Laplace and Schrödinger operators without eigenvalues on homogeneous amenable graphs
aa r X i v : . [ m a t h . SP ] F e b LAPLACE AND SCHR ¨ODINGER OPERATORSWITHOUT EIGENVALUES ON HOMOGENEOUSAMENABLE GRAPHS
R. GRIGORCHUK AND CH. PITTET
Dedicated to the memory of Mikhail A. Shubin (1944–2020)
Abstract.
A one-by-one exhaustion is a combinatorial/geometriccondition which excludes eigenvalues from the spectra of Laplaceand Schr¨odinger operators on graphs. Isoperimetric inequalitiesin graphs with a cocompact automorphism group provide an up-per bound on the von Neumann dimension of the space of eigen-functions. Any finitely generated indicable amenable group has aCayley graph without eigenvalues.
Contents
1. Introduction 21.1. Infinite connected graphs without eigenvalues 21.2. The integrated density of states of ∆ 61.3. Large-scale geometry to bound jumps in the IDS 81.4. Perspectives and questions 91.5. Acknowledgments 92. Schr¨odinger operators on graphs 92.1. Graphs 92.2. The path-metric on a connected graph 102.3. The Hilbert spaces associated to a weighted graph 112.4. The Laplace operator of a weighted graph 122.5. Adjacency and Markov operators 132.6. Schr¨odinger operators 153. Boundary conditions 163.1. Uniqueness of eigenfunctions 16
Date : February 25th, 2021.2020
Mathematics Subject Classification.
Primary: 47A10 ; Secondary: 31C20.
Key words and phrases.
Amenable group, Cayley graph, continuous spectrum,discrete Laplace operator, discrete Schr¨odinger operator, eigenvalue, Følner se-quence, integrated density of states, pure point spectrum.The first author acknowledges partial support from the University of Geneva andfrom the Simons Foundation through Collaboration Grant 527814.
Introduction
Infinite connected graphs without eigenvalues.
Let Γ be aweighted connected graph with infinite vertex set (see Definition 2.1,Subsection 2.2, and Subsection 2.3 below). Let ∆ be the associatedLaplacian on Γ (see Definition 2.6). Which geometric or combinato-rial properties of Γ imply that ∆ has no eigenvalue? A necessary andsufficient condition, based on Bloch analysis, is known for graphs ad-mitting a cocompact free action by a finitely generated abelian groupof automorphisms, see [11, Proposition 4.2]. See also [12, Theorem 4].Applying this condition, Higuchi and Nomura deduce that the combi-natorial Laplacian (see Item (3) in Subsection 2.5) on a graph which isthe maximal abelian covering of (the realization of) a connected finitegraph having a 2-factor, has no eigenvalue [11, Theorem 2]. (A finiteconnected graph Γ has a 2-factor if and only if there exists a finitenumber of oriented simplicial circles which disjointly embed in Γ insuch a way that any vertex of Γ lies on one of the embedded circles.The maximal abelian cover of the geometric realization T of a finiteconnected graph Γ is the Galois cover of T defined by the commutatorsubgroup of the fundamental group of T , hence the Galois group is iso-morphic to the first homology group H ( T, Z ).) With the help of theseresults, Higuchi and Nomura are able to decide, for several examplesof planar graphs, wether the Laplacian admits an eigenvalue or not,see [11, 6. Examples]. They also ask for new geometric or combinato-rial properties implying the NEP (no eigenvalue property) for graphswhose automorphism group contains a finite index finitely generatedabelian subgroup [11, Problem 6.11]. A similar question is raised in[5, page 93] for a general finitely generated group. The spectrum ofthe combinatorial Laplacian ∆ has been studied on Cayley graphs (seeDefinition 2.3) of some finitely generated metabelian groups by several ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 3 authors (see [1], [7], [9], [13]). We recall the example of the so called“lamplighter group”. Consider the ring F [ X, X − ] of Laurent polyno-mials in the variable X with coefficients in the field F with 2 elements.Its group of units F [ X, X − ] × = { X n : n ∈ Z } ∼ = Z acts by multiplication on F [ X, X − ]. The lamplighter group L is thecorresponding semi-direct product L = F [ X, X − ] ⋊ Z . The elements a = (0 , X ) and c = (1 ,
1) = (1 , X ) together generate L (right multiplication by a increases the position of the lamplighter by 1,right multiplication by c switches the lamp the lamplighter stands at).Putting b = ac , it is obvious that the set S = { a ; b } also generates L .In [9] it is proved that the eigenvalues of the combinatorial Laplacian∆ of the Cayley graph C ( L , S ) form a dense countable subset of thespectrum [0 ,
2] of ∆ and in [7] and [1], an orthonormal Hilbert basisof finitely supported eigenfunctions is constructed. Does the structureof the spectrum depend on the set of generators? The question wasbrought up in [9, page 210]. A positive answer has been given byGrabowski and Virag in an unpublished preprint from 2015 entitled“Random walks on Lamplighters via random Schr¨odinger operators”.Grabowski and Virag use the work of Martinelli and Micheli [16] todeduce that L has a system of generators with singular continuousspectral measure; we refer the reader to [8, page 655] and to [10, pages2, 4, 22, 29] for more details. The following theorem provides manyexamples of finitely generated amenable groups with Cayley graphshaving no eigenvalues. Theorem 1.1. (Cayley graphs of indicable amenable groups withouteigenvalue.) Let G be a group with an epimorphism h : G → Z from G to Z the infinite cyclic group (in other words G is indicable).Assume G is finitely generated. Consider a finite generating set S of G of the form S = { t } [ K such that h ( t ) generates Z and K ⊂ Ker( h ) . If G is amenable then thecombinatorial Laplace operator on the Cayley graph of G with respectto S has no eigenvalue. R. GRIGORCHUK AND CH. PITTET
There are two main steps in the proof of Theorem 1.1. These twosteps are carefully explained in the remaining part of this introduc-tion, after the formal proof that we give now, assuming all the neededdefinitions and preliminary results.
Proof.
First step: the homomorphism h is a “height function” (see thehypothesis of Theorem 3.7 for the properties of a height function) onthe Cayley graph of G with respect to S hence Theorem 3.7 applies(the special form of S is essential here) and implies the λ -unicity forany finite set of vertices and any λ ∈ R (see Definition 3.1). Secondstep: the amenability of G is equivalent to the existence of a Følnersequence (see Definition 6.6) in the Cayley graph, hence implication(1) = ⇒ (4) from Theorem 6.7 is true and we conclude that the λ -unicity for any finite set of vertices from step one implies that the onlyeigenfunction is the zero function. (cid:3) Corollary 1.2.
The combinatorial Laplace operator on the Cayleygraph of the lamplighter group F [ X, X − ] ⋊ Z with respect to the generating set { a = (0 , X ); c = (1 , X ) } has noeigenvalue.Proof. The projection h : F [ X, X − ] ⋊ Z → Z sends (0 , X ) to the generator h (0 , X ) = X of the quotient Z ∼ = { X n : n ∈ Z } and h (1 , X ) is the trivial element of Z ∼ = { X n : n ∈ Z } . (cid:3) Theorem 1.1 applies to several classes of groups (strongly polycyclicgroups, free solvable groups, some HNN-extensions, some wreath prod-ucts, etc.) and is valid for a general class of operators described in Def-inition 2.8. In particular it applies to the operators on the lamplightergroup considered by Virag and Grabowski [8, page 655]. Althoughthe theorem says nothing about the singular spectrum, its conclusionis equivalent to the continuity of the integrated density of states (seeProposition 4.5 and Proposition 4.4). Here are some examples. Takeany finitely generated amenable indicable group G , choose a finitegroup H , form the wreath product G = H ≀ G . According to Theorem1.1 there is a finitely generating set of G such that the combinatorialLaplacian on the associated Cayley graph has no eigenvalue. Accord-ing to [13] there exists a Cayley graph of G with weights (defined by ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 5 a symmetric probability measure on G ) so that the spectrum of itsassociated Laplace operator admits a dense subset of eigenvalues.The first step in the proof of Theorem 1.1 is to exclude the existenceof a non-zero finitely supported eigenfunction. Given a real number λ , we say that the Laplacian ∆ of a weighted graph Γ satisfies λ -uniqueness on a subset Ω of vertices of Γ if the only λ -eigenfunction of∆ whose support is included in Ω is the zero function (see Definition3.1). If a finite or infinite countable subset of vertices Ω admits a one-by-one exhaustion (see Definition 3.4) then ∆ satisfies λ -uniqueness onΩ for any λ ∈ R (see Theorem 3.6). A one-by-one exhaustion of Ω canbe understood as an inductive process. First we look for a vertex v ofthe graph Γ which is not in Ω and which has exactly one neighbor w belonging to Ω and we remove w from Ω. Then we look for a vertex v not in Ω \ { w } which has exactly one neighbor w belonging toΩ \ { w } and we remove w . And so on. IfΩ = [ n { w n } , where the union is finite or infinite countable, then we say that Ω admitsa one-by-one exhaustion. For example, the combinatorial Laplacian ofthe Cayley graph of Z × Z / Z (the direct product of the infinite cyclicgroup with the group of cardinality 2) with respect to the generatingset (cid:8)(cid:0) , (cid:1) ; (cid:0) , (cid:1)(cid:9) satisfies λ -uniqueness on any finite subset becausefor any integer n ≥ n ⊂ G defined asΩ n = { ( k, x ) : | k | ≤ n, x ∈ Z / Z } admits a one-by-one exhaustion. See Figure 1 and 2, which illustratethe case n = 1. The dashed rectangle from Figure 1 encloses thevertices of Ω . The dashed polygon from Figure 2 encloses the verticesof Ω \ { w } . The same subset Ω viewed in the Cayley graph of thesame group but with respect to the generating set (cid:8)(cid:0) , (cid:1) ; (cid:0) , (cid:1) ; (cid:0) , (cid:1) ; (cid:0) − , (cid:1)(cid:9) admits no one-by-one exhaustion: Figure 3 shows an eigenfunctionfor the combinatorial Laplacian of this graph which takes exactly threevalues which are − , , ⊂ Ω . The eigenvalue equals 6 /
5. The hypothesis of Theorem1.1 imply the existence of an height function (see Theorem 3.7) whichin turn implies the existence of a one-by-one exhaustion and of the λ -unicity for any finite set of vertices and any λ ∈ R . R. GRIGORCHUK AND CH. PITTET w v Figure 1.
A one-by-one exhaustion (step 1). w v w v Figure 2.
A one-by-one exhaustion (step 2).0 0 1 − Figure 3.
An eigenfunction with finite support.The second step in the proof of Theorem 1.1 is an application ofthe localization principle for eigenfunctions: the existence of a non-zero square summable λ -eigenfunction implies the existence of a non-zero finitely supported λ -eigenfunction (a proof of this implication inour setting follows from the equivalence between conditions (4) and(5) in Theorem 6.7). (An instance of this localization principle in adiscrete setting can already be found in [12, Theorem 3].) This secondstep is achieved with the help of the integrated density of states of ∆(see Subsection 1.2) and a Følner sequence (see Subsection 1.3). Thisidea goes back to a short note of Delyon and Souillard [6]. See also[3, Proposition 4.1.4] and its proof and [2, Question 2 page 116] aboutthe continuity of the integrated density of states, as well as Shubin’sformula [2, Appendix pages 146-148] for computing the von Neumanntrace with the help of usual traces and a Følner sequence.1.2. The integrated density of states of ∆ . We consider the spec-tral resolution ( E λ ) λ ∈ R of the Laplace operator ∆ of a weighted graphΓ (see Subsection 4.1). In the case Γ admits a group of automorphismswith finitely many orbits of vertices (see Subsection 6.2) - no otherhypothesis about the action is needed - we may use the von Neumann ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 7 trace τ (see Definition 6.2) to define the integrated density of statesof ∆ R → [0 , λ N ( λ ) = τ ( E λ ) . The function N is a non-decreasing right-continuous function whichhas a jump at λ if and only if λ is an eigenvalue of ∆ (see Proposition4.5). Lebesgue’s theorem for the differentiability of monotone functions[18, Ch. 1, no. 2] implies that N is a.e. differentiable and the Darboux-Froda’s theorem implies that its set of points of discontinuity is at mostcountable. The exact computation of N is probably hopeless for mostgraphs. The easiest case is when Γ is the Cayley graph of the infinitecyclic group Z with respect to the generating set S = { } . In thiscase, the integrated density of states of the combinatorial Laplacian isexpressed with the help of the level sets of the function g : [ − π, π ] → [0 , g ( θ ) = 1 − cos θ. More precisely, for each λ ∈ R , we write { g ≤ λ } = { θ : g ( θ ) ≤ λ } and denote { g ≤ λ } the characteristic function of this set. One finds:(1) τ ( E λ ) = 12 π Z π − π { g ≤ λ } ( θ ) dθ. This function is obviously analytic in λ . Analogous computations arewell-known in any dimension (i.e. on Z d , the free abelian group of rank d ∈ N ). For the convenience of the reader we prove the above formulaand compute it explicitly when d = 1 in Example 6.4 and Proposition4.3. For the general case (which is handled in the same manner) werefer to [15, 1.4, 2.6, 2.47, 9.6]. For completeness we have included theproofs of several well-known facts about spectral projections and traces(Proposition 4.2, Proposition 4.4, Lemma 4.7, Proposition 6.3) whichare needed for the proofs of the main results.What about non-abelian cases? According to Corollary 1.2 above,the integrated density of states of the combinatorial Laplacian of theCayley graph of the lamplighter group with respect to the generatingset { a ; c } is continuous, whereas according to [9], if we consider the gen-erating set { a ; bc } instead, then the corresponding integrated densityof states restricted to [0 ,
2] has as a dense set of points of discontinuity.Even though this example shows that the regularity of the integrateddensity of states is sensitive to the choice of the generating set, it is
R. GRIGORCHUK AND CH. PITTET known [4, Theorem 1.1] that its asymptotic behavior near 0 (more pre-cisely its dilatational equivalence class near zero) is an invariant of thegroup (and more generally of its quasi-isometry class). In the caseof the lamplighter group the dilatational equivalence class of N ( λ ) isrepresented by the function λ e − √ λ . A general formula [4, Theorem 1.2] relating the asymptotic behavior of N near zero to the asymptotic behavior of the l -isoperimetric profilenear infinity brings estimates of N for several families of finitely gener-ated amenable groups [4, 1.7 Explicit computations]. (Good estimatesfor the l -isoperimetric profile can be obtained with the help of optimalFølner sequences.) It follows from the technics explained in [4] that forany group G , the zero-dimensional Novikov-Schubin invariant (as de-fined in [14, Definition 1.8]) of a positive self-adjoint element ∆ of thegroup algebra R [ G ] of the form ∆ = e − P g a g g , where e ∈ G is theneutral element, with a g ≥ P g a g = 1 is bounded bellow by 1 / ∞ + if the support of P g a g g generates a finite group). The lowest possible value 1 / Z with respect to the generating set { } .1.3. Large-scale geometry to bound jumps in the IDS.
TheBorel functional calculus applied to the Laplace operator ∆ of a con-nected weighted graph Γ associates to any λ ∈ R the orthogonal pro-jection E { λ } onto the subspace of λ -eigenfunctions (see Proposition 4.2and Proposition 4.4). In the case Γ admits a cocompact group of auto-morphisms, we may use the von Neumann trace τ ( E { λ } ) to measure thevon Neumann dimension of the space of λ -eigenfunctions (see Propo-sition 6.3). The projection E { λ } vanishes if and only if the integrateddensity of states N of ∆ is continuous at λ (see Proposition 4.5). If ∆satisfies λ -uniqueness on a finite subset Ω of the set V of vertices of Γ,then τ ( E { λ } ) ≤ | ∂ Ω | d | Ω | , where d is the cardinality of a fundamental domain for the action of theautomorphism group of Γ on V , where | ∂ Ω | is the cardinality of the 2-boundary of Ω (that is the set of vertices of Ω at distances less or equalto 2 from V \ Ω, see Definition 3.2), and where | Ω | is the cardinalityof Ω. The above inequality is true in the general setting of Theorem6.5. When Γ has a Følner sequence (see Definition 6.6) and each set ofthe Følner sequence admits a one-by-one exhaustion, then the above ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 9 inequality is a tool to prove the vanishing of all the projections E { λ } , λ ∈ R , or, in other words, a tool to prove the continuity of N . If thereis no Følner sequence, the inequality still provides an upper bound (al-ways a bad one?) for the von Neumann dimensions of the spaces ofeigenfunctions. In the presence of a Følner sequence, the inequalityleads to a proof of the localization principle: the existence of a non-zero λ -eigenfunction implies that the above inequality fails for at leastone set, say Ω, from the Følner sequence. Hence λ -unicity on Ω has tofail. It means that there is a non-zero λ -eigenfunction whose supportis included in Ω. Theorem 6.7 formalizes these ideas in a general set-ting (no hypothesis on the structure of the involved groups are needed;neither on the acting group nor on the stabilizer subgroups). It gener-alizes [11, Theorem 3.2]. Similar ideas involving random Schr¨odingeroperators are presented in [22] and references therein.1.4. Perspectives and questions.
Apply one-by-one exhaustions toshow continuity of the IDS for more operators, e.g. random Schr¨odingeroperators. In the presence of a one-by-one exhaustion, when is thespectral measure absolutely continuous, when is it singular continuous?Compute von Neumann dimensions τ ( E { λ } ) for family of graphs, e.g.planar Cayley graphs of crystallographic groups, Cayley graphs of F r × Z / Z , the direct product of the (non-abelian if r ≥
2) free group ofrank r with the group of cardinality 2, relative to generating sets of thekind of the one involved in Figure 3.1.5. Acknowledgments.
We are grateful to Cosmas Kravaris for help-ing us to gain a better understanding of λ -uniqueness through exam-ples. Rostislav Grigorchuk is grateful to Jean Bellissard and PeterKuchment for numerous discussions about periodic graphs and inte-grated densities of states. Christophe Pittet is grateful to Jean Bellis-sard, Alexander Bendikov and Roman Sauer for sharing their knowl-edge about integrated densities of states, spectral measures and l -invariants. 2. Schr¨odinger operators on graphs
Graphs.
We recall the definition of a graph (in the sense of Serre[20, 2.1]).
Definition 2.1. A graph Γ = ( V, E, o, t, ι ) consists in:(1) a set V ,(2) a set E , (3) a map E → V × Ve ( o ( e ) , t ( e )) (4) a fixed point free involution ι defined on the set E ,with compatibility conditions: o ( ι ( e )) = t ( e ) , t ( ι ( e )) = o ( e ) , ∀ e ∈ E. An element x ∈ V is a vertex of Γ, an element e ∈ E is an orientededge of Γ, o ( e ) is the origin of the vertex x and t ( e ) its terminus . Theinvolution “flips the orientation of each oriented edge” and is usuallywritten as ι ( e ) = e .(Notice that the above definition allows loops, multiple edges, anddoes not imply local finiteness.)2.2. The path-metric on a connected graph.
Let x and y be twovertices of Γ. A path of Γ of finite length n ∈ N ∪ { } with origin x and terminus y is a sequence of vertices v , . . . , v n of Γ, such that v = x and v n = y , together with a sequence of n edges e k , ≤ k ≤ n of Γ, such that v k − = o ( e k ) , v k = t ( e k ) , ∀ ≤ k ≤ n. The graph Γ is connected if any two vertices of Γ are the origin andterminus of a path of Γ of finite length.
Definition 2.2. (The path-metric.) If Γ is connected, the function d : V × V → N ∪ { } ( x, y ) d ( x, y ) , defined as the minimum of the lengths of the paths of Γ with origin x and terminus y , is a distance on V , also called the path-metric asso-ciated to Γ . Definition 2.3. (Cayley graph.) Let G be a group and let S ⊂ G bea generating set of G (i.e. the only subgroup of G containing S is G itself ). The Cayley graph Γ = C ( G, S ) of G relative to (the non-necessary symmetric) subset S is defined asthe graph with vertex set V = G , with edge set the disjoint union E = ( G × S ) G ( G × S ) , ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 11 and origin and terminus maps ( o (( g, s )) , t (( g, s ))) = ( g, gs ) and hence (cid:16) o (cid:16) ( g, s ) (cid:17) , t (cid:16) ( g, s ) (cid:17)(cid:17) = ( gs, g ) . Notice that C ( G, S ) is connected because S is a generating set (sym-metric or not). Definition 2.4.
Let Γ be a connected graph with path-metric d definedon its vertex set V . For any x ∈ V and r ≥ , we define the closed ballof radius r with center x as B ( x, r ) = { y ∈ V : d ( x, y ) ≤ r } . The Hilbert spaces associated to a weighted graph.
Weconsider a strictly positive weight on the vertex set m V : V → ]0 , ∞ [ , and the associated Hilbert space l ( V, m V ) = { ϕ : V → C : X x | ϕ ( x ) | m V ( x ) < ∞} of square summable functions on V . The hermitian product of ϕ, ψ ∈ l ( V, m V ) is: h ϕ, ψ i = X x ϕ ( x ) ψ ( x ) m V ( x ) , and k ϕ k = p h ϕ, ϕ i denotes the l -norm of ϕ . We consider a strictly positive weight on theoriented edge set m E : E → ]0 , ∞ [ , which is symmetric in the sense that m E ( e ) = m E ( e ) , ∀ e ∈ E. A function α : E → C is anti-symmetric if α ( e ) = − α ( e ) , ∀ e ∈ E . Let l ( E, m E ) = { α : E → C : ∀ e ∈ E, α ( e ) = − α ( e ) , X e ∈ E | α ( e ) | m ( e ) < ∞} denote the Hilbert space of anti-symmetric square summable functionson E . (We may thing of l ( E, m E ) as a space of 1-forms on Γ.) Bydefinition, the hermitian product of α, β ∈ l ( E, m E ) is: h α, β i = 12 X e α ( e ) β ( e ) m E ( e ) . The l -norm of α ∈ l ( E, m E ) is k α k = p h α, α i . The Laplace operator of a weighted graph.
For each vertex x ∈ V , we denote E x = { e ∈ E : o ( e ) = x } the set of edges of Γ whose origin is x . Definition 2.5. (The Sunada-Sy necessary and sufficient condition forthe boundedness of the Laplace operator [21] .) We say that a weightedgraph satisfies the
Sunada-Sy condition if sup x ∈ V m V ( x ) X e ∈ E x m E ( e ) < ∞ . In this work, when we consider a weighted graph, we always assume itsatisfies the Sunada-Sy condition. Notice that the Sunada-Sy conditionimplies that E x is at most countable for any vertex x . Let f be afunction on V . Assuming the Sunada-Sy condition, it is easy to checkthat the formulae df ( e ) = f ( t ( e )) − f ( o ( e )) , ∀ e ∈ E define a bounded operator d : l ( V, m V ) → l ( E, m E ) . Hence its adjoint d ∗ : l ( E, m E ) → l ( V, m V ) , is also bounded. We conclude that the composition d ∗ d : l ( V, m V ) → l ( V, m V ) , is bounded, self-adjoint, and positive. Definition 2.6.
The
Laplace operator ∆ associated to the weightedgraph (Γ , m V , m E ) is defined as ∆ = d ∗ d . Proposition 2.7. (The key formulae.) Let α ∈ l ( E, m E ) and f ∈ l ( V, m V ) . Let x ∈ V .(1) d ∗ α ( x ) = − m V ( x ) X e ∈ E x α ( e ) m E ( e ) . (2) ∆ f ( x ) = m V ( x ) X e ∈ E x m E ( e ) ! f ( x ) − m V ( x ) X e ∈ E x f ( t ( e )) m E ( e ) . ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 13
Notice for further use that it is transparent from the formulae thatif we choose a constant c > m V and m E by cm V and cm E then neither d ∗ nor ∆ changes. The Sunada-Sycondition also remains unaltered. Before proving the above propositionwe mention that the Sunada-Sy condition defined above is equivalentto the boundedness of ∆ (see [21] for a proof). Proof.
Let δ x be the characteristic function of the singleton { x } ⊂ V .Recalling that α is anti-symmetric and m E is symmetric, we find: d ∗ α ( x ) m V ( x ) = h d ∗ α, δ x i = h α, dδ x i = 12 X e α ( e )( δ x ( t ( e )) − δ x ( o ( e ))) m E ( e )= 12 X t ( e )= x α ( e ) m E ( e ) − X o ( e )= x α ( e ) m E ( e )= 12 X o ( e )= x α ( e ) m E ( e ) − X o ( e )= x α ( e ) m E ( e )= − X o ( e )= x α ( e ) m E ( e ) − X o ( e )= x α ( e ) m E ( e )= − X e ∈ E x α ( e ) m E ( e ) . This proves the first formula. Applying it with α = df establishes thesecond formula. (cid:3) Adjacency and Markov operators.
As explained in [11], sev-eral familiar operators on Γ may be seen as different avatars of thegeneral Laplace operator defined above.(1)
Adjacency operator.
In the case sup x | E x | < ∞ , choosing m V and m E identically equal to 1, we obtain∆ = D − A where, for any f ∈ l ( V, m V ) and x ∈ V , Df ( x ) = | E x | f ( x )is the diagonal multiplication operator by the degree at thevertex x (i.e. the number of oriented edges with origin x ) of Γ,and Af ( x ) = X e ∈ E x f ( t ( e )) is the adjacency operator.(2) Markov operator.
Suppose we are given a w -reversible ran-dom walk p on Γ: that is a function p : E → [0 , X e ∈ E x p ( e ) = 1 , ∀ x ∈ V, and a strictly positive function w : V → ]0 , ∞ [such that w ( o ( e )) p ( e ) = w ( o ( e )) p ( e ) , ∀ e ∈ E. (The random walk defined by p is a Markov chain with statespace V and the formula P x,y = X { e : o ( e )= x, t ( e )= y } p ( e ) . expresses the probability of transition from the state x to thestate y .) Choosing m V = w and defining the following weighton the edges: m E ( e ) = w ( o ( e )) p ( e ) , ∀ e ∈ E (notice that the w -reversibility of p implies that m E is symmet-ric), we obtain ∆ = I − M, where I is the identity operator and M is the Markov operatorassociated to p , i.e. if f ∈ l ( V, m V ) and x ∈ V , M f ( x ) = X e ∈ E x f ( t ( e )) p ( e ) . Here we have the freedom to choose a constant c > w = m V and m E by cm V and cm E : this neither affects thereversibility of p nor the symmetry of the weight on the edges,moreover - as already explained - such a normalization keepsthe operator ∆ unchanged.(3) The simple random walk and the combinatorial Lapla-cian.
A special case of the previous setting is when p is thesimple random walk, i.e. p ( e ) = 1 | E o ( e ) | , ∀ e ∈ E, ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 15 and w ( x ) = c | E x | , ∀ x ∈ V, where c > f ∈ l ( V, m V )and x ∈ V ,∆ f ( x ) = f ( x ) − | E x | X e ∈ E x f ( t ( e )) . In the case the degree E x does not depend on x , we may choosethe constant c = | E x | − so that the weight m V = w is constantequal to 1.2.6. Schr¨odinger operators.Definition 2.8. (Schr¨odinger operator.) Let Γ be a weighted graphsatisfying the Sunada-Sy condition. Let ∆ be its associated Laplacian.Let q : V → R be a bounded function (which will be called a potential )on the vertex set V of Γ : sup x ∈ V | q ( x ) | < ∞ . The
Schr¨odinger operator H associated to Γ and the potential q , is thebounded self-adjoint operator H = ∆ + q : l ( V, m V ) → l ( V, m V ) , i.e. if f ∈ l ( V, m V ) and x ∈ V , then Hf ( x ) = ∆ f ( x ) + q ( x ) f ( x ) . When working with an eigenfunction ϕ of a Schr¨odinger operator H ,it will be convenient for us to apply an “associated adjacency operator” L to ϕ and to control the result Lϕ . The aim of the next propositionis to make this idea precise. Proposition 2.9. (From Schr¨odinger to adjacency and back.) Let Γ be a weighted graph satisfying the Sunada-Sy condition. Let V be thevertex set of Γ . Let q be a real bounded potential on V and let H be thecorresponding Schr¨odinger operator on l ( V, m V ) . Let L be the boundedoperator on l ( V, m V ) , defined on each ϕ ∈ l ( V, m V ) and x ∈ V as: Lϕ ( x ) = 1 m V ( x ) X e ∈ E x ϕ ( t ( e )) m E ( e ) . For each real λ , consider the bounded potential p λ on V , defined oneach x ∈ V as: p λ ( x ) = q ( x ) − λ + 1 m ( x ) X e ∈ E x m E ( e ) . Let ϕ ∈ l ( V, m V ) . Then Hϕ = λϕ, if and only if Lϕ = p λ ϕ Proof.
Let ϕ ∈ l ( V, m V ). Let x ∈ V . We have:[( H − λ ) ϕ ]( x ) = [(∆ + q − λ ) ϕ ]( x )= [( p λ − L ) ϕ ]( x ) . In other words, we have the following equality between operators: H − λ = p λ − L. Hence for any ϕ ∈ l ( V, m V ), ϕ ∈ Ker( H − λ ) ⇐⇒ ϕ ∈ Ker( L − p λ ) . This proves the proposition. (cid:3) Boundary conditions
Uniqueness of eigenfunctions.Definition 3.1. ( λ -uniqueness for H on Ω .) Let Γ be a weighted graphsatisfying the Sunada-Sy condition. Let Ω ⊂ V be a subset of verticesof Γ . Let H be a Schr¨odinger operator on Γ and λ be a real number.The operator H satisfies λ -uniqueness on Ω if the only λ -eigenfunctionof H which vanishes outside of Ω is the zero function. Formally: if ϕ ∈ l ( V, m V ) is such that Hϕ = λϕ, and if ϕ ( x ) = 0 for all x ∈ V \ Ω , then ϕ ( x ) = 0 for all x ∈ V . Notice that if Ω ′ ⊂ Ω and if H satisfies λ -uniqueness on Ω then H satisfies λ -uniqueness on Ω ′ . On the empty set, H satisfies λ -uniquenessfor any λ .Let m Ω be the restriction of m V to Ω. Let l (Ω , m Ω ) = { f : Ω → C : X x ∈ Ω | f ( x ) | m Ω ( x ) < ∞} be the Hilbert space of square summable functions on Ω. The inclusionΩ ⊂ V induces a natural linear isometric embedding U Ω : l (Ω , m Ω ) → l ( V, m V ) ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 17 which is defined as “the extension by zero outside of Ω”, more formally:if f ∈ l (Ω , m Ω ), then U Ω f ( x ) = f ( x ) in the case x ∈ Ω, and U Ω f ( x ) =0 in the case x ∈ V \ Ω. Let U ∗ Ω : l ( V, m V ) → l (Ω , m Ω )be the adjoint of U Ω . Using δ -functions (i.e. characteristic functions ofsingletons) for x ∈ V and ω ∈ Ω we have: h U ∗ Ω δ x , δ ω | Ω i l (Ω ,m Ω ) = h δ x , U Ω δ ω | Ω i l ( V,m V ) . Hence:(2) x ∈ V \ Ω = ⇒ U ∗ Ω δ x = 0 , (3) x ∈ Ω = ⇒ U ∗ Ω δ x = δ x | Ω . Notice that the composition P Ω = U Ω U ∗ Ω is the orthonormal projectiononto the subspace U Ω ( l (Ω , m Ω )) ⊂ l ( V, m V ). In other words, for any ϕ ∈ l ( V, m V ), we have: P Ω ϕ = X x ∈ Ω ϕ ( x ) δ x . Definition 3.2. (Thick boundary.) Let ( X, d ) be a metric space. For E ⊂ X and x ∈ X , we write d ( x, E ) = inf y ∈ E d ( x, y ) . For r ≥ , the r - boundary of a subset Ω ⊂ X is the subset of X definedas: ∂ r Ω = { x ∈ Ω : d ( x, X \ Ω) ≤ r } . In other words ∂ r Ω consists in points of X lying inside Ω at depth lessor equal to r . This definition will mainly be applied to the vertex set of a connectedgraph with its path-metric. In this case, all distances take integralvalues. The case r = 2 is relevant for uniqueness properties in theDirichlet problem: roughly speaking we try to control λ -eigenfunctionson a domain Ω with the help of a condition on ∂ Ω. The followingtechnical lemma will be useful.
Lemma 3.3. (Cutting and pasting eigenfunctions.) Let Γ be a weightedgraph with vertex set V . Assume Γ is connected and satisfies theSunada-Sy condition. Let H = ∆ + q be a Schr¨odinger operator on Γ whose potential q is bounded. Consider Ω ⊂ V and λ ∈ R andsuppose ϕ ∈ l ( V, m V ) is a λ -eigenfunction of H : Hϕ = λϕ. Assume ϕ vanishes on ∂ Ω . Then:(1) the function P Ω ϕ ∈ l ( V, m V ) is also a λ -eigenfunction of H ,(2) and in the case H satisfies λ -uniqueness on Ω , the function P Ω ϕ vanishes everywhere (i.e. is the zero function on V ).Proof. Let L and p λ be the bounded operator and the bounded poten-tial defined in Proposition 2.9. We know that the equations Hϕ = λϕ, and Lϕ = p λ ϕ, are equivalent. Hence, in order to prove the first implication of thelemma, it is enough to show that the equality Lϕ = p λ ϕ implies LP Ω ϕ ( x ) = p λ ( x ) P Ω ϕ ( x ) , ∀ x ∈ V. We consider two cases.(1) Assume d ( x, V \ Ω) ≥ x lies in Ω at depth 2or more). When restricted to Ω, the functions P Ω ϕ and ϕ areequal. Hence, on one hand we have: P Ω ϕ ( x ) = ϕ ( x ) , and on the other hand, LP Ω ϕ ( x ) = Lϕ ( x )(because t ( e ) ∈ Ω for any e ∈ E x ).(2) Assume d ( x, V \ Ω) < x does notbelong to Ω or x lies at depth 1 in Ω). If x does not belong toΩ then P Ω ϕ ( x ) = 0. If x lies at depth 1 in Ω, then ϕ ( x ) = 0by hypothesis (notice, for later use - when proving below that LP Ω ϕ ( x ) vanishes - that the same conclusion holds if x lies atdepth 2) and, as already mentioned, P Ω ϕ and ϕ are equal onΩ. So again P Ω ϕ ( x ) = 0. We claim that LP Ω ϕ ( x ) = 0 too. Toprove this claim, notice first that if e ∈ E x , then t ( e ) belongseither to the thick boundary of Ω, or to V \ Ω. In both cases P Ω ϕ ( t ( e )) = 0 as explained before the claim. This proves theclaim.The second implication in the lemma follows immediately from the firstimplication. (cid:3) ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 19
One-by-one exhaustions.Definition 3.4. (One-by-one exhaustion.) Let Γ be a connected graphwith vertex set V and path-metric d . Let Ω ⊂ V . A one-by-one exhaus-tion ( V n ) n ≥ of Ω is a countable non decreasing sequence of subsets of V , V ⊂ · · · ⊂ V n ⊂ V n +1 ⊂ · · · satisfying the following conditions:(1) V = V \ Ω ,(2) for each n , there exists x n ∈ V n , such that V n +1 = V n ∪ B ( x n , , (3) either | V n +1 \ V n | = 1 or V n = V ,(4) S n V n = V . Lemma 3.5. (One way into Ω .) Let Γ be a connected graph withvertex set V and path-metric d . Let Ω ⊂ V . Let ( V m ) m ≥ be a one-by-one exhaustion of Ω . Assume that for a given integer n ≥ , the set V n is strictly contained in V (that is there exists at least one vertex of Γ which is not in V n ). Let x n ∈ V n , such that V n +1 = V n ∪ B ( x n , . Let y n +1 be the unique element of V n +1 \ V n . Then E x n = { e ∈ E x n : t ( e ) = y n +1 } ∪ { e ∈ E x n : t ( e ) ∈ V n } . Proof.
The inclusion E x n ⊃ { e ∈ E x n : t ( e ) = y n +1 } ∪ { e ∈ E x n : t ( e ) ∈ V n } is tautological. In order to prove the other inclusion, let e ∈ E x n . Wehave: t ( e ) ∈ B ( x n , ⊂ V n ∪ B ( x n ,
1) = V n +1 = V n ∪ { y n +1 } . (cid:3) Theorem 3.6. (One-by-one exhaustion implies λ -uniqueness.) Let Γ be a weighted graph with vertex set V . Assume Γ is connected andsatisfies the Sunada-Sy condition. Let H be a Schr¨odinger operatoron Γ . If Ω ⊂ V admits a one-by-one exhaustion, then H satisfies λ -uniqueness on Ω for any λ ∈ R .Proof. Let ϕ ∈ l ( V, m V ) such that Hϕ = λϕ . We have to prove theimplication:( ∀ x ∈ V \ Ω ϕ ( x ) = 0) = ⇒ ( ∀ x ∈ V ϕ ( x ) = 0) . The implication is obviously true if V \ Ω is empty. Hence we mayassume V = V \ Ω is non-empty. Let ( V n ) n ≥ be a one-by-one exhaus-tion of Ω. We proceed by induction on n . By hypothesis, we knowthat the restriction of ϕ to V is identically equal to zero. Assume that V n is strictly included in V and assume that the restriction of ϕ to V n is identically equal to zero. Let us show that this implies that therestriction of ϕ to V n +1 is identically equal to zero. Notice that this willfinish the proof because of Conditions 4 and 3 in Definition 3.4. Let x n ∈ V n as in Condition 1 in Definition 3.4. Our induction hypothesis,implies that ϕ ( x n ) = 0 . Let L and p λ be defined as in Proposition 2.9. Applying Proposition2.9 to the hypothesis Hϕ = λϕ , we obtain:0 = ϕ ( x n ) = p λ ( x n ) ϕ ( x n ) = Lϕ ( x n ) . We may rewrite this equality as: X e ∈ E xn ϕ ( t ( e )) m E ( e ) = 0 . As V n is strictly included in V , Lemma 3.5 applies and implies that E x n = { e ∈ E x n : t ( e ) = y n +1 } ∪ { e ∈ E x n : t ( e ) ∈ V n } . Hence: X e ∈ E xn : t ( e )= y n +1 ϕ ( t ( e )) m E ( e ) = − X e ∈ E xn : t ( e ) ∈ V n ϕ ( t ( e )) m E ( e ) . According to the induction hypothesis, the right-hand side vanishes.This proves that ϕ ( y n +1 ) X e ∈ E xn : t ( e )= y n +1 m E ( e ) = 0 . As |{ e ∈ E x n : t ( e ) = y n +1 }| ≥ d ( x n , y n +1 ) = 1), and as theweight m E is strictly positive on all edges, we deduce that ϕ ( y n +1 ) = 0.Hence ϕ vanishes on V n +1 = V n ∪ { y n +1 } . (cid:3) Theorem 3.7. (Height functions bring λ -uniqueness and exhaustions.)Let Γ be a connected graph with vertex set V and path-metric d . Let Z be the set of integers with the usual metric d ( m, n ) = | m − n | . Assumethere exists a function h : V → Z with the following properties:(A) for any x, y ∈ V , d ( h ( x ) , h ( y )) ≤ d ( x, y ) ,(B) for each b ∈ V there exists exactly one element c ∈ V such that d ( b, c ) = 1 and such that h ( c ) = h ( b ) + 1 , ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 21 (C) for each b ∈ V there exists at least one vertex a ∈ V such that d ( a, b ) = 1 and such that h ( a ) = h ( b ) − .Then the following holds true.(1) Any finite subset Ω ⊂ V admits a one-by-one exhaustion.(2) If Γ admits weights satisfying the Sunada-Sy condition and if H = ∆ + q is a Schr¨odinger operator on the weighted graph Γ ,defined by a bounded potential q , then H satisfies λ -uniqueness,for any λ ∈ R , on any subset Ω ⊂ V such that min ω ∈ Ω h ( ω ) > −∞ . Proof.
We first prove that any finite subset Ω ⊂ V admits a one-by-oneexhaustion. We proceed by induction on k = | Ω | . If k = 0 then Ω isempty and V = V is a one-by-one exhaustion. Assume we are given Ωwith | Ω | = k ≥ k elements admits a one-by-one exhaustion. Let V = V \ Ω. Let y ∈ Ω such that h ( y ) = min ω ∈ Ω h ( ω ) . Let V = V ∪{ y } . By hypothesis there exists x ∈ V such that d ( x, y ) =1 and such that h ( x ) = h ( y ) −
1. Obviously, x ∈ V . Let us check that V = V ∪ B ( x, . The inclusion V ⊂ V ∪ B ( x,
1) is obvious. In order to prove the otherinclusion, it is enough to prove that if z ∈ B ( x, z ∈ V or z = y . We know that d ( h ( z ) , h ( x )) ≤ d ( z, x ) ≤ . Hence h ( z ) = h ( x ) + ǫ where ǫ ∈ {−
1; 0; 1 } . But h ( z ) = h ( x ) + ǫ = h ( y ) − ǫ. If ǫ ∈ {−
1; 0 } , then z ∈ V . If ǫ = 1, then h ( z ) = h ( y ). In thiscase, both y and z are at distance exactly one from x , at equal height h ( x ) + 1. This forces z = y . By induction hypothesis, the set Ω \ { y } admits a one-by-one exhaustion W = V \ (Ω \ { y } ) ⊂ · · · ⊂ W n . Wehave V ⊂ V = W . Setting V i = W i − for 1 ≤ i ≤ n + 1 defines aone-by-one exhaustion of Ω.Now we prove that H satisfies λ -uniqueness on any subset Ω ⊂ V such that min ω ∈ Ω h ( ω ) > −∞ . Let λ ∈ R and let ϕ ∈ l ( V, m V ). Assume that Hϕ = λϕ . We have toprove the implication:( ∀ x ∈ V \ Ω ϕ ( x ) = 0) = ⇒ ( ∀ x ∈ V ϕ ( x ) = 0) . We proceed by contradiction: suppose ϕ is not identically equal to zero.Then we may choose ω ∈ Ω such that ϕ ( ω ) = 0 and such that h ( ω ) = min { h ( ω ) : ω ∈ Ω , ϕ ( ω ) = 0 } . By hypothesis, there exists a ∈ V such that d ( a, ω ) = 1 and such that h ( a ) = h ( ω ) −
1. We have ϕ ( a ) = 0. We proceed as in the proof ofTheorem 3.6. Namely, we apply Proposition 2.9 to obtain:0 = ϕ ( a ) = p λ ( a ) ϕ ( a ) = Lϕ ( a ) = X e ∈ E a ϕ ( t ( e )) m E ( e ) . We claim that E a = { e ∈ E a : t ( e ) = ω } ∪ { e ∈ E a : t ( e ) ∈ V \ Ω } . One inclusion is obvious. In order to prove the other, let e ∈ E a . Wehave: d ( h ( t ( e )) , a ) ≤ d ( t ( e ) , a ) ≤ . Hence there exists ǫ ∈ {−
1; 0; 1 } such that h ( t ( e )) = h ( a ) + ǫ . If ǫ ∈ {−
1; 0 } then h ( t ( e )) ≤ h ( a ) < h ( ω ) , hence t ( e ) ∈ V \ Ω. In the case ǫ = 1, h ( t ( e )) = h ( a ) + 1 = h ( ω ) . Hence, as t ( e ) and ω are two vertices at distance exactly 1 from thevertex a , this forces t ( e ) = ω . This finishes the proof of the claim.Applying the claim to Equation 3.2, we obtain: X e ∈ E a : t ( e )= ω ϕ ( t ( e )) m E ( e ) = − X e ∈ E a : t ( e ) ∈ V \ Ω ϕ ( t ( e )) m E ( e ) . The right-hand side vanishes by hypothesis. The left-hand side equals ϕ ( ω ) X e ∈ E a : t ( e )= ω m E ( e ) . As the weights m E ( e ) are all strictly positive and as |{ e ∈ E a : t ( e ) = ω }| ≥ , (because d ( a, ω ) = 1), we deduce that ϕ ( ω ) = 0. This is a contradic-tion. Hence ϕ has to be identically zero. (cid:3) ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 23 Dimensions of eigenspaces
Spectral projections.
Let H be a Hilbert space. Let B ( H ) bethe C ∗ -algebra of bounded operators on H . If A ∈ B ( H ), let A ∗ denoteits adjoint. Let k A k = sup k v k≤ k A ( v ) k , be the operator norm of A . Assume A ∈ B ( H ) is self-adjoint. Thespectrum σ ( A ) of A is a compact subset of the real line. Let B ( σ ( A ))be the C ∗ -algebra of bounded Borel function on σ ( A ), with involutiondefined by the equalities f ∗ ( x ) = f ( x ) , ∀ x ∈ σ ( A ), and norm k f k = sup x ∈ σ ( A ) | f ( x ) | . We need the following form of the spectral theorem (see for example[17, Theorem VII.2] or [19, 12.24]).
Theorem 4.1. (Borel functional calculus.) There is a unique map
Φ : B ( σ ( A )) → B ( H ) , with the following properties:(1) Φ is ∗ -morphism of algebras,(2) ∀ f ∈ B ( σ ( A )) , k Φ( f ) k ≤ k f k ,(3) let f be the function defined as f ( x ) = x, ∀ x ∈ σ ( A ) , then Φ( f ) = A ,(4) if f n ∈ B ( σ ( A )) , n ∈ N , is a sequence which converges point-wise to f ∈ B ( σ ( A )) , then the sequence Φ( f n ) , n ∈ N , convergesstrongly to Φ( f ) . Let λ ∈ R . Consider the Borel sets] − ∞ , λ ] ∩ σ ( A ) , { λ } ∩ σ ( A ) , ] − ∞ , λ [ ∩ σ ( A ) , their characteristic functions ] −∞ ,λ ] ∩ σ ( A ) , { λ }∩ σ ( A ) , ] −∞ ,λ [ ∩ σ ( A ) , and the corresponding operators defined by the Borel functional calcu-lus: E ] −∞ ,λ ] = Φ (cid:0) ] −∞ ,λ ] ∩ σ ( A ) (cid:1) ,E { λ } = Φ (cid:0) { λ }∩ σ ( A ) (cid:1) ,E ] −∞ ,λ [ = Φ (cid:0) ] −∞ ,λ [ ∩ σ ( A ) (cid:1) . It is customary to use the short notation: E λ = E ] −∞ ,λ ] . (To the interested reader, we recommend the elementary construc-tion of E λ and E { λ } explained in [18, 106. Fonctions d’une transfor-mation sym´etrique born´ee].) The following proposition gathers somewell-known properties of E { λ } we need. Proposition 4.2. (Spectral projections.) Let A ∈ B ( H ) be self-adjoint.Let λ ∈ R and let E { λ } = Φ (cid:0) { λ }∩ σ ( A ) (cid:1) .(1) The operator E { λ } is an orthogonal projection: E { λ } = E ∗{ λ } = E { λ } . (2) The operator E { λ } is positive: ∀ v ∈ H , h E { λ } v, v i ≥ . (3) Let B ∈ B ( H ) . Assume that AB = BA . Then: BE { λ } = E { λ } B. Proof.
In order to prove the first statement in the proposition noticethat the values of the function { λ }∩ σ ( A ) are either 0 or 1, hence: { λ }∩ σ ( A ) = ∗{ λ }∩ σ ( A ) = { λ }∩ σ ( A ) . So the first statement in Theorem 4.1 above implies: E { λ } = E ∗{ λ } = E { λ } . The second statement in the proposition follows from the first: if v ∈ H then h E { λ } v, v i = h E { λ } v, v i = h E { λ } v, E { λ } v i ≥ . In order to prove the third statement, notice first that if L n ∈ B ( H ) isa sequence which converges strongly to L ∈ B ( H ) and if BL n = L n B then BL = LB . This is because BL − LB = BL − BL n + L n B − LB, hence, if v ∈ H , k ( BL − LB ) v k ≤ k B kk ( L − L n ) v k + k ( L n − L ) Bv k . Let P n be a sequence of polynomial functions which converges point-wise to { λ } on σ ( A ). It follows from the last implication in Theorem4.1 that Φ( P n ) converges strongly to Φ( { λ } ) = E { λ } . As AB = BA ,and as Φ is a morphism of algebras, we have Φ( P n ) B = B Φ( P n ). Wededuce that the strong limitlim n →∞ Φ( P n ) = E { λ } also commutes with B . (cid:3) ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 25
The following proposition gives concrete examples of spectral projec-tions. It is a well-known consequence of the Borel functional calculusand the Lebesgue dominated convergence theorem. It is useful in orderto prove Formula 1 from the Introduction.
Proposition 4.3. (Spectral projections of a multiplication operator.)Let (Ω , µ ) be a measured space. Let g : Ω → R be a real valued boundedmeasurable function. We denote M g : L (Ω , µ ) → L (Ω , µ ) the corresponding multiplication operator; it is defined by the conditions ( M g ϕ ) = gϕ, ∀ ϕ ∈ L (Ω , µ ) . For any λ ∈ R , consider the level set { g ≤ λ } = { ω : g ( ω ) ≤ λ } ⊂ Ω , its characteristic function { g ≤ λ } , and the corresponding multiplicationoperator M { g ≤ λ } . This multiplication operator is the spectral projection E λ associated to the bounded self-adjoint operator M g : E λ = M { g ≤ λ } . Proof.
Let λ ∈ R be given. We choose a sequence of polynomial func-tions P n ( X ) ∈ R [ X ], n ∈ N , such that their restrictions to the compactinterval [inf g, sup g ] satisfy:(1) 2 ≥ P n ≥ P n +1 ≥ ] −∞ ,λ ] ,(2) P n converges point-wise to ] −∞ ,λ ] .Notice that σ ( M g ) ⊂ [inf g, sup g ] ⊂ R . Notice also for later use thatfor all n ∈ N , sup x ∈ [inf g, sup g ] | P n ( x ) − ] −∞ ,λ ] ( x ) | ≤ . According to the Borel functional calculus, the point-wise convergence(2) implies the strong convergencelim n →∞ Φ( P n ) = Φ (cid:0) ] −∞ ,λ ] (cid:1) . By definition, E λ = Φ (cid:0) ] −∞ ,λ ] (cid:1) . Hence, in order to prove the proposition, it is enough to show the strongconvergence lim n →∞ Φ( P n ) = M { g ≤ λ } . A key algebraic fact is that the operator Φ( P n ) ∈ B ( L (Ω , µ )) is themultiplication operator defined by the function ω P n ( g ( ω )) . We have:sup ω ∈ Ω | P n ( g ( ω )) − { g ≤ λ } ( ω ) | = sup ω ∈ Ω | P n ( g ( ω )) − ] −∞ ,λ ] ( g ( ω )) |≤ sup x ∈ [inf g, sup g ] | P n ( x ) − ] −∞ ,λ ] ( x ) |≤ . Let ϕ ∈ L (Ω , µ ). The sequence of measurable functions definedalmost everywhere as f n ( ω ) = | P n ( g ( ω )) ϕ ( ω ) − { g ≤ λ } ( ω ) ϕ ( ω ) | converges almost everywhere to 0 and is bounded above by the inte-grable function (2 | ϕ | ) . Lebesgue dominated convergence Theorem implies thatlim n →∞ k (Φ( P n ) − M { g ≤ λ } ) ϕ k L (Ω ,µ ) = lim n →∞ Z Ω f n ( ω ) dµ ( ω )= Z Ω lim n →∞ f n ( ω ) dµ ( ω )= 0 . This proves the proposition. (cid:3)
Proposition 4.4. (An eigenspace is the range of a spectral projection.)Let A and E { λ } be as above. Then the image of the operator E { λ } isthe λ -eigenspace of A : Im (cid:0) E { λ } (cid:1) = { v ∈ H : Av = λv } . Proof.
In order to save notation, any function f ( x ), of a real variable x , we consider in this proof, has to be understood as a function definedon σ ( A ). In other words, were we should write f | σ ( A ) , we allow ourselfto write only f . The inclusion Im (cid:0) E { λ } (cid:1) ⊂ { v ∈ H : Av = λv } isequivalent to the following identity in B ( H ):( A − λI ) E { λ } = 0 . To prove it, notice that the function x ( x − λ ) { λ } ( x ) is identicallyequal to zero. Applying the Borel functional calculus, we deduce:( A − λI ) E { λ } = Φ( x ( x − λ ))Φ( { λ } )= Φ( x ( x − λ ) { λ } ( x ))= Φ(0) = 0 . ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 27
We now prove the inclusion { v ∈ H : Av = λv } ⊂ Im (cid:0) E { λ } (cid:1) . For eachinteger n ≥
1, we define the function f n ( x ) = ( x − λ if | x − λ | > n ,0 otherwise.The sequence of bounded Borel functions { λ } + f n · ( x ( x − λ )) , n ∈ N , converges point-wise to the function R identically equal to 1. Hencefor any v ∈ H ,lim n →∞ Φ (cid:0) { λ } + f n · ( x ( x − λ )) (cid:1) v = Φ( R ) v. We deduce that E { λ } v + lim n →∞ [Φ ( f n ) ( A − λI ) v ] = v. In the case v ∈ Ker ( A − λI ), we obtain E { λ } v = v. (cid:3) We will need the following fact.
Proposition 4.5. (Continuity of the integrated density ⇔ vanishingof spectral projections.) With the same notation as above, consider anyfinite set D ⊂ H of vectors of H and the function N ( λ ) = X v ∈ D h E λ v, v i . The function N ( λ ) is continuous at λ if and only if X v ∈ D h E { λ } v, v i = 0 . Proof.
Assume that X v ∈ D h E { λ } v, v i = 0 . In order to prove the continuity of N ( λ ) at λ , it is enough to provethat for any v ∈ H , lim t> ,t → h E λ + t v, v i = h E λ v, v i and lim t> ,t → X v ∈ D h E λ − t v, v i = X v ∈ D h E λ v, v i . The continuity from the right always holds: according to the Borelfunctional calculus, it follows from the obvious fact that for any x ∈ R ,lim t> ,t → ] −∞ ,λ + t ] ( x ) = 1 ] −∞ ,λ ] ( x ) . In order to prove the continuity from the left, first notice that for any x ∈ R , lim t> ,t → ] −∞ ,λ − t ] ( x ) = 1 ] −∞ ,λ [ ( x ) . Hence the Borel functional calculus implies that for any v ∈ H ,lim t> ,t → h E ] −∞ ,λ − t ] v, v i = h E ] −∞ ,λ [ v, v i . Applying the hypothesis, we conclude thatlim t> ,t → X v ∈ D h E ] −∞ ,λ − t ] v, v i = X v ∈ D h E ] −∞ ,λ [ v, v i = X v ∈ D h E ] −∞ ,λ [ v, v i + X v ∈ D h E { λ } v, v i = X v ∈ D h E λ v, v i . This proves the continuity of N ( λ ) at λ . If N ( λ ) is continuous at λ then X v ∈ D h E ] −∞ ,λ [ v, v i = X v ∈ D lim t> ,t → h E ] −∞ ,λ − t ] v, v i = lim t> ,t → X v ∈ D h E ] −∞ ,λ − t ] v, v i = X v ∈ D h E λ v, v i = X v ∈ D h E ] −∞ ,λ [ v, v i + X v ∈ D h E { λ } v, v i . This proves that P v ∈ D h E { λ } v, v i = 0. (cid:3) Boundaries to bound eigenspaces dimensions.
In a Dirichletproblem one seeks for a function which solves a specified equation on agiven region and which takes prescribed values on the boundary of theregion. Lemma 4.6 together with Proposition 4.8 below are suitableformalizations of a pervading idea of harmonic analysis: in a Dirichletproblem the “size” of a λ -eigenspace is often controlled by the “shape”of the boundary. ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 29
Lemma 4.6. (Boundaries to bound eigenspaces dimensions.) Let Γ be a weighted connected graph satisfying the Sunada-Sy condition. Let H = ∆ + q be a Schr¨odinger operator on Γ defined by a real boundedpotential q . Let Ω be a finite subset of vertices of Γ . Let λ ∈ R . Suppose H satisfies λ -uniqueness on Ω . Then the dimension of the image of theendomorphism U ∗ Ω E { λ } U Ω of the finite-dimensional space l (Ω , m Ω ) , isbounded above by the cardinality of the thick boundary of Ω : rank (cid:0) U ∗ Ω E { λ } U Ω (cid:1) ≤ | ∂ Ω | . Proof.
Consider a family of n functions g , . . . , g n , belonging toIm (cid:0) U ∗ Ω E { λ } U Ω (cid:1) . Assume that n > | ∂ Ω | . The lemma will be proved if we show that the family g , . . . , g n islinearly dependent. Asdim l ( ∂ Ω , m ∂ Ω ) = | ∂ Ω | < n, there exists ( α , . . . , α n ) ∈ C n \ { } , such that n X i =1 α i ( g i | ∂ Ω ) = 0 . Consider the corresponding linear combination: g = n X i =1 α i g i ∈ Im (cid:0) U ∗ Ω E { λ } U Ω (cid:1) . Let V denotes the vertex set of Γ. Let f ∈ l (Ω , m Ω ) such that g = U ∗ Ω E { λ } U Ω f, and let h = E { λ } U Ω f ∈ l ( V, m V ) . According to Proposition 4.4, Hh = λh .Claim: h | ∂ Ω = 0 . In order to prove the claim, we expend h with the help of the Hilbertbasis δ x , x ∈ V , of l ( V, m V ). Namely: h = X x ∈ V h ( x ) δ x . Hence, according to Implications 2 and 3, U ∗ Ω h = X x ∈ V h ( x ) U ∗ Ω δ x = X ω ∈ Ω h ( ω ) δ ω | Ω . In particular, for any ω ∈ Ω, h ( ω ) = ( U ∗ Ω h )( ω ) = ( U ∗ Ω E { λ } U Ω f )( ω ) = g ( ω ) . In the case ω ∈ ∂ Ω, remembering that g | ∂ Ω = 0, we obtain h ( ω ) = 0.This finishes the proof of the claim. We can now finish the proof ofthe lemma. Indeed, according to the second statement of Lemma 3.3(which applies because of the claim, the fact that Hh = λh , and thehypothesis of λ -uniqueness), P Ω h = 0 . We then have: 0 = P Ω h = P Ω E { λ } U Ω f = U Ω U ∗ Ω E { λ } U Ω f = U Ω g. As U Ω is one-to-one we deduce that g = 0. We conclude that n X i =1 α i g i = g = 0 , is a non-trivial linear combination, showing that the family g , . . . , g n is linearly dependent. (cid:3) We will need the following basic result from linear algebra.
Lemma 4.7.
Let H be a finite dimensional Hilbert space. Let A ∈ B ( H ) be an endomorphism of H (viewed as an operator). Then theusual trace of A is bounded by the rank of A times its norm: | T r ( A ) | ≤ rank ( A ) k A k . Proof.
Let r be the rank of A . There exists an orthonormal basis v , . . . , v r of Im( A ), an orthonormal family u , . . . , u r of H , and strictlypositive numbers λ , . . . , λ r (the singular values of A ), such that, forany w ∈ H ,(4) A ( w ) = r X i =1 λ i h w, u i i v i , (see for example [17, Theorem VI.17] and its proof, which is elementarywhen the Hilbert space is of finite dimension). If A = 0, the lemmais obvious and the above formula is correct with the convention thata sum over the empty set equals zero. Let w ∈ H be orthogonal to ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 31 all vectors u , . . . , u r . It follows from Formula 4 above that A ( w ) = 0.Hence T r ( A ) = r X j =1 h A ( u j ) , u j i . We obtain: | T r ( A ) | = | r X j =1 h A ( u j ) , u j i| = | r X j =1 h r X i =1 λ i h u j , u i i v i , u j i| = | r X j =1 λ j h v j , u j i| ≤ r X j =1 λ j ≤ r max ≤ j ≤ r λ j . The proof of the lemma will be finished if the following claim is true.Claim: k A k = max ≤ j ≤ r λ j . To prove the claim, on one-hand we choose 1 ≤ j m ≤ r such that λ j m = max ≤ j ≤ r λ j . Formula 4 shows that k A ( u j m ) k = λ j m . Hence k A k ≥ max ≤ j ≤ r λ j . On the other-hand, for any w ∈ H , weconsider its orthogonal projection r X j =1 α j u j , α j ∈ C , into the subspace spanned by u , . . . , u r . We have: k A ( w ) k = r X i =1 λ i |h w, u i i v i | = r X i =1 λ i |h r X j =1 α j u j , u i i v i | = r X i =1 λ i | α i | ≤ ( max ≤ j ≤ r λ j ) k w k . This proves that k A k ≤ max ≤ j ≤ r λ j . (cid:3) Proposition 4.8. (Finite-dimensional reduction.) Let Γ be a weightedconnected graph satisfying the Sunada-Sy condition. Let H = ∆ + q bea Schr¨odinger operator on Γ defined by a real bounded potential q . Let V be the vertex set of Γ and let Ω ⊂ V be a finite subset. Let λ ∈ R .Then X x ∈ Ω h E { λ } δ x , δ x i ≤ rank( U ∗ Ω AU Ω ) . Proof.
Let A ∈ B ( l ( V, m V )). For any x ∈ V , it is obvious that U Ω ( δ x | Ω ) = δ x . Hence, Lemma 4.7 leads to the following upper bound: X x ∈ Ω h Aδ x , δ x i = X x ∈ Ω h AU Ω ( δ x | Ω ) , U Ω ( δ x | Ω ) i = X x ∈ Ω h U ∗ Ω AU Ω ( δ x | Ω ) , δ x | Ω i = T r ( U ∗ Ω AU Ω ) ≤ rank( U ∗ Ω AU Ω ) k U ∗ Ω AU Ω k . According to Theorem 4.1, k E { λ } k = k Φ( { λ } ) k ≤ k { λ } ) k = 1 , hence we see that k U ∗ Ω E { λ } U Ω k ≤
1. Choosing A = E { λ } , we obtain X x ∈ Ω h E { λ } δ x , δ x i ≤ rank( U ∗ Ω E { λ } U Ω ) . (cid:3) Proposition 4.9. (Boundaries to bound traces.) Let Γ be a weightedconnected graph satisfying the Sunada-Sy condition. Let H = ∆ + q bea Schr¨odinger operator on Γ defined by a real bounded potential q . Let V be the vertex set of Γ and let Ω ⊂ V be a finite subset. Let λ ∈ R .Assume H satisfies λ -uniqueness on Ω . Then X x ∈ Ω h E { λ } δ x , δ x i ≤ | ∂ Ω | . Proof.
The inequality follows from the combination of Proposition 4.8with Lemma 4.6. (cid:3) Large-scale geometry
Definition 5.1. (Packing number relative to a family.) Let X be aset. Suppose F is a given family of non-empty subsets of X . For anysubset Ω of X , we define the packing number of Ω relative to the family F as the maximal number of disjoint elements of F which are included ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 33 in Ω , with the convention that this number is zero if no element of F is included in Ω : P (Ω , F ) = 0 if there is no F ∈ F such that F ⊂ Ω ,P (Ω , F ) = max { n : G ≤ i ≤ n F i ⊂ Ω , F i ∈ F } otherwise . Notice that in the case F contains each singleton of X , then P (Ω , F )coincides with the cardinality of Ω. Definition 5.2. (Nets and maximal nets.) Let ( X, d ) be a metric space.Let r ≥ . An r -net of X is a subset N ( X, r ) ⊂ X such that ∀ x, y ∈ N ( X, r ) , ( d ( x, y ) ≤ r ) = ⇒ ( x = y ) . A maximal r -net of X is an r -net of X which is maximal with respectto the inclusion relation among all r -nets of X . Let (
X, d ) be a metric space. Let x ∈ X and let r ≥
0. Consider theclosed ball in X with center x and radius r : B ( x, r ) = { y ∈ X : d ( y, x ) ≤ r } . Definition 5.3. (Space with uniform growth.) A metric space ( X, d ) has uniform growth if for each r ≥ , Vol( r ) = sup x ∈ X | B ( x, r ) | < ∞ . Lemma 5.4. (Bounding a set with a net.) Let ( X, d ) be a metric spacewith uniform growth. Let Ω ⊂ X be a subset. For r ≥ , let N (Ω , r ) be a r -net of Ω . If N (Ω , r ) is maximal then | Ω | ≤ Vol( r ) · | N (Ω , r ) | . Proof. As N (Ω , r ) is a maximal r -net of Ω there exists a map p : Ω → N (Ω , r )whose maximal displacement is bounded by r : ∀ x ∈ Ω , d ( x, p ( x )) ≤ r. Hence: ∀ y ∈ N (Ω , r ) , p − ( { y } ) ⊂ B ( y, r ) . We conclude that | p − ( { y } ) | ≤ Vol( r ) and asΩ ⊂ G y ∈ N (Ω ,r ) p − ( { y } )we deduce that | Ω | ≤ Vol( r ) · | N (Ω , r ) | . (cid:3) Definition 5.5. (The r -interior of a subset.) Let ( X, d ) be a metricspace. Let Ω ⊂ X and r ≥ . The r -interior of Ω is the set I (Ω , r ) ofpoints of Ω which lies at “depth” strictly greater than r . More precisely I (Ω , r ) = Ω \ ∂ r Ω = { x ∈ Ω : d ( x, X \ Ω) > r } . Definition 5.6. (Inclusive radius.) Let ( X, d ) be a metric space. Let x ∈ X . Let F be a family of non-empty subsets of X . The inclusiveradius r ( x, F ) at x relative to the family F is the minimal r ≥ suchthat there exists a set F ∈ F such that F ⊂ B ( x, r ) . Proposition 5.7. (Interior points as a lower bound for the packingnumber.) Let ( X, d ) be a metric space with uniform growth Vol . Let F be a family of non-empty subsets of X . Assume ≤ r < ∞ is auniform upper bound for the inclusive radii relative to F : ∀ x ∈ X, ∃ F x ∈ F : F x ⊂ B ( x, r ) . Then the number | I (Ω , r ) | of r -interior points of Ω satisfies | I (Ω , r ) | ≤ Vol(2 r ) · P (Ω , F ) . Proof.
Let N ( I (Ω , r ) , r ) be a 2 r -net of I (Ω , r ). As r is a uniformupper bound for the inclusive radii relative to F , we may choose foreach point x ∈ N ( I (Ω , r ) , r ) a set F x ∈ F such that F x ⊂ B ( x, r ) . Hence, as x ∈ I (Ω , r ), it is obvious that F x ⊂ Ω . Let x, y ∈ N ( I (Ω , r ) , r ). By definition of a 2 r -net,( x = y ) = ⇒ ( B ( x, r ) ∩ B ( y, r ) = ∅ ) . Therefore, we obtain a disjoint union of subsets of Ω, G x ∈ N ( I (Ω ,r ) , r ) F x ⊂ Ω , where each subset F x belongs to the family F . This proves that thecardinal of the net N ( I (Ω , r ) , r ) is a lower bound for the packingnumber of Ω relative to F : | N ( I (Ω , r ) , r ) | ≤ P (Ω , F ) . If we choose the net N ( I (Ω , r ) , r ) maximal among 2 r -nets of I (Ω , r )then Lemma 5.4 implies that | I (Ω , r ) | ≤ Vol(2 r ) · | N ( I (Ω , r ) , r ) | . ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 35
Finally: | I (Ω , r ) | ≤ Vol(2 r ) · P (Ω , F ) . (cid:3) Definition 5.8. (Geodesic distance with integer values.) Let ( X, d ) bea metric space. We say that the distance d is geodesic with integervalues if the following conditions hold true:(1) ∀ x, y ∈ X, d ( x, y ) ∈ N ∪ { } ,(2) for all x, z ∈ X , for all integer k such that ≤ k ≤ d ( x, z ) ,there exists y ∈ X such that d ( x, y ) = k and d ( x, z ) = d ( x, y ) + d ( y, z ) . Lemma 5.9. (Comparing boundaries.) Let ( X, d ) be a metric spacewith uniform growth Vol . Assume d is geodesic with integer values.Let Ω ⊂ X . Let r > . Then | ∂ r Ω | ≤ Vol( r − · | ∂ Ω | . Proof.
Let x ∈ ∂ r Ω. By definition of ∂ r Ω the set Z = { z ∈ X \ Ω : d ( x, z ) ≤ r } is non-empty. As d takes integer values, there exists z ∈ Z such that d ( x, z ) = d ( x, Z ) . Notice that x = z because x ∈ Ω and z ∈ X \ Ω. Hence d ( x, z ) ≥ d is geodesic with integral values, there exists y ∈ X such that d ( y, z ) = 1 and such that d ( x, z ) = d ( x, y ) + d ( y, z ) . The minimality of z implies that y ∈ Ω. From the facts that z ∈ X \ Ωand d ( y, z ) = 1, we conclude that y ∈ ∂ Ω. Notice also that d ( x, y ) = d ( x, z ) − d ( x, Z ) − ≤ r − . Denoting y = p ( x ) we see that we have constructed a map p : ∂ r Ω → ∂ Ωsuch that d ( x, p ( x )) ≤ r −
1. As in the proof of Lemma 5.4, we boundthe cardinalities of the fibres of p and obtain | ∂ r Ω | ≤ Vol( r − · | ∂ Ω | . (cid:3) The next lemma brings a lower bound on the number of interiorpoints in a finite set. It will be applied to some finite sets of verticesin a graph which have a “a relatively small boundary”. The idea of afinite set with “a relatively small boundary” is formalized through thedefinition of a Følner sequence (see Definition 6.6 below).
Lemma 5.10. (Følner to bound below interior points.) Let ( X, d ) bea metric space with uniform growth Vol . Assume d is geodesic withinteger values. Let Ω ⊂ X . Let r > . Assume ǫ ≥ is such that Vol( r ) · | ∂ Ω | ≤ ǫ | Ω | . Then the number of r -interior points is bounded below as follows: | I (Ω , r ) | ≥ (1 − ǫ ) | Ω | . Proof.
According to Lemma 5.9, | ∂ r Ω | ≤ Vol( r ) · | ∂ Ω | . Hence, | I (Ω , r ) | = | Ω \ ∂ r Ω | = | Ω | − | ∂ r Ω |≥ | Ω | − Vol( r ) · | ∂ Ω | ≥ | Ω | − ǫ | Ω | = (1 − ǫ ) | Ω | . (cid:3) Groups and quasi-homogeneous graphs
Groups acting on graphs.
Let G be a group acting (on the left)on a graph (Γ , V, E, o, t, ι ). It means that G acts both on V and on E and that the two actions are compatible in the sense that ∀ g ∈ G, ∀ e ∈ E, o ( ge ) = go ( e ) , t ( ge ) = gt ( e ) . For example, a group G acts on any of its Cayley graph C ( G, S ) andthe action preserves the path metric. If the group G acts on a weightedgraph, we always assume that the weights are invariant: ∀ g ∈ G, ∀ x ∈ V, m V ( gx ) = m V ( x ) , ∀ e ∈ E, m E ( ge ) = m E ( e ) . The permutation representation associated to a m V -preserving G -actionon V is the unitary representation defined by the following conditions: π : G → U ( l ( V, m V )) , ∀ g ∈ G, ∀ f ∈ l ( V, m V ) , ∀ x ∈ V, ( π ( g ) f )( x ) = f ( g − x ) . ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 37
If the weighted graph Γ satisfies the Sunada-Sy condition, and if theSchr¨odinger operator H = ∆+ q is defined with the help of a G -invariantpotential q , then ∀ g ∈ G, π ( g ) H = Hπ ( g ) . Hence H belongs to the commutant π ( G ) ′ = { A ∈ B ( l ( V, m V ) : ∀ g ∈ G, π ( g ) A = Aπ ( g ) } of π . It is straightforward to check that π ( G ) ′ is an algebra, whichis stable under conjugation, contains the identity, and which is closedwith respect to the strong operator topology. (In other words, it is avon Neumann algebra.)6.2. The von Neumann trace.Definition 6.1.
A graph Γ is quasi-homogeneous if there exists a group G acting on Γ (in the sense explained in Subsection 6.1 above) such thatthe action of G on the vertex set of Γ has a finite number of orbits.(We don’t require freeness of the action.) Here and in what follows, no hypothesis on the stabilizers of theaction is needed.
Definition 6.2. (The von Neumann trace of a positive operator.) Let Γ be a weighted graph and let G be a group acting on Γ . We assumethat the weights m V and m E are G -invariant. Assume the vertex set V of Γ decomposes as a finite union of G -orbits (in the case the degrees ofthe vertices of Γ are finite, this is equivalent to assume that the actionof G is cocompact). Let D be a fundamental domain for this action.In other words, the vertex set V is the disjoint union of the orbits ofthe vertices of the fundamental domain: V = G x ∈ D Gx.
Let A ∈ B ( l ( V, m V ) be in the commutant π ( G ) ′ of the permutationrepresentation π ( G ) as defined in Subsection 6.1 above. Assume that A is a positive operator, that is: ∀ f ∈ l ( V, m V ) , h Af, f i ≥ . We define τ ( A ) = X x ∈ D h Aδ x , δ x i . We normalize τ and define the von Neumann trace of A as τ ( A ) = 1 | D | X x ∈ D h Aδ x , δ x i . Proposition 6.3. (Some properties of the von Neumann trace.) Withthe notation as above, let A ∈ π ( G ) ′ be a positive operator. The fol-lowing properties are true.(1) Neither τ ( A ) nor τ ( A ) depends on the choice of the fundamen-tal domain,(2) ≤ τ ( A ) ≤ k A k ,(3) τ ( A ) = 0 if and only if A = 0 .Proof. Il order to prove the firt property, let D and F be fundamentaldomains. Let x ∈ D . The orbit Gx meets F in a single point y = y ( x ).Let us choose for each x ∈ D an element g x ∈ G such that y ( x ) = g x x .We have: F = G x ∈ D { g x x } . We obtain: X y ∈ F h Aδ y , δ y i = X x ∈ D h Aδ g x x , δ g x x i = X x ∈ D h Aπ ( g x ) δ x , π ( g x ) δ x i = X x ∈ D h π ( g x ) Aδ x , π ( g x ) δ x i = X x ∈ D h Aδ x , δ x i = τ ( A ) . This finishes the proof of the first property. The fact that τ ( A ) and τ ( A ) are positive follows from the positivity of A . The Cauchy-Schwartz inequality implies τ ( A ) = X x ∈ D |h Aδ x , δ x i| ≤ | D |k A k . This proves the second property. The fact that τ (0) = 0 is obvious.As h Aδ x , δ x i ≥ x , the hypothesis τ ( A ) = 0 and the fact that τ ( A ) does not depend on the choice of the fundamental domain implies: ∀ x ∈ V, h Aδ x , δ x i = 0 . As A is positive, there exists a self-adjoint operator B such that A = B . Hence, ∀ x ∈ V, h Aδ x , δ x i = k Bδ x k . This implies B = 0 hence A = 0. (cid:3) ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 39
Example 6.4. (Integrated density of states of the combinatorial Lapla-cian on Z .) Let Γ = C ( Z , { } ) be the Cayley graph of the infinite cyclicgroup Z with respect to the generating set { } . The simple randomwalk on Γ is reversible with respect to the measure on Z which takesthe value on each singleton. The corresponding combinatorial Lapla-cian ∆ : l ( Z ) → l ( Z ) is defined by the conditions ∆ ϕ ( n ) = ϕ ( n ) −
12 ( ϕ ( n + 1) + ϕ ( n − , ∀ n ∈ Z , ∀ ϕ ∈ l ( Z ) . The integrated density of state of ∆ can be computed as τ ( E λ ) = 12 π Z π − π { h ≤ λ } ( θ ) dθ, where for any θ ∈ [ − π, π ] , h ( θ ) = 1 − cos θ , and where { h ≤ λ } denotesthe characteristic function of the level set { h ≤ λ } = { θ : h ( θ ) ≤ λ } .More explicitly: τ ( E λ ) = if λ ≤ , arccos(1 − λ ) π if ≤ λ ≤ , if ≤ λ. Proof.
It is obvious that the degree of the vertices is constant equalto 2 and that the simple random walk is reversible with respect tothe measure on Z which takes the value 1 on each singleton. Hence,applying the formula from Item (3) of Subsection 2.5, we deduce thatthe combinatorial Laplacian ∆ : l ( Z ) → l ( Z ) satisfies∆ ϕ ( n ) = ϕ ( n ) −
12 ( ϕ ( n + 1) + ϕ ( n − . We denote δ n ∈ l ( Z ) the characteristic function of the singleton { n } ⊂ Z . The Fourier transform F : L (cid:0) T (cid:1) → l ( Z )satisfies F (cid:0)(cid:0) θ e iπθn (cid:1)(cid:1) = δ n for any n ∈ Z . It is a unitary transformation of Hilbert spaces. Theconjugation by F c ( F ) : B (cid:0) L (cid:0) T (cid:1)(cid:1) → B ( l ( Z ))defined on any A ∈ B ( L ( T )) as c ( F ) A = F A F − is an isomorphismof C ∗ -algebras. The function h : [ − π, π ] → [0 , h ( θ ) = 1 − cos θ defines the multiplication operator M h : L (cid:0) T (cid:1) → L (cid:0) T (cid:1) ϕ hϕ. It satisfies ∆ = c ( F ) M h . The operator M h is bounded self-adjointpositive with spectrum [0 , C ∗ -algebrasΦ ∆ : B ([0 , → B ( l ( Z ))which sends the identity function on [0 ,
2] to ∆. The Borel functionalcalculus applied to the operator M h produces the morphism of C ∗ -algebras Φ M h : B ([0 , → B (cid:0) L (cid:0) T (cid:1)(cid:1) which sends the identity function on [0 ,
2] to M h . The composition c ( F ) Φ M h : B ([0 , → B ( l ( Z ))is a morphism of C ∗ -algebras sending the identity function on [0 ,
2] to∆ = c ( F ) M h . Hence, the unicity property in the Borel functionalcalculus implies that Φ ∆ = c ( F ) Φ M h . In particular, for any λ ∈ R , the spectral projections E ∆ λ = Φ ∆ (cid:0) ] −∞ ,λ ] (cid:1) and E M h λ = Φ M h (cid:0) ] −∞ ,λ ] (cid:1) satisfy E ∆ λ = c ( F ) E M h λ . Applying Proposition 4.3 above, we obtain: τ ( E ∆ λ ) = h E ∆ λ δ , δ i l ( Z ) = h c ( F ) E M h λ δ , δ i l ( Z ) = h E M h λ T , T i L ( T ) = h { h ≤ λ } T , T i L ( T ) = 12 π Z π − π { h ≤ λ } ( θ ) dθ. The explicit expression for τ ( E ∆ λ ) easily follows. (cid:3) Theorem 6.5. (Boundaries to bound von Neumann traces.) Let Γ be a connected weighted graph satisfying the Sunada-Sy condition. Let G be a group acting on Γ (as defined in Subsection 6.1). (We makeno freeness hypothesis.) We assume the weigths m V and m E are G -invariant. Let H = ∆ + q be a Schr¨odinger operator on Γ defined by areal valuated G -invariant potential q . Suppose the action of G on thevertex set V of Γ has a finite number of orbits. Let F be the family ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 41 of all fundamental domains for the action of G on V . Let Ω ⊂ V bea finite subset. Assume the packing number P (Ω , F ) of Ω with respectto the family F is nonzero: in other words, Ω contains as a subset atleast one fundamental domain. Let λ ∈ R . Let E { λ } be the spectralprojection onto the λ -eigenspace of H . If H satisfies λ -unicity on Ω then τ (cid:0) E { λ } (cid:1) ≤ | ∂ Ω | P (Ω , F ) . Proof.
Let n = P (Ω , F ). Let F , . . . , F n be a collection of disjointfundamental domains included in Ω. According to Proposition 4.2,the spectral projection E { λ } is positive and belongs to the commutant π ( G ) ′ . Therefore, we may apply Proposition 6.3 to deduce that thevon Neumann trace of E { λ } does not depend on the choice of the fun-damental domain. We therefore have: n · τ (cid:0) E { λ } (cid:1) = n X x ∈ F h E { λ } δ x , δ x i = X x ∈ F ni =1 F i h E { λ } δ x , δ x i≤ X x ∈ Ω h E { λ } δ x , δ x i . As H satisfies λ -unicity, Proposition 4.9 applies hence we deduce that X x ∈ Ω h E { λ } δ x , δ x i ≤ | ∂ Ω | . (cid:3) Quasi-homogeneous graphs with a Følner sequence.Definition 6.6. (Følner sequence in a graph.) Let Γ be a connectedgraph. A Følner sequence in Γ is a sequence (Ω n ) n ≥ of finite subsetsof the vertex set of Γ such that lim n →∞ | ∂ Ω n || Ω n | = 0 . Theorem 6.7. (Continuity of the integrated density of states.) Let Γ be a connected weighted graph satisfying the Sunada-Sy condition andadmitting a Følner sequence. Let G be a group acting on Γ (as definedin Subsection 6.1). (We make no freeness hypothesis.) We assume theweights m V and m E are G -invariant. Let H = ∆ + q be a Schr¨odingeroperator on Γ defined by a real valuate G -invariant potential q . Suppose the action of G on the vertex set V of Γ has a finite number of orbits.Let λ ∈ R . The following conditions are equivalent.(1) The operator H satisfies λ -uniqueness on any finite subset Ω of V .(2) There exists a Følner sequence Ω n in V such that H satisfies λ -uniqueness on each Ω n .(3) The spectral projection E { λ } of H is equal to zero.(4) The only λ -eigenfunction of H is identically equal to zero.(5) The only λ -eigenfunction of H with finite support is identicallyequal to zero.(6) The integrated density of states λ τ ( E λ ) of H is continuousat λ .Proof. We first show(1) = ⇒ (2) = ⇒ (3) = ⇒ (4) = ⇒ (5) = ⇒ (1) , then (3) ⇐⇒ (6). The implication (1) = ⇒ (2) is obvious. In order toprove (2) = ⇒ (3), assume we have a Følner sequence Ω n such that H satisfies λ -unicity on each Ω n . Let us consider the family F ⊂ P ( V )of all fundamental domains for the action of G on V . According toTheorem 6.5, τ (cid:0) E { λ } (cid:1) ≤ | ∂ Ω n | P (Ω n , F ) . Let us check that the hypothesis of Proposition 5.7 are fulfilled. Thegroup G acts by isometries on the vertex set ( V, d ) of Γ endowed withits path metric and the action has a finite number of orbits. Thisimplies that the growth of (
V, d ) is uniform (define Vol as the maximalgrowth over all vertices belonging to a finite fundamental domain forthe action of G on V ). This also implies the existence of a uniformupper bound 1 < r < ∞ for the inclusive radii relative to F : ∀ x ∈ X, ∃ F x ∈ F : F x ⊂ B ( x, r ) . (Choose a fundamental domain D for the action of G on V . For each x ∈ V , choose r x big enough so that D ⊂ B ( x, r x ). As D is finite, r = max x ∈ D r x < ∞ . If g ∈ G then gD ⊂ B ( gx, r x ). This showsthat r is a (bad) uniform upper bound for the inclusive radii relative to F .) Applying Proposition 5.7, we obtain that the number of r -interiorpoints of any set Ω n is bounded in terms of the uniform growth andthe packing number: | I (Ω n , r ) | ≤ Vol(2 r ) · P (Ω n , F ) . ISCRETE LAPLACE OPERATORS WITHOUT EIGENVALUES 43
The metric space (
V, d ) is geodesic in the sense of Definition 5.8. Ap-plying Lemma 5.10 to ǫ = 1 / n such that(5) Vol( r ) · | ∂ Ω n | ≤ | Ω n | , we obtain: | I (Ω n , r ) | ≥ | Ω n | . According to Lemma 5.9, | ∂ r Ω | ≤ Vol( r − · | ∂ Ω | . Eventually, we come to the conclusion that(6) τ (cid:0) E { λ } (cid:1) ≤ r )Vol(1) | ∂ Ω n || Ω n | , providing Ω n satisfies Inequality (5). By definition of a Følner sequence,lim n →∞ | ∂ Ω n || Ω n | = 0Hence Inequality (5) above holds if n is big enough. Letting n goesto infinity in Inequality (6) above we deduce that τ (cid:0) E { λ } (cid:1) = 0. Ac-cording to Proposition (4.2), the operator E { λ } is positive and belongsto the commutant π ( G ) ′ because H ∈ π ( G ) ′ . Hence Proposition 6.3applies to the operator E { λ } :(7) τ (cid:0) E { λ } (cid:1) = 0 ⇐⇒ E { λ } = 0 . This finishes the proof of (2) = ⇒ (3). According to Proposition 4.4,condition (3) and condition (4) in the theorem are equivalent. Theimplication (4) = ⇒ (5) is obvious. In order to prove (5) = ⇒ (1), letΩ be a finite subset of V and let ϕ be a λ -eigenfunction of H whichvanishes outside of Ω. Its support is contained in the finite set Ω. Byhypothesis ϕ is identically zero on V . This proves that H satisfies λ -uniqueness on Ω. In order to prove the equivalence (3) ⇐⇒ (6),notice that according to Proposition 4.5, the function λ τ ( λ )is continuous at λ if and only if X x ∈ D h E { λ } δ x , δ x i = 0 , where D is a fundamental domain for the action of G on V . Accordingto Equivalence (7) above, this last condition is equivalent to E { λ } =0. (cid:3) References [1] Laurent Bartholdi and Wolfgang Woess,
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