐ด ๎ and that ๐ โ ๐ผ (๐บ) is an odd number. Figure 3 shows two instances where these conditions aremet. If {0} is an isolated point of Spec ๐ด ๎ then any band in Spec ๐ด ๎ is either disjoint from (0, โ) or fully contained in this infinite interval. Hence, by Sunadaโs gap labeling theorem ๐(0, โ) = ๐/๐ for some integer ๐ โ [๐] . On the other hand, since ๐ is symmetric, ๐(โโ, 0) = ๐/๐ . Finally, byTheorem 3.1 ๐{0} = ๐ผ (๐บ)/๐ . Putting all these observations together we get (๐บ)๐ , which contradicts the assumption that ๐ โ ๐ผ (๐บ) is odd.Figure 3. As in Section 3.1, a combination of Observation 4.2 and Theorem 3.3yields that the red vertices are the Aomoto set associated to 0 for each of the graphsdisplayed above. In both cases ๐ผ (๐บ) = 1 while |๐ (๐บ)| is even.4. Acyclic Nature of Aomoto SetsIn this section we will prove the first two assertions of Theorem 3.1, namely that if ๐ โ Spec ๐ ๎ ,then the Aomoto set ๐ ๐ (๐บ) is acyclic, and ๐ is an eigenvalue of the induced Jacobi operator oneach of its connected components. We begin by generalizing to the infinite case a result of Fielderregarding eigenvectors of finite trees [Fie75, Proposition 1]. Lemma 4.1.
Let ๐ be a locally finite tree with Hermitian edge weights and potential ๐ โถ ๐ธ(๐ ) โ โ and ๐ โถ ๐ (๐ ) โ โ respectively, and Jacobi operator ๐ด ๐ . If ๐ โ Ker(๐ โ ๐ด ๐ก ) and ๐(๐ฃ) โ 0 for everyvertex ๐ฃ โ ๐ (๐ ) , then dim Ker(๐ โ ๐ด ๐ ) = 1 .Proof. Choose a root ๐ for ๐ , and for each vertex ๐ฃ , write ๐(๐ฃ) for its unique parent, ๐ ๐ฃ the infinitesub-tree emanating from ๐ฃ away from its parent and ๐| โฅ๐ฃ for the restriction of ๐ to the subtree ๐ ๐ฃ .As ๐ is acyclic, it has no multi-edges or self-loops, and there is no ambiguity in writing ๐ ๐ฃโ๐ข for OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 12 the weight of the unique edge with source ๐ข and terminal ๐ฃ . We then have (๐ โ ๐ด ๐ )๐| โฅ๐ฃ = โ ๐ขโ๐ (๐ ๐ฃ ) ๐๐(๐ข)๐ฟ ๐ข โ โ ๐ขโ๐ (๐ ๐ฃ )โงต{๐ฃ} (๐ด ๐ ๐)(๐ข)๐ฟ ๐ข โ (๐ ๐ฃ ๐(๐ฃ) + โ ๐ฅโถ๐(๐ฅ)=๐ฃ ๐ ๐ฃโ๐ฅ ๐(๐ฅ)) ๐ฟ ๐ฃ โ ๐ ๐(๐ฃ)โ๐ฃ ๐(๐ฃ)๐ฟ ๐(๐ฃ) = ๐๐(๐ฃ)๐ฟ ๐ฃ โ ((๐ด ๐ ๐)(๐ฃ) โ ๐ ๐ฃโ๐(๐ฃ) ๐(๐(๐ฃ))) ๐ฟ ๐ฃ โ ๐ ๐(๐ฃ)โ๐ฃ) ๐(๐ฃ)๐ฟ ๐(๐ฃ) = ๐ ๐ฃโ๐(๐ฃ) ๐(๐(๐ฃ))๐ฟ ๐ฃ โ ๐ ๐(๐ฃ)โ๐ฃ ๐(๐ฃ)๐ฟ ๐(๐ฃ) for any ๐ฃ โ ๐ . Now, let ๐ โ Ker(๐ โ ๐ด ๐ ) . As ๐ด ๐ is self-adjoint and ๐ real, ๐ )๐| โฅ๐ฃ โฉ = ๐ ๐ฃโ๐(๐ฃ) ๐(๐(๐ฃ))๐ (๐ฃ) โ ๐ ๐(๐ฃ)โ๐ฃ ๐(๐ฃ)๐ (๐(๐ฃ)), which implies ๐ (๐ฃ)๐ (๐(๐ฃ)) = ๐ ๐(๐ฃ)โ๐ฃ ๐ ๐ฃโ๐(๐ฃ) ๐(๐ฃ)๐(๐(๐ฃ)) . This identity holds for every ๐ โ Ker(๐ โ ๐ด ๎ ) , including ๐ itself, so we obtain ๐ (๐ฃ)๐ (๐(๐ฃ)) = ๐ ๐(๐ฃ)โ๐ฃ ๐ ๐ฃโ๐(๐ฃ) ๐(๐ฃ)๐(๐(๐ฃ)) = ๐ ๐(๐ฃ)โ๐ฃ ๐ ๐ฃโ๐(๐ฃ) ๐ ๐(๐ฃ)โ๐ฃ ๐ ๐ฃโ๐(๐ฃ) ๐(๐ฃ)๐(๐(๐ฃ)) = |๐ ๐(๐ฃ)โ๐ฃ | |๐ ๐ฃโ๐(๐ฃ) | ๐(๐ฃ)๐(๐(๐ฃ)) = ๐(๐ฃ)๐(๐(๐ฃ)) ; in the final equality we have used conjugate symmetry of the edge weights. Since ๐| โฅ๐ = ๐ โKer(๐ โ ๐ด ๎ ) , ๐ is unconstrained at the root, and the above equation propagates a condition downthe tree that ๐ = ๐ โ ๐ (๐ )/๐(๐ ) . (cid:3) We now prove that the subgraph of ๐บ induced by ๐ ๐ (๐บ) is a forest, and that ๐ is an eigenvalue,with multiplicity one, of the induced Jacobi operator of each of its connected components. Proof of Theorem 3.1(i-ii).
Assume ๐ is in the point spectrum of ๐ด ๎ , and let ๐บ โฒ be a connectedcomponent of the subgraph induced by ๐ ๐ (๐บ) . Let ๎ โฒ be the universal cover of ๐บ โฒ . If we view ๎ โฒ as a subgraph of ๎ then any vector in Ker(๐ โ ๐ด ๎ ) vanishes on the boundary of ๎ โฒ in ๎ , and thusrestricts to a ๐ -eigenvector of ๎ โฒ . Hence ๐ ๐ (๐บ โฒ ) = ๐ (๐บ โฒ ) by Proposition 2.1(v), and we can nowuse the following observation, which follows from Zornโs lemma and appeared as [Nyl98, Lemma7]. Observation 4.1. If ๐ ๐ (๐บ โฒ ) = ๐ (๐บ โฒ ) then there is an ๐ โ Ker(๐ โ ๐ด ๎ โฒ ) satisfying ๐(๐ข) โ 0 for every ๐ข โ ๐ ( ๎ โฒ ) .Combining Observation 4.1 and Lemma 4.1 we conclude finally that dim Ker(๐ โ ๐ด ๎ โฒ ) = 1 , andthus, by Proposition 2.1(i), that ๐บ โฒ is acyclic. This further implies that ๎ โฒ = ๐บ โฒ , which proves thesecond assertion. (cid:3) In the course of the proof above we showed the following fact, which will be of repeated usethroughout the paper.
Lemma 4.2.
Let ๐บ be a finite graph with Hermitian edge weights and potential, with ๐ โ Spec ๐ ๐ด ๎ and ๐ , โฆ , ๐ ๐ the Aomoto trees of ๐บ associated to ๐ . Then, for every ๐ โ [๐] there is a unique (up tophase) unit vector ๐ ๐ โ Ker(๐ โ ๐ด ๐ ๐ ) satisfying ๐ (๐ข) โ 0 for every ๐ข โ ๐ (๐ ๐ ) . For use in the next section, we record one consequence of the above lemma.
OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 13
Observation 4.2.
Let ๐บ be a graph with ๐ โก 0 , ๎ its universal cover, and assume ๐ ๐ด ๎ .Then ๐ (๐บ) is an independent set in ๐บ . Proof.
By Lemma 4.2, each Aomoto tree of ๐บ must have a unique, everywhere nonzero eigenvectorin the kernel of its Jacobi operator. On the other hand, a vector in the kernel of a Jacobi operatorwith potential zero for a tree cannot be nonzero at the parent of a leaf. Thus every Aomoto tree of ๐บ is an isolated vertex as desired. (cid:3)
5. Aomotoโs Index FormulaIn this section we complete the proof of Theorem 3.1 by verifying the formula in equation (2): if ๐ โ Spec ๐ ๐ด ๎ , then |๐ (๐บ)| โ ๐{๐} = ๐ผ ๐ (๐บ). Our strategy will be to reduce the problem to the proof of an analogous result on an auxiliarybipartite graph ๐บ โฒ .5.1. Constructing the Auxilliary Graph.
Let ๐ , โฆ , ๐ ๐ be the Aomoto trees of ๐บ = (๐ , ๐ธ, ๐, ๐) associated to ๐ , write ๎ฒ ๐ for the set of disjoint copies of ๐ ๐ in ๎ = ( ๎ , ๎ฑ , (cid:97) , (cid:98) ) obtained by lifting ๐ ๐ , and let ๎ฒ = โ ๐๐=1 ๎ฒ ๐ . Note that ๎ฒ is a subforest of ๎ and all of its subtrees are isomorphic tosome Aomoto tree of ๐บ .By Lemma 4.2 there is for each ๐ ๐ a unique (up to phase) vector ๐ ๐ โ Ker(๐ โ ๐ด ๐ ๐ ) with unit normand nonzero entries. Take any ๐ โ Ker(๐ โ ๐ด ๎ ) . For every ๐ โ ๎ฒ , by definition of the Aomoto setit holds that ๐ is zero on all vertices in ๐๐ (๐) . Hence, the restriction of ๐ to any ๐ โ ๎ฒ ๐ induces aneigenvector of ๐ด ๐ ๐ . This implies that ๐ can be decomposed as ๐ = โ ๐โ ๎ฒ ๐ผ ๐ ๐ ๐ , (4)where ๐ผ ๐ โ โ are coefficients and the ๐ ๐ โ ๐ (๐ ( ๎ )) are inclusions of the ๐ -eigenvectors of eachAomoto tree: ๐ ๐ (๐ฃ) = {๐ ๐ (ฮ(๐ฃ)) if ๐ฃ โ ๐ (๐) and ๐ โ ๎ฒ ๐ otherwise . We now construct ๐บ โฒ = (๐ โฒ , ๐ธ โฒ , ๐ โฒ , ๐ โฒ ) ; the process is summarized in Figure 4. First, ๐ โฒ is obtainedfrom ๐ by deleting every vertex outside ๐ ๐ (๐บ) โช ๐๐ ๐ (๐บ) , and contracting each Aomoto tree ๐ ๐ to a single vertex ๐ก ๐ . Write {๐ก , ..., ๐ก ๐ } = ๐ โ ๐ โฒ , and identify ๐๐ ๐ (๐บ) with ๐๐ . Now, for each ๐ฃ โ ๐๐ = ๐๐ ๐ (๐บ) and each edge ๐ โ ๐ (๐ฃ) โ ๐ธ whose source is in a tree ๐ ๐ , include an edge ๐ โฒ โ ๐ธ โฒ with ๐ (๐ โฒ ) = ๐ฃ and ๐ (๐ โฒ ) = ๐ก ๐ , and set its weight as ๐ ๐ โฒ = ๐ ๐ ๐ ๐ (๐ (๐)). (5)This process is mirrored to construct an edge ๐ โฒ โ ๐ธ โฒ from any ๐ โ ๐ (๐ฃ) โ ๐ธ whose terminal is in ๐ ๐ ; no other edges are included. Finally, the potential ๐ โฒ is identically zero.We have arranged things so that ๐บ โฒ is bipartite, with connected components ๐บ โฒ1 , ..., ๐บ โฒ๐ , whoserespective covers we will denote ๎ โฒ1 , ..., ๎ โฒ๐ . We may also construct a new infinite graph ๎ โฒ =( ๎ โฒ , ๎ฑ โฒ , (cid:97) โฒ , (cid:98) โฒ ) from ๎ analogously to the construction of ๐บ โฒ from ๐บ : by deleting the fibres over anyvertex outside ๐ ๐ (๐บ) โช ๐๐ ๐ (๐บ) , contracting each tree ๐ โ ๎ฒ โ ๎ into a single vertex ๐ข ๐ , includingfor each ๐ โ ๎ฑ with ends in ฮ โ1 (๐ ๐ (๐บ)) and ฮ โ1 (๐๐ ๐ (๐บ)) a corresponding edge ๐ โฒ โ ๎ฑ โฒ with endsin the contraction if ฮ โ1 (๐ ๐ (๐บ)) and its boundary, and reweighting any such edge according to (5). OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 14
Figure 4. On the left an example of a graph ๐บ with Aomoto trees in red. On theright its auxiliary graph ๐บ โฒ , where each tree ๐ ๐ has been contracted into a vertex ๐ก ๐ and the blue edges have been removed.This ๎ โฒ consists of countably many copies of each ๎ โฒ๐ , and is a cover of ๐บ โฒ via a map ฮ โฒ ; note that (ฮ โฒ ) โ1 (๐ ) = {๐ข ๐ โถ ๐ โ ๎ฒ } , the contraction of ฮ โ1 (๐ ๐ (๐บ)) . With this setup and the decompositionin (4), any ๐ โ Ker(๐ โ ๐ด ๎ ) gives rise to a vector ๐ โฒ โ ๐ (๐ ( ๎ โฒ )) in a natural way: ๐ = โ ๐โ ๎ฒ ๐ผ ๐ ๐ ๐ โฆ ๐ โฒ = โ ๐โ ๎ฒ ๐ผ ๐ ๐ฟ ๐ข ๐ . Observation 5.1.
The map ๐ โฆ ๐ โฒ is an isometric inclusion of Ker(๐ โ ๐ด ๎ ) in Ker ๐ด ๎ โฒ . Proof.
Preservation of norm is immediate since ๐ ๐ is a unit vector, and since the map is an isometryit is injective; it remains only to show that Ker(๐ โ ๐ด ๎ ) is mapped to Ker ๐ด ๎ โฒ . The vector ๐ โฒ isidentically zero on the fibre over ๐๐ and thus (๐ด ๐ โฒ ๐ โฒ )(๐ข) = 0 for any ๐ข โ (ฮ โฒ ) โ1 (๐ ) . It remains onlyto consider ๐ฃ โ (ฮ โฒ ) โ1 (๐๐ ) , which as above we may identify with ฮ โ1 (๐๐ ๐ (๐บ)) โ ๎ . For each edge ๐ โ ๐ (๐ฃ) โ ๎ฑ write ๐ ๐ for the tree in ๎ฒ to which ๐ (๐) belongs, so that the reweighting in (5) gives ๐ โฒ๐ โฒ = ๐ ๐ ๐ ๐ ๐ (๐ (๐)) . As the potential ๐ โฒ is identically zero and ๐ and ๐ โฒ vanish outside the fibres over ๐ ๐ (๐บ) and ๐ respectively, we have (๐ด ๎ โฒ ๐ โฒ )(๐ฃ โฒ ) = (cid:98) ๐ฃ โฒ ๐ โฒ (๐ฃ โฒ ) + โ ๐ โฒ โ๐(๐ฃ โฒ )โ ๎ฑ โฒ (cid:97) โฒ๐ โฒ ๐ โฒ (๐ (๐ โฒ ))= โ ๐โ๐(๐ฃ)โ ๎ฑ โถ๐(๐)โฮ โ1 (๐ ๐ (๐บ)) (cid:97) ๐ ๐ ๐ ๐ (๐ (๐))๐ผ ๐ ๐ = (cid:98) ๐ฃ ๐(๐ฃ) + โ ๐โ๐(๐ฃ)โ ๎ฑ (cid:97) ๐ ๐(๐ (๐))= (๐ด ๎ ๐)(๐ฃ)= ๐๐(๐ฃ) = 0. In the third line, note that some edges in ๐ (๐ฃ) โ ๎ฑ have a source outside of the fibre over ๐ ๐ (๐บ) ,but that ๐ is identically zero there. (cid:3) Immediately from this observation, we can conclude that ๐ ๐ด ๎ โฒ . Moreover, as ๎ โฒ iscomprised of disjoint copies of the ๎ โฒ๐ โs, ๐ด ๎ โฒ restricts to ๐ด ๎ โฒ๐ on each one, and thus ๐ ๐ด ๎ โฒ๐ for at least one ๎ โฒ๐ . Our next observation characterizes the associated Aomoto set on ๐บ โฒ๐ . Recallthat ๐บ โฒ is bipartite with vertex classes ๐ and ๐๐ , and let us write ๐ ๐ and ๐๐ ๐ for the correspondingclasses of vertices in each connected component ๐บ โฒ๐ . Observation 5.2. ๐ (๐บ โฒ๐ ) = ๐ ๐ . OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 15
Proof.
By Proposition 2.1, the definition of the map ๐ โฆ ๐ โฒ , and Observation 5.1, we immediatelyhave the inclusion ๐ ๐ โ ๐ (๐บ โฒ๐ ) , since any ๐ โ Ker(๐ โ ๐ด ๎ ) maps to ๐ โฒ supported only on the fibreover ๐ ๐ . On the other hand, from Lemma 4.2, we know that ๐ (๐บ โฒ๐ ) is an independent set, and ๐ ๐ isa maximal independent set in ๐บ โฒ๐ by definition of ๐๐ ๐ . (cid:3) Finally, we can strengthen Observation 5.1
Observation 5.3.
The map ๐ โฆ ๐ โฒ gives an isomorphism between Ker(๐ โ ๐ด ๎ ) and Ker ๐ด ๎ โฒ . Proof.
We noted above that ๐ด ๎ โฒ decomposes as a direct sum of the Jacobi operators on the copiesof ๎ โฒ๐ comprising ๎ โฒ . By applying Observation 5.2 separately to each copy, any ๐ โ Ker ๐ด ๎ โฒ issupported only on (ฮ โฒ ) โ1 (๐ ) . Thus the adjoint of the map ๐ โฆ ๐ โฒ takes any vector ๐ โ Ker ๐ด ๎ โฒ toone in ๐ (๐ ( ๎ )) : ๐ = โ ๐โ ๎ฒ ๐ผ ๐ ๐ฟ ๐ข ๐ โฆ โ ๐โ ๎ฒ ๐ผ ๐ ๐ ๐ . Once again this is clearly injective and norm-preserving, and a parallel argument to Observation 5.1shows that it takes
Ker ๐ด ๎ โฒ into Ker(๐ โ ๐ด ๎ ) . (cid:3) We can finally relate the density of states of ๐ด ๎ to those of the ๐ด ๎ โฒ๐ . Observation 5.4.
Let ๐ be an Aomoto tree in ๐บ , and ๐ก โ ๐ (๐บ โฒ๐ ) its contraction in a component ๐บ โฒ๐ of ๐บ โฒ . Writing ๐ โฒ๐ก for the spectral measure of ๐ก in ๐ด ๎ โฒ๐ , and for ๐ ๐ฃ for the spectral measure of ๐ด ๎ for each ๐ฃ โ ๐ (๐ ) โ ๐ (๐บ) , we have โ ๐ฃโ๐ (๐ ) ๐ ๐ฃ {๐} = ๐ โฒ๐ก {0}. Proof.
Choose a copy ฬ๐ of ๐ in its fibre in ๎ , and let ฬ๐ก be the contraction of ฬ๐ in ๎ โฒ๐ โ ๎ โฒ . Byconstruction, for each ๐ โ ker(๐ โ ๐ด ๎ ) , ๐ โฒ ( ฬ๐ก) = โ ฬ๐ฃโ๐ ( ฬ๐ ) ๐( ฬ๐ฃ) . Now, let B โฒ๐ be an orthonormal basis of Ker ๐ด ๎ โฒ๐ . By Observation 5.3 this is the image of someorthonormal set B ๐ in Ker(๐ โ ๐ด ๎ ) . In particular, recalling our construction of ๎ โฒ from ๎ bydeleting vertices and contracting Aomoto trees, our chosen copy of ๎ โฒ๐ in ๎ โฒ pulls back to a subtree ๎ ๐ of ๎ containing ฬ๐ . Moreover, B ๐ is an orthonormal basis for the orthogonal projection of Ker(๐ โ ๐ด ๎ ) to the subspace of ๐ ( ๎ ) supported on the vertices of ๎ ๐ , and we can therefore augment B ๐ to an orthonormal basis B of Ker(๐ โ ๐ด ๎ ) , whose additional vectors vanish on ๎ ๐ .We now use Lemma 2.4 to compute โ ๐ฃโ๐ (๐ ) ๐ ๐ฃ {๐} = โ ฬ๐ฃโ๐ ( ฬ๐ ) โ ๐โ B ๐( ฬ๐ฃ) = โ ฬ๐ฃโ๐ ( ฬ๐ ) โ ๐โ B ๐ ๐( ฬ๐ฃ) = โ ๐โ B ๐ ๐ โฒ ( ฬ๐ก) = โ ๐ โฒ โ B โฒ๐ ๐ โฒ ( ฬ๐ก) = ๐ โฒ๐ก {0}. (cid:3) Analyzing the Auxiliary Graph.
This section is devoted to the final observation of ourproof:
Observation 5.5.
Fix ๐ โ [๐] and assume that ๐ ๐ด ๎ โฒ๐ . Then โ ๐กโ๐ ๐ ๐ โฒ๐ก {0} = ๐ผ (๐บ โฒ๐ ). OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 16
This will finish the proof, as combining Observations 5.4 and 5.5 and recalling the construction of ๐บ โฒ gives |๐ (๐บ)| โ ๐{๐} = โ ๐ขโ๐ ๐ (๐บ) ๐ ๐ข {๐} = โ ๐กโ๐ ๐ โฒ๐ก {0}= โ ๐โ[๐] ๐ผ (๐บ โฒ๐ ) = โ ๐โ[๐] |๐ ๐ | โ |๐๐ ๐ |= |๐ | โ |๐๐ | = cc ๐ ๐ (๐บ) โ |๐๐ ๐ (๐บ)|= ๐ผ ๐ (๐บ). Proof of Observation 5.5.
Let ๐ โฒ be the DOS of ๐ด ๎ โฒ๐ . Let ๐ฟ , ๐ฟ , โฆ be a sequence of finite lifts of ๐บ โฒ๐ with covering maps ๐ ๐ โถ ๐ฟ ๐ โ ๐บ โฒ๐ . By Lemma 2.3 we may choose the ๐ฟ ๐ with girth goingto infinity. Since ๐บ โฒ๐ is bipartite with zero potential, the Jacobi matrices ๐ด ๐ฟ ๐ and ๐ด ๎ โฒ๐ have thefollowing block structure ๐ด ๐ฟ ๐ = ( 0 ๐ ๐๐ ๐ ๐ and ๐ด ๎ โฒ๐ = ( 0 ๐ ๐โ ๐ โ where for ๐ โ โ โช {โ} the domain and range of ๐ ๐ correspond to the fibers of ๐๐ ๐ and ๐ ๐ respectively. Note that ๐ด ๐ = ๐ ๐๐ ๐ ๐ โ ๐ ๐ ๐ ๐๐ and ๐ด ๎ โฒ๐ = ๐ ๐โ ๐ โ โ ๐ โ ๐ ๐โ .Let ๐ ๐ ๐ ๐ ๐๐ and ๐ ๐ ๐๐ ๐ ๐ be the empirical spectral distributions of ๐ ๐ ๐ ๐๐ and ๐ ๐๐ ๐ ๐ respectively. Fixa positive integer ๐ and note that, since ๐ฟ ๐ is bipartite, the terms in tr(๐ ๐๐ ๐ ๐ ) ๐ are in one-to-onecorrespondence with the closed walks of length in ๐ฟ ๐ that start and end at the same vertexin ๐ โ1๐ (๐๐ ๐ ) . Moreover, by the girth assumption, for large enough ๐ it holds that the value of thediagonal entries of ๐ด ๐ are constant on each fiber ๐ โ1๐ (๐ฃ) for every ๐ฃ โ ๐ (๐บ โฒ๐ ) and coincide withthe respective diagonal entries of ๐ด ๎ โฒ๐ . Hence, if we write ๐ ๐ฃ for the spectral measure of ๐ข for theoperator ๐ด ๎ โฒ๐ , then by the method of moments ๐ ๐ ๐๐ ๐ ๐ and ๐ ๐ ๐ ๐ ๐๐ converge weakly to ๐ ๐๐ ๐ = 1|๐๐ ๐ | โ ๐ฃโ๐๐ ๐ ๐ ๐ฃ and ๐ ๐ ๐ = 1|๐ ๐ | โ ๐ฃโ๐ ๐ ๐ ๐ฃ . (6)Since ๐ (๐บ โฒ๐ ) = ๐ ๐ , equation (6) implies ๐ ๐๐ ๐ {0} = 0 . If it were the case that |๐๐ ๐ | > |๐ ๐ | , we wouldhave by standard properties of matrices that the spectrum of ๐ ๐๐ ๐ ๐ is equal to that of ๐ ๐ ๐ ๐๐ , plusthe eigenvalue zero with multiplicity at least |๐๐ ๐ | โ |๐ ๐ | , and thus |๐๐ ๐ | โ |๐ ๐ ||๐๐ ๐ | โค lim sup ๐โโ ๐ ๐ ๐ ๐ ๐๐ {0} โค ๐ ๐๐ ๐ {0} = 0, a contradiction. Thus ๐ผ (๐บ โฒ๐ ) โฅ 0 . Applying the same matrix property a second time, we have ๐ ๐ ๐ ๐ ๐๐ = (1 โ ๐ผ (๐บ โฒ๐ )|๐ ๐ | ) ๐ ๐ ๐๐ ๐ ๐ + ๐ผ (๐บ โฒ๐ )|๐ ๐ | ๐ฟ . By weak convergence, and as compact measures are determined by their moments, ๐ ๐ ๐ = (1 โ ๐ผ (๐บ โฒ๐ )|๐ ๐ | ) ๐ ๐๐ ๐ + ๐ผ (๐บ โฒ๐ )|๐ ๐ | ๐ฟ , and thus โ ๐กโ๐ ๐ ๐ โฒ๐ก {0} = |๐ ๐ |๐ ๐ ๐ {0} = (|๐ ๐ | โ ๐ผ (๐บ โฒ๐ )) ๐ ๐๐ ๐ {0} + ๐ผ (๐บ โฒ๐ )๐ฟ {0} = ๐ผ (๐บ โฒ๐ ). (cid:3) OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 17
6. A Generalized Converse to Aomotoโs TheoremIn this section we will prove the following generalization of Theorem 3.3, and use it to proveour other main contribution, Theorem 3.2.
Theorem 6.1.
Let ๐บ be a fintite graph, ๎ its universal cover, ๐ โถ ๐ธ(๐บ) โ U (๐) a set of unitary-valued edge weights satisfying ๐ โ๐ = ๐ ฬ๐ for every ๐ โ ๐ธ(๐ ) , and ๐ด ๐บ,๐ the unitary-weighted Jacobioperator acting on ๐ โ ๐ (๐ (๐บ)) โ โ ๐ as (๐ด ๐บ,๐ ๐)(๐ฃ) = ๐ ๐ฃ ๐(๐ฃ) + โ ๐โ๐(๐ฃ) ๐ ๐ ๐ ๐ ๐(๐ (๐)) โ โ ๐ . If some set of vertices
๐ โ ๐ (๐บ) induces an acyclic subgraph, every component of which has ๐ in thespectrum of its induced unitary-weighted Jacobi operator and cc (๐ ) โ |๐๐ | > 0 , then ๐ โ Spec ๐ด ๐บ,๐ with multiplicity at least ๐( cc (๐ ) โ |๐๐ |) . We begin with a lemma regarding unitary-weighted Jacobi operators of finite trees.
Lemma 6.1.
Let ๐ be a finite tree, ๐ โถ ๐ธ(๐ ) โ U (๐) a set of unitary-valued edge weights satisfying ๐ โ๐ = ๐ ฬ๐ for every ๐ โ ๐ธ(๐ ) , and ๐ด ๐ ,๐ the associated unitary-weighted Jacobi operator. If ๐ โ Spec ๐ด ๐ ,then ๐ โ Spec ๐ด ๐ ,๐ with multiplicity at least ๐ .Proof. As in the proof of Lemma 4.1, we will choose a root ๐ of ๐ , for each vertex ๐ฃ write ๐(๐ฃ) for its unique parent and ๐(๐ฃ) for its set of children, and, since ๐ is acyclic, write ๐ฃ โ ๐ข for theunique edge with source ๐ข and terminal ๐ฃ . By absorbing ๐ into the potential, it suffices to studythe case when ๐ = 0 . So, let ๐ โ Ker ๐ด ๐ ; we will produce a subspace of dimension ๐ contained in Ker ๐ด
๐ ,๐ .Fix a vector ๐ โ โ ๐ and set ๐ (๐ ) = ๐ . For each vertex ๐ฃ โ ๐ (๐ ) , letting ๐พ ๐ฃ denote the directededges in the unique shortest path from ๐ฃ to ๐ , set ๐ (๐ฃ) = โ ๐โ๐พ ๐ฃ ๐ โ๐ โ ๐(๐ฃ) โ ๐ We claim that ๐ โ Ker ๐ด
๐ ,๐ ; since ๐ was arbitrary, this will complete the proof.At the root, we have (๐ด ๐ ,๐ ๐ )(๐ ) = ๐ ๐ ๐(๐ )๐ + โ ๐ขโ๐(๐) ๐ ๐โ๐ข ๐ ๐โ๐ข ๐ ๐ขโ๐ ๐(๐ข)๐ = (๐ ๐ ๐(๐ ) + โ ๐ขโ๐(๐) ๐ ๐โ๐ข ๐(๐ข)) ๐ = 0, since ๐ ๐โ๐ข ๐ ๐ขโ๐ = 1 and ๐ โ Ker ๐ด ๐ . Similarly, for any other vertex ๐ฃ โ ๐ (๐ ) , conjugate symmetryof the unitary weights gives us (๐ด ๐ ,๐ ๐ )(๐ฃ) = ๐ ๐ฃ โ ๐โ๐พ ๐ฃ ๐ โ๐ ๐(๐ฃ)๐ + ๐ ๐ฃโ๐(๐ฃ) ๐ ๐ฃโ๐(๐ฃ) โ ๐โ๐พ๐(๐ฃ) ๐ โ๐ ๐(๐(๐ฃ))๐ + โ ๐ขโ๐(๐ฃ) ๐ ๐ฃโ๐ข ๐ ๐ฃโ๐ข โ ๐โ๐พ ๐ข ๐ โ๐ ๐(๐ข)๐ = (๐ ๐ฃ + ๐ ๐ฃโ๐(๐ฃ) ๐(๐(๐ฃ)) + โ ๐ขโ๐(๐ฃ) ๐ ๐ฃโ๐ข ๐(๐ข)) โ ๐โ๐พ ๐ฃ ๐ โ๐ ๐ = 0. (cid:3) We can now proceed with the proof.
OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 18
Proof of Theorem 6.1.
For any Aomoto tree ๐ of ๐บ , the induced Jacobi operator ๐ด ๐ has ๐ in itsspectrum. By Lemma 6.1, the induced unitary-weighted Jacobi operator ๐ด ๐ ,๐ thus satisfies dim Ker(๐ โ ๐ด
๐ ,๐ ) โฅ ๐ , and therefore the space โจ ๐ โ๐ ๐ (๐บ) Ker(๐ โ ๐ด
๐ ,๐ ) โ ๐ (๐ ๐ (๐บ)) โ โ ๐ โ ๐ (๐ ) โ โ ๐ has dimension ๐ cc ๐ ๐ (๐บ) . We will show that it contains a subspace of dimension ๐๐ผ ๐ (๐บ) which isitself contained in Ker(๐ โ ๐ด ๐บ ๐ ) .For each ๐ฃ โ ๐ ๐ (๐บ) , let ฮ ๐ฃ โถ ๐ (๐ ) โ โ ๐ โ ๐ (๐ฃ) โ โ ๐ be the orthogonal projection to the โ ๐ -valued functions in ๐ (๐ ) โ โ ๐ supported on ๐ฃ . For each ๐ข โ ๐๐ ๐ (๐บ) , there is an operator ๐ ๐ข = โ ๐โ๐(๐ข)๐(๐)โ๐ ๐ (๐บ) ๐ ๐ ๐ ๐ ฮ ๐(๐) โถ โจ ๐ โ๐ ๐ (๐บ) Ker(๐ โ ๐ด
๐ ,๐ ) โ ๐ (๐ฃ) โ โ ๐ and we define ๐ = โจ ๐ขโ๐๐ ๐ (๐บ) ๐ ๐ข โถ โจ ๐ โ๐ ๐ (๐บ) Ker(๐ โ ๐ด
๐ ,๐ ) โ ๐ (๐๐ ๐ (๐บ)) โ โ ๐|๐๐ ๐ (๐บ)| . Counting dimensions, dim Ker ๐ โฅ ๐๐ผ ๐ (๐บ) , and we will show that Ker ๐ โ Ker(๐ โ ๐ด
๐บ,๐ ) .Let ๐ โ Ker ๐ ; since the latter is a subspace of ๐ (๐ ๐ (๐บ)) โ โ ๐ โ ๐ (๐ ) โ โ ๐ , we have ๐ (๐ข) = 0 forevery ๐ข โ ๐ ๐ (๐บ) . This immediately gives ((๐ โ ๐ด ๐บ,๐ )๐ ) (๐ข) = 0 for any ๐ข outside the Aomoto setand its boundary, as ๐ is identically zero on ๐ข and its neighbors. On the other hand, if ๐ข belongsto some tree ๐ in the Aomoto set, then because Ker ๐ โ โจ
๐ โ๐ ๐ (๐บ) Ker(๐ โ ๐ด
๐ ,๐ ) and ๐ vanishes on ๐๐ ๐ (๐บ) , we have ((๐ โ ๐ด ๐บ,๐ )๐ )(๐ข) = ๐๐ (๐ข) + ๐ ๐ข ๐ (๐ข) + โ ๐โ๐(๐ข) ๐ ๐ ๐ ๐ ๐ (๐ (๐))= ๐๐ (๐ข) + ๐ ๐ข ๐ (๐ข) + โ ๐โ๐(๐ข)๐(๐)โ๐ ๐ ๐ ๐ ๐ ๐ (๐ (๐)) = ((๐ โ ๐ด ๐ ,๐ )๐ )(๐ข) = 0
It remains to check that ((๐ โ ๐ด
๐บ,๐ )๐ )(๐ข) = 0 when ๐ข โ ๐๐ ๐ (๐บ) , which will follow from ๐ โ Ker ๐ .In particular, using a final time that ๐ is supported only on the Aomoto set, if ๐ข โ ๐๐ ๐ (๐บ) we have ((๐ โ ๐ด ๐บ,๐ )๐ )(๐ข) = ๐๐ (๐ข) + ๐ ๐ข ๐ (๐ข) + โ ๐โ๐(๐ข) ๐ ๐ ๐ ๐ ๐ (๐ (๐))= โ ๐โ๐(๐ข)๐(๐)โ๐ ๐ (๐บ) ๐ ๐ ๐ ๐ ๐ (๐ (๐)) = (๐๐ )(๐ข) = 0. (cid:3) Theorems 3.2 and 3.3 now follow easily.
Proof of Theorem 3.2.
By Theorem 3.1, the Aomoto set satisfies the hypotheses of Theorem 6.1, andif ๐ป is an ๐ -lift of ๐บ , both ๐ด ๐บ and ๐ด ๐ป/๐บ are unitary-weighted Jacobi operators for ๐บ โthe formerwith weights taking values in ๐ (1) and the latter in
๐ (๐ โ 1) by the discussion in Section 2.2. Thus ๐ โ Spec ๐ด ๐บ with multiplicity at least cc (๐ ๐ (๐บ)) โ |๐๐ ๐ (๐บ)| = ๐ผ ๐ (๐บ) = |๐ (๐บ)| โ ๐{๐} and similarly ๐ โ Spec ๐ด ๐ป/๐บ with multiplicity at least (๐ โ 1)|๐ (๐บ)| โ ๐{๐} , as desired. (cid:3)
OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 19
Proof of Theorem 3.3.
Assertion (i) is a special case of Theorem 6.1. For (ii), let ๐บ ๐ be the sequenceof lifts of ๐บ promised in Lemma 2.3, whose empirical spectral measures ๐ ๐บ ๐ converge weakly to thedensity of states ๐ . Applying Theorem 6.1 to each ๐ด ๐บ ๐ , viewed again as a unitary-weighted Jacobioperator on ๐บ , the empirical spectral measures ๐ ๐บ ๐ satisfy ๐ ๐บ ๐ {๐} โฅ cc (๐)โ|๐๐|๐ (๐บ) . As these convergeweakly to ๐ , we have ๐{๐} โฅ cc (๐ ) โ |๐๐ |๐ (๐บ) . (cid:3)
7. Spectral Delocalization for ๐ด ๎ In this section we will prove Theorem 3.4. The fact that the set ๎ผ mentioned in the theorem is aclosed set will follow from Theorem 3.3, while the fact that it has large codimension will followfrom Theorem 3.1.Let ๐บ = (๐ , ๐ธ) be a fixed finite and unweighted graph, for which we will vary the parameters ๐ ๐ and ๐ ๐ฃ . In what follows we will identify โ |๐ธ|/2 with โ |๐ธ| so that the parameter space for the ๐ ๐ andthe ๐ ๐ฃ is a subset of โ |๐ธ|+|๐ | , and we will denote elements of โ |๐ธ|+|๐ | by (๐, ๐) , where ๐ = (๐ ๐ ) ๐โ๐ธ and ๐ = (๐ ๐ฃ ) ๐ฃโ๐ . To ease notation define ๐ = |๐ธ| + |๐ | .Let ๎ญ (๐บ) be the family of vertex sets ๐ โ ๐ that induce an acyclic subgraph of ๐บ with theproperty that cc (๐ ) โ |๐๐ | > 0 . For ๐ โ ๎ญ (๐บ) let ๎ผ ๐ โ โ ๐ be the set of parameters for which allthe Jacobi matrices of the trees induced by ๐ have a common eigenvalue. Note that Theorems 3.1(ii) and 3.3 (ii) imply that ๎ผ = โ ๐โ ๎ญ (๐บ) ๎ผ ๐ . (7)To compute the dimension of ๎ผ we will analyze each ๎ผ ๐ individually. This will require basictechniques and concepts from real algebraic geometry, which we condense below. The readerfamiliar with real algebraic geometry may jump directly to Section 7.2.7.1. Real Algebraic Geometry Preliminaries.
We will need some elementary facts aboutalgebraic and semialgebraic sets, as well as appropriate notions of dimension for each of these. Athorough introduction can be found, for instance, in Sections 2 and 3 of [Cos00].An algebraic set (or, more formally, a real affine algebraic set ) is a subset of โ ๐ defined as thezero set of a family of polynomials with real coefficients. It is easy to see from the definitionthat any finite union or finite intersection of algebraic sets is still an algebraic set. Similarly, a semialgebraic set is a subset of โ ๐ defined by a family of polynomial inequalities. Any algebraicset is semialgebraic, but the reverse need not be true.An algebraic set ๎ is irreducible if it cannot be expressed as a disjoint union of two algebraicsets strictly contained in ๎ . It well known [BCR13, Section 2.8] that any algebraic set ๎ admitsa unique decomposition of the form ๎ = โ ๐๐=1 ๎ ๐ where each ๎ ๐ is an irreducible algebraic setand such that for no ๐ โ ๐ is ๎ ๐ contained in ๎ ๐ . If ๎ is an irreducible algebraic set we definethe algebraic dimension of ๎ , denoted dim ๎ , as the maximum integer ๐ such that there existsa chain of the form ๎ โ ๎ โ โฏ โ ๎ ๐ = ๎ , where each ๎ ๐ is an irreducible algebraic set andeach containment is strict. If ๎ is any algebraic set and ๎ = โ ๐๐=1 ๎ ๐ is its decomposition intoirreducible sets, we define the algebraic dimension of ๎ as dim ๎ = max ๐โ[๐] dim ๎ ๐ . It follows OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 20 from these definitions that if ๎ and ๎ are two algebraic sets, with ๎ irreducible, and ๎ is notcontained in ๎ , then dim ๎ โฉ ๎ < dim ๎ .The notion of algebraic dimension for algebraic sets can be extended to a notion of dimensionfor semialgebraic sets via the cylindrical algebraic decomposition , which we describe here. Anysemialgebraic set admits a decomposition of the form ๎ฟ = โ ๐๐=1 ๎ฏ ๐ , where the ๎ฏ ๐ are disjointsemialgebraic subsets dieffeomorphic to the open hypercube (0, 1) ๐ ๐ for some nonnegative integer ๐ ๐ [Cos00, Corollary 3.8]. With this setup we define the dimension of ๎ฟ as dim ๎ฟ = max ๐โ[๐] ๐ ๐ . Wealso remind the reader that the Hausdorff dimension of a semialgebraic set coincides with thenotion of dimension described here.Finally, we will use a fundamental fact about projections of semialgebraic sets. If ๎ฟ โ โ ๐+1 issemialgebraic and ฮ โถ โ ๐+1 โ โ ๐ is the projection onto the first ๐ coordinates, then ฮ ๎ฟ is asemialgebraic subset of โ ๐ [Cos00, Theorem 2.3], and moreover dim ฮ ( ๎ฟ ) โค dim ๎ฟ [Cos00, Lemma3.17].7.2. The Dimension of ๎ผ . We begin by proving the main technical result of this section.
Proposition 7.1.
For any
๐ โ ๎ญ (๐บ) , ๎ผ ๐ is a semialgebraic set of dimension at most ๐ โ cc ๐ + 1 .Proof.
Let ๐ = cc ๐ and ๐ , โฆ , ๐ ๐ be the trees induced by ๐ . For any Hermitian edge weights ๐ โถ ๐ธ โ โ , let ๐ฅ = โ(๐) and ๐ฆ = โ(๐) , that is, for every ๐ โ ๐ธ we write ๐ ๐ = ๐ฅ ๐ + ๐๐ฆ ๐ , with ๐ฅ ๐ , ๐ฆ ๐ โ โ . View the characteristic polynomials of the ๐ ๐ as polynomials in the ๐ฅ ๐ , ๐ฆ ๐ , ๐ ๐ฃ and ๐ง ,namely, define ๐ ๐ (๐ฅ, ๐ฆ, ๐, ๐ง) = det(๐ง โ ๐ด ๐ ๐ ) . We will first show that each ๐ ๐ (๐ฅ, ๐ฆ, ๐, ๐ง) is a polynomialwith real coefficients. Remember that โ๐ ๐ (๐ฅ, ๐ฆ, ๐, ๐ง) is a polynomial with real coefficients in theaformentioned variables. Now, since ๐ด ๐ ๐ is Hermitian, for any choice of ๐ฅ, ๐ฆ โ โ |๐ธ|/2 and ๐ โ โ |๐ | ,we have that det(๐ง โ ๐ด ๐ ๐ ) โ โ , so โ๐ ๐ โก 0 on โ ๐+1 and hence โ๐ ๐ is the zero polynomial. It thenfollows that ๐ ๐ = โ๐ ๐ , which means that ๐ ๐ โ โ[๐ฅ, ๐ฆ, ๐, ๐ง] .Now, for ๐, ๐ โ [๐] define the algebraic sets ๎ ๐ = {(๐ฅ, ๐ฆ, ๐, ๐ง) โ โ ๐+1 โถ ๐ ๐ (๐ฅ, ๐ฆ, ๐, ๐ง) = 0} and ๎ โค๐ = ๐ โ ๐=1 ๎ ๐ . We will now show that ๎ โค๐ has codimension at least ๐ for all ๐ โ [๐] , which implies in particularthat that ๎ โค๐ has algebraic dimension at most ๐ + 1 โ ๐ . For the base case note that since ๐ isnot the zero polynomial, ๎ is a proper algebraic subset of โ ๐+1 , and since โ ๐+1 is irreducible dim( ๎ ) < dim(โ ๐+1 ) = ๐ + 1 . Now, for assume that dim( ๎ โค๐ ) โค ๐ โ ๐ + 1 , define thesubspace ๐ ๐ = {(๐ฅ, ๐ฆ, ๐, ๐ง) โ โ ๐+1 โถ ๐ ๐ฃ = 0 for ๐ฃ โ ๐ (๐บ) โงต ๐ โ ๐=1 ๐ (๐ ๐ )} , and let ๎ = ๎ โค๐ โฉ ๐ ๐ .The set ๎ is itself algebraic. Moreover, since for each ๐ โค ๐ the set ๐ ๐ depends only on those ๐ ๐ฃ โs with ๐ฃ โ โ ๐๐=1 ๐ (๐ ๐ ) , we have that ๎ โค๐ = ๎ ร ๐ โ๐ . Let โ ๐ ๐=1 ๎ ๐ be the decomposition of ๎ intoirreducible components. Since ๎ ๐ and ๐ โ๐ are both irreducible ๎ ๐ ร ๐ โ๐ is as well, and hence โ ๐โ[๐ ] ( ๎ ๐ ร ๐ โ๐ ) is in fact the decomposition of ๎ โค๐ into irreducible components. On the otherhand ๎ โค๐+1 = ๎ โค๐ โฉ ๎ ๐+1 = ๐ โ ๐=1 ( ๎ ๐ ร ๐ โ๐ ) โฉ ๎ ๐+1 . We will now show that, for every ๐ โ [๐ ] , ๎ ๐ ร ๐ โ๐ is not contained in ๎ ๐+1 . OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 21
Indeed, fix (๐ฅ, ๐ฆ, ๐, ๐ง) โ ๎ ๐ ร ๐ โ๐ . Adding a constant ๐ โ โ to the ๐ ๐ฃ with ๐ฃ โ ๐ (๐ ๐+1 ) has theeffect of shifting the spectrum of ๐ด ๐ ๐+1 by ๐ . We can then find ๐ โฒ โ โ |๐ | with the property that ๐ โฒ๐ฃ = ๐ ๐ฃ for all ๐ฃ โ ๐ (๐บ) โงต ๐ (๐ ๐+1 ) and such that ๐ ๐+1 (๐ฅ, ๐ฆ, ๐ โฒ , ๐ง) โ 0 . By construction we have (๐ฅ, ๐ฆ, ๐ โฒ , ๐ง) โ ๎ ๐ โฉ๐ โ๐ and (๐ฅ, ๐ฆ, ๐ โฒ , ๐ง) โ ๎ ๐+1 as we wanted to show. This implies that ( ๎ ๐ ร๐ โ๐ )โฉ ๎ ๐+1 is a proper subset of ๎ ๐ ร ๐ โ๐ , and since the latter is irreducible we get dim (( ๎ ๐ ร ๐ โ๐ ) โฉ ๎ ๐+1 ) < dim( ๎ ๐ ร ๐ โ๐ ) โค dim( ๎ โค๐ ) โค ๐ + 1 โ ๐ , concluding the inductive step. Finally, let ฮ โถ โ ๐+1 โ โ ๐ be the projection defined by ฮ (๐ฅ, ๐ฆ, ๐, ๐ง) = (๐ฅ, ๐ฆ, ๐) , and note that ฮ ๎ โค๐ = ๎ผ ๐ . From the results mentioned in Section 7.1, ฮ ๎ โค๐ is a semialgebraic set whose dimension is less or equal to that of ๎ โค๐ , and in turn dim ๎ โค๐ โค๐ โ ๐ + 1 . (cid:3) We are now ready to prove Theorem 3.4.
Proof of Theorem 3.4.
By Proposition 7.1 and Equation (7), ๎ผ is semialgebraic with codim ๎ผ โฅ min ๐โ ๎ญ (๐บ) cc ๐ โ 1 โฅ min ๐โ ๎ญ (๐บ) ๐๐ , and we want to further lower bound the latter quantity. As ๐บ has at least one cycle, ๐๐ โ โ for all ๐ โ ๎ญ (๐บ) , so ๎ผ has codimension at least 1. Now, if ๐ min โฅ 2 take any tree ๐ induced by ๐ . Any vertex ๐ฃ of ๐ must be connected to at least ๐ min โ 1 distinct vertices in ๐๐ , and hence cc ๐ โ 1 โฅ ๐๐ โฅ ๐ min โ 1 . This proves the bound codim ๎ผ โฅ max{๐ min โ 1, 1} .We show finally that ๎ผ ๐ is open. For every ๐ โ ๎ญ (๐บ) denote the forest induced by ๐ by ๎ฒ ๐ .Fix (๐, ๐) โ ๎ผ ๐ . By Theorem 3.3, for every ๐ โ ๎ญ (๐บ) the Jacobi matrices of the trees in ๎ฒ ๐ (withweights and potentials given by ๐ and ๐ ), do not have a common eigenvalue. Now, define ๐ = โ ๐โ ๎ญ (๐บ) โ ๐ โ ๎ฒ ๐ Spec ๐ด ๐ . As ๎ฒ ๐ is finite for each of the finitely many ๐ โ ๎ญ (๐บ) , we may safely define ฮ > 0 to be thesmallest distance between two distinct points in ๐ . We will show that if (๐ โฒ , ๐ โฒ ) โ โ ๐ satisfies โ(๐, ๐) โ (๐ โฒ , ๐ โฒ )โ < ฮ/2 then (๐ โฒ , ๐ โฒ ) โ ๎ผ ๐ .Assume otherwise. Then there exists an ๐ โ ๎ญ (๐บ) such that the Jacobi matrices with parametersin (๐ โฒ , ๐ โฒ ) of the trees in ๎ฒ ๐ have a common eigenvalue ๐ . Let ๐ , โฆ , ๐ ๐ be the trees in ๎ฒ ๐ withparameters in (๐, ๐) and let ๐ โฒ1 , โฆ , ๐ โฒ๐ denote the same trees but with parameters in (๐ โฒ , ๐ โฒ ) . Forevery ๐ let ๐ ๐ be the closest point in Spec ๐ด ๐ ๐ to ๐ . Since (๐, ๐) โ ๎ผ ๐ we have ๐ ๐ โ ๐ ๐ for some ๐, ๐ .On the other hand since โ๐ด ๐ ๐ โ ๐ด ๐ โฒ๐ โ โค โ๐ด ๐ ๐ โ ๐ด ๐ โฒ๐ โ ๐น โค โ(๐, ๐)โ < ฮ/2 and similarly โ๐ด ๐ ๐ โ ๐ด ๐ โฒ๐ โ < ฮ/2 ,the triangle inequality and Weylโs inequality together imply |๐ ๐ โ ๐ ๐ | โค |๐ ๐ โ ๐| + |๐ ๐ โ ๐| < ฮ/2 + ฮ/2 = ฮ, contradicting the definition of ฮ . (cid:3) Acknowledgements.
We are grateful to Nikhil Srivastava for many helpful discussions. We alsothank Barry Simon for his many helpful comments and suggestions on a previous version of thismanuscript, and in particular for his guidance in showing the current version of Theorem 3.4. JGValso thanks Irit Huq-Kuruvilla for patiently clarifying many basic concepts in geometry. JB issupported by the NSF Graduate Research Fellowship Program under Grant DGE-1752814. JGVand SM are supported by NSF Grants CCF-1553751 and CCF-2009011.
OINT SPECTRUM OF PERIODIC OPERATORS ON UNIVERSAL COVERING TREES 22
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