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THE ESSENTIAL SPECTRUM OF THE LAPLACIAN ON RAPIDLY BRANCHING TESSELLATIONS MATTHIAS KELLER 8 0 Abstract. Inthispaperwecharacterizeemptinessoftheessenti- 0 alspectrumofthe Laplacianunderahyperbolicityassumptionfor 2 general graphs. Moreover we present a characterizationfor empti- n ness of the essential spectrum for planar tessellations in terms of a curvature. J 7 1 ] 0. Introduction and main results h p - The paper is dedicated to investigate the essential spectrum of the h t Laplacian on graphs. More precisely the purpose is threesome. Firstly a m we give a comparison theorem for the essential spectra of the Laplaci- an ∆ used in the Mathematical Physics community (see for instance [ [ASW, AF, AV, Br, CFKS, FHS, GG, Go, Kl, KLPS]) and the com- 2 v binatorial Laplacian ∆ used in Spectral Geometry (see for instance 6 [DKa, DKe, Fu, Wo1]) on general graphs. 1 8 Secondly we consider egraphs which are rapidly branching, i.e. the ver- 3 tex degree is growing uniformly as one tends to infinity. We establish . 2 a criterion under which absence of essential spectrum of the Laplacian 1 ∆ is completely characterized. This criterion will be positivity of the 7 0 Cheeger constant at infinity introduced in [Fu], based on [Che, D1]. It : turns out that in the case of planar tessellating graphs this positivity v i will be implied automatically by uniform growth of vertex degree. Mo- X reover we can interpret the rapidly branching property as a uniform r a decrease of curvature. An immediate consequence is that these operators have no continuous spectrum. The third purpose is to demonstrate that ∆ and ∆ may show a very different spectral behavior. Therefore we discuss a particular class of rapidly branching graphs. This discussion will alsoeprove independence of our assumptions in the results mentioned above. In the following introduction we will give an overview. We refer to Section 1 for precise definitions. There is a result of H. Donnelly and P. Li [DL] on negatively curved manifolds. It shows that the Laplacian ∆ ona rapidly curving manifold has a compact resolvent, i.e. empty essential spectrum. 1 2 M. KELLER Theorem (Donnelly, Li) Let M be a complete, simply connected, negatively curved Riemannian manifold and K(r) = sup K(x,π) { | d(p,x) r the sectional curvature for r 0, where d is the distan- ≥ } ≥ ce function on the manifold, p M and π is a two plane in T M. x ∈ If lim K(r) = , then ∆ on M has no essential spectrum i.e. r→∞ −∞ σ (∆) = . ess ∅ A remarkable result of K. Fujiwara [Fu] provides an analogue in the graph case for the combinatorial Laplacian ∆. Theorem (Fujiwara) Let G = (V,E) be an infinite graph. Then e σ (∆) = 1 if and only if α = 1. ess ∞ { } Hereeα is a Cheeger constant at infinity. Since the combinatorial ∞ Laplacian ∆ is a bounded operator the essential spectrum can not be empty. Yet it shrinks to one point for α = 1. ∞ e We will show that an analogue result holds for the Laplacian ∆, which is used in the community of mathematical physicists. Let G = (V,E) be an infinite graph. For compact K V denote by Kc its complement ⊂ V K and let \ m = inf deg(v) v Kc and M = sup deg(v) v Kc , K K { | ∈ } { | ∈ } where deg : V N is the vertex degree. Denote → m = lim m and M = lim M . ∞ K ∞ K K→∞ K→∞ In the next section we will be precise about what we mean by the limits. We call a graph rapidly branching if m = . We will prove ∞ ∞ the following theorems. Theorem 1. Let G be infinite. For all λ σ (∆) it holds ess ∈ m infσ (∆) λ M supσ (∆) ∞ ess ∞ ess ≤ ≤ and e e infσ (∆) min m ,M infσ (∆) . ess ∞ ∞ ess ≤ { } In the first statement we have the convention thaet if infσ (∆) = 0 ess and m = we set m infσ (∆) = 0. The first part of the theorem ∞ ∞ ess ∞ shows that the essential spectra of the operators ∆ and ∆ correespond in terms of the minimal and maeximal vertex degree at infinity. The second part gives two options to estimate the infimeum of the essential spectrum of ∆ from above. Oursecond theoremisthecharacterizationofemptiness oftheessential spectrum. THE ESSENTIAL SPECTRUM ON RAPIDLY BRANCHING TESSELLATIONS 3 Theorem 2. Let G = (V,E) be infinite and α > 0. Then σ (∆) = ∞ ess ∅ if and only if m = . ∞ ∞ Note that m = does not imply α > 0 or σ (∆) = . This will ∞ ∞ ess ∞ ∅ be discussed in Section 4. We may interpret α > 0 as an assumption on the graph to be hyper- ∞ bolic at infinity. (See discussion in [Hi] and the references [GH, Gr, LS] found there.) Moreover the growth of the vertex degree can be inter- preted as decrease of the curvature. In this way we may understand Theorem 2 as an analogue of Donnelly and Li for ∆ on graphs. For tessellating graphs this analogy will be even more obvious. Since the continuous spectrum of an operator is always contained in the essential spectrum there is an immediate corollary. Corollary 1. Let G = (V,E) be infinite, α > 0 and m = . Then ∞ ∞ ∞ ∆ has pure point spectrum. The class of examples for which [Fu] shows absence of essential spectrum are rapidly branching trees. We will show that the result is also valid for rapidly branching tessellations. We will formulate the statement in terms of the curvature because this makes the analogy to Don- nelly and Li more obvious. For this sake we define the combinatorial curvature function κ : V R for a vertex v V as it is found in → ∈ [BP1, BP2, Hi, Wo2] by deg(v) 1 κ(v) = 1 + , − 2 deg(f) f∈F,v∈f X where deg(f) denotes the number of vertices contained in a face f F. ∈ For compact K V let ⊂ κ = sup κ(v) v Kc K { | ∈ } and κ = lim κ . Obviously κ = is equivalent to m = . ∞ K→∞ K ∞ ∞ −∞ ∞ Here is our main theorem. Theorem 3. Let G be a tessellation. Then σ (∆) = if and only if ess ∅ κ = . Moreover κ = implies σ (∆) = 1 . ∞ ∞ ess −∞ −∞ { } The theorem is a special case of Theorem 2. Tehe hyperbolicity assumption α > 0 follows from κ = in the case of tessellating graph. ∞ ∞ −∞ In particular it even holds α = 1 whenever the curvature tends uni- ∞ formly to . −∞ Klassert, Lenz, Peyerimhoff, Stollmann [KLPS] proved the absence of compactly supported eigenfunctions for non-positively curved tessellations. Since κ = implies non-positive curvature outside of a ∞ −∞ 4 M. KELLER certain set K the result applies here. Hence we have pure point spectrum such that all eigenfunctions are either supported in K or on an infinite set. To end this section we introduce a technical proposition which is used almost throughout all the proofs of the paper. It uses quite standard technics and may be of independent interest. For a linear operator B on a space of functions on V, we write B for its restriction to the K space of functions on Kc with Dirichlet boundary conditions, where K V is compact set. Let l2(V,g) be the space of square summable ⊂ functions with respect to the weight function g and c (V) the space of c compactly supported functions on V. Proposition 1. Let G = (V,E) be infinite and B a self adjoint operator with c (V) D(B) l2(V,g) which is bounded from below. Then c ⊆ ⊆ Bϕ,ϕ g infσ (B) = lim inf h i = lim infσ(B ), ess K K→∞ ϕ∈cc(V) hϕ,ϕig K→∞ suppϕ⊆Kc Bϕ,ϕ g supσ (B) lim sup h i = lim supσ(B ). ess K ≤ K→∞ ϕ∈cc(V) hϕ,ϕig K→∞ suppϕ⊆Kc If B is bounded, we have equality in the second formula. The paper is structured as follows. In Section 1 we will define the versi- onsoftheLaplacianwhich appearindifferent contexts oftheliterature. We discuss Fujiwara’s Theorem which can be understood as a result on compact operators. In Section 2 we prove Proposition 1 and Theorem 1 and 2. In Section 3 we give an estimate of the Cheeger constant at infinity for planar tessellations andprove Theorem 3.Finally in Section 4 we discuss a class of examples which shows that for general graphs ∆ and ∆ can have a quite different spectral behavior. 1. Thee combinatorial Laplacian ∆ in terms of compact operators e Let G = (V,E) be a connected graph with finite vertex degree in each vertex. For a positive weight function g : V R let + → l2(V,g) = ϕ : V R ϕ,ϕ = g(v) ϕ(v) 2 < , g { → | h i | | ∞} v∈V X c (V) = ϕ : V R supp ϕ < c { → | | | ∞} where supp is the support of a function. For g = 1 we write l2(V). Notice that l2(V,g) is the completion of c (V) under , . For g = c g h· ·i deg it is clear that l2(V,deg) l2(V) and if sup deg(v) < then ⊆ v∈V ∞ l2(V) = l2(V,deg). We occasionally write l2(G,g) for l2(V,g). THE ESSENTIAL SPECTRUM ON RAPIDLY BRANCHING TESSELLATIONS 5 For ϕ c (V) and v V define the operators c ∈ ∈ (Aϕ)(v) = ϕ(u) and (Dϕ)(v) = deg(v)ϕ(v). u∼v X The operator A is often called the adjacency matrix. Since we assumed that the graph has no isolated vertices the operator D has a bounded inverse. The Laplace operator plays an important role in different areas of ma- thematics. Yet there occur different versions of it. To avoid confusion we want to discuss them briefly. We start with the Laplacian used in the Mathematical Physicist community in the context of Schrödinger operators. For reference see e.g. [CFKS, D1] (and the references there) or in more recent publications like [AF, ASW, AV, Br, D2, FHS, GG, Go, Kl, KLPS]. (1.) The operator D A defined on c (V) yields the following form c − 1 dϕ,dϕ = ϕ(u) ϕ(v) 2. h i 2 | − | v∈V u∼v XX The self adjoint operator on l2(V) corresponding to this form will be denoted by ∆. It gives for ϕ D(∆) and v V ∈ ∈ (∆ϕ)(v) = deg(v)ϕ(v) ϕ(u). − u∼v X Notice that ∆ is unbounded if there is no bound on the vertex degree. WenextintroducethecombinatorialLaplacian.Twounitaryequivalent versions are found in the literature. They are given as follows. (2.) Let ∆ = I A = I D−1A − − be defined on l2(V,deg), where I is the identity operator. It is easy to see that ∆ is bounded aend self aedjoint. For ϕ l2(V,deg) and v V ∈ ∈ it gives 1 e (∆ϕ)(v) = ϕ(v) ϕ(u). − deg(v) u∼v X The matrix A is ofteen called the transition matrix. This version of the combinatorial Laplacian can be found for instance in [DKa, DKe, Fu, Wo1] and maeny others. (3.) There is a unitary equivalent version as discussed e.g. in [Chu]. Let ∆ = I A = I D−21AD−21 − − be defined on l2(V). It gives for ϕ l2(V) and v V b b ∈ ∈ 1 (∆ϕ)(v) = ϕ(v) ϕ(u). − deg(u)deg(v) u∼v X b p 6 M. KELLER Notice that the operator 1 D2 : l2(V,deg) l2(V), ϕ deg ϕ, 1,deg → 7→ · p −1 is an isometric isomorphism and we denote its inverse by D 2 . Then deg,1 1 −1 ∆ = D2 ∆D 2 . 1,deg deg,1 Moreover on c (V) c b e 1 1 ∆ = D2∆D2. Furthermore we define the Dirichlet restrictions of these operators. For b a set K V let P : l2(V,g) l2(Kc,g) be the canonical projection K ⊆ → and i : l2(Kc,g) l2(V,g) its dual operator, which is the continuation K → by 0 on K. For an operator B on l2(V,g) we write B = P Bi . K K K Hence we can speak of ∆ , ∆ or ∆ on Kc with Dirichlet boundary K K K conditions. Mostly K will be a compact set. e b For a graph G and compact K V define the Cheeger constant, see ⊆ [Che, DKe, Fu], ∂ W E α = inf | |, K W⊆Kc,|W|<∞ A(W) where ∂ W is the set of edges which have one vertex in W and one E outside and A(W) = deg(v). Let W Kc, for K compact and v∈W ⊆ χ the characteristic function of W. Two simple calculations, mentioned P in [DKa] yield ∆ χ,χ = ∆ χ,χ = ∂ W K deg K E h i h i | | and e χ,χ = Dχ,χ = A(W). deg h i h i This gives ∆χ χ W W deg (1) α = inf h i . K W⊆Kc,|W|<∞ χW,χW deg h i e The set K(V) of compact subsets of V is a net under the partial order . We say a function F : K(V) R, K F converges to F R K ∞ ⊆ → 7→ ∈ if for all ǫ > 0 there is a K K(V) such that F F < ǫ for all ǫ K ∞ ∈ | − | K K . We then write lim F = F . With this convention we ǫ K→∞ K ∞ ⊇ define the Cheeger constant at infinity like [Fu] by α = lim α . ∞ K K→∞ The limit always exists since α α 1 for compact K L V. K L ≤ ≤ ⊆ ⊆ Therefore we can think of taking the limit over distance balls of an arbitrary vertex. THE ESSENTIAL SPECTRUM ON RAPIDLY BRANCHING TESSELLATIONS 7 The next part is dedicated to a discussion of [Fu]. We will look at the result from the perspective of compact operators. The proof is based on two propositions which hold for general graphs. We present them here as norm estimates on the transition matrix. The essential part for the proof of the first proposition was remarked in [DKa]. Proposition 2. For any compact set K V ⊆ A 1 α . K K k k ≥ − In particular infσ(∆ ) α . K K ≤ e Proof By equation (1) we receive infσ(∆ ) α . Since A is self K K K ≤ adjoint we get infσ(e∆ ) = infσ(I A ) = 1 supσ(A ) = 1 A K K K K − − −k k and thus A 1 α . e e (cid:3) K K k k ≥ − e e e e Proposition 3. For any compact set K V e ⊆ A 1 α2 . k Kk ≤ − K q The second proposition is deerived from the proof of the Theorem in [DKe]. Alternatively it may be derived from Proposition 1 in [Fu]. The essential of the proof of this proposition goes back to Dodziuk, Kendall but the statement can be found explicitly in Fujiwara. We will use it later, so we state it here as a Theorem. Theorem 4. For K V compact ⊆ 1 1 α2 ∆ 1+ 1 α2 . − − K ≤ K ≤ − K q q Remark. (1.) For K = we get these estimates for the operator ∆. e ∅ We can also take the limits over K. This is one implication in Fujiwa- ra’s Theorem, since σ (∆) = σ (∆ ) σ(∆ ). e ess ess K K ⊆ (2.) Since the operators ∆ and ∆ are unitary equivalent, similar state- ments hold for ∆ and A.e e e e b The essential pabrts of tbhe next theorem are already found in [Fu]. The implications (i.) (ii.), (iii.) (ii.) and (iii.) (iv.) are minor ex- ⇒ ⇒ ⇔ tensions. Theorem 5. Let G be infinite. The following are equivalent. (i.) σ (∆) consists of one point. ess (ii.) σ (∆) = 1 . ess e { } (iii.) A is compact. e (iv.) lim A = 0. e K→∞k Kk (v.) α = 1. ∞ e 8 M. KELLER Proof The implication (i.) (ii.) is a consequence of Proposition ⇒ 1 and Theorem 4. The implication (ii.) (i.) is trivial. Furthermore ⇒ (ii.) is equivalent to σ (A) = 0 which is equivalent to (iii.). As- ess { } sume (iii.). Let (K ) be a growing sequence of compact sets. Choose n f l2(Kc,deg), f e= 1 such that 2 A f A . Be- n ∈ n k nkdeg k Kn nkdeg ≥ k Knk cause f is supported on Kc the sequence (f ) tends weakly to 0 as n n n n . The compactness of A implies lim eAi f = 0eand thus n n deg →∞ k k lim A = 0, which is (iv.). We assume (iv.), take the limit over K k Knk in Proposition 2 and concludee(v.). Suppose e(v.). For compact K we have eA = i A P +i A P +C , where C is a compact opera- Kc Kc Kc K K K K K tor.ByProposition 3 we have lim A 0.Moreover A is compact, K Kc k k ≤ since Ke is comepact. Thus Aeis the norm limit of compact operators and hence compact, which is (iii.). e e (cid:3) e Remark. We may think of the problem in an alternative way. For compactK V let G = (V ,E ) bethegraphinduced by thevertex K K K ⊆ set Kc, added by loops in the following way. To each vertex v Kc we ∈ add as many loops as there are edges in ∂ K = ∂ Kc which contain E E v. (We say an edge is a loop if its beginning and end vertex coincides.) We can define projections and embeddings for l2(G,g) and l2(G ,g) K as above. Note that the projected operators from l2(V,g) to l2(Kc,g) and l2(G,g) to l2(G ,g) are unitary equivalent. Thus we can separate K the proof explicitly in graph and operator theory. Proposition 2 and 3 hold for general graphs in particular also for G . On the other hand K Theorem 5 is only operator theory, which uses the estimates on the operator norm of A . K 2.eThe essential spectrum of ∆ In this section we compare the operators ∆ and ∆. We will establish bounds on the essential spectrum of ∆ by bounds obtained for ∆. Therefore we will firstly prove Propositione1. Then we prove two propositions which estimate the infimum of the essential spectrum of e∆ from below and above. This will be the ingredients for the proofs of Theorem 1 and 2. Proof of Proposition 1. Without loss of generality we can assume B 0. Let λ = infσ (B). Because σ (B) = σ (B ) σ(B ) 0 ess ess ess K K ≥ ⊆ it holds λ σ(B ) for any compact K V. To show the other 0 K ∈ ⊂ direction we prove that if there is an λ σ(B ) σ (B) then there is K ess ∈ \ L K such that λ σ(B ). It follows λ σ(B ) for L L , since 0 ⊃ 6∈ L0 6∈ L ⊇ 0 infσ(B ) infσ(B ) for L L . L ≥ L0 ⊇ 0 For compact K V let λ σ(∆ ) such that λ < λ . Choose λ such K 0 1 ⊂ ∈ that λ < λ < λ . 1 0 THE ESSENTIAL SPECTRUM ON RAPIDLY BRANCHING TESSELLATIONS 9 The spectral projection E is a finite rank operator since B 0. ]−∞,λ1] ≥ Let f ,...,f be an orthonormal basis of the finite dimensional sub- 1 n space E l2(V,g). Now for arbitrary ǫ > 0 choose a compact ]−∞,λ1] L V so large that for L L ǫ ǫ ⊂ ⊇ max P f 2 ǫ. j=1,...,nk L jkg ≤ Let L L . For ϕ l2(Lc,g) with ϕ = 1 there are β ,...,β R ǫ g 1 n ⊇ ∈ k k ∈ with β2+...+β2 1 such that E ϕ = β P f +...+β P f , 1 n ≤ ]−∞,λ1],L 1 L 1 n L n where E = P E i . Remember P was the projection of ]−∞,λ1],L L ]−∞,λ1] L L l2(V,g) onto l2(Lc,g) and i its dual. Thus K (2) E ϕ 2 = β2 P f 2 +...+β2 P f 2 ǫ. k ]−∞,λ1],L kg 1k L 1kg nk L nkg ≤ Now let ψ l2(Lc,g) such that B ψ,ψ (infσ(B )+ǫ) ψ,ψ and L g L g ∈ h i ≤ h i let dρ ( ) = d B E ψ,E ψ be a spectral measure of B . ψ L ]−∞,·],L ]−∞,·],L g L · h i Then B ψ,ψ = B E ψ,E ψ h L ig h L ]−∞,λ1],L ]−∞,λ1],L ig + B E ψ,E ψ h L ]λ1,∞[,L ]λ1,∞[,L ig t dρ (t) ψ ≥ Z]λ1,∞[ λ 1 dρ (t) 1 ψ ≥ Z]λ1,∞[ = λ ( ψ,ψ E ψ,E ψ ) 1 h ig −h ]−∞,λ1],L ]−∞,λ1],L ig λ (1 ǫ) ψ,ψ . 1 g ≥ − h i In the second step we used that B is positive and in the fifth step equation (2). Now we choose δ > 0 such that λ+δ < λ . Moreover let 1 λ (λ+δ) 1 ǫ = − λ +1 1 and L = L . By our choice of ψ and ǫ we get for all L L 0 ǫ 0 ⊇ B ψ,ψ L g infσ(B ) h i ǫ λ (1 ǫ) ǫ = λ+δ > λ. L 1 ≥ ψ,ψ − ≥ − − g h i If the operator B is bounded, we can do a similar estimate from above. Otherwise it still holds supσ (B) = supσ (B ) supσ(B ). (cid:3) ess ess K K ≤ Since ∆and∆areunitaryequivalent it makes no difference tocompare to operators ∆ and ∆ or the operators ∆ and ∆. Yet ∆ and ∆ are definede on thbe same space, so it seems to be easier with notation to compare theme. However to do this the fobllowing identity is vitabl. For ϕ c (Kc) one can calculate c ∈ 1 1 1 1 ∆ ϕ,ϕ D2 ∆ D2 ϕ,ϕ ∆ D2 ϕ,D2 ϕ D ϕ,ϕ (3) h K i = h K K K i = h K K K ih K i. ϕ,ϕ ϕ,ϕ 1 1 ϕ,ϕ D2ϕ,D2ϕ h i bh i bh K K i h i 10 M. KELLER Proposition 4. Let G be infinite. Then for λ σ (∆) ess ∈ m infσ (∆) λ M supσ (∆). ∞ ess ∞ ess ≤ ≤ Proof LetK V becompact.Byequation(3)wehaveforϕ c (Kc) c ⊂ b b ∈ 1 1 ∆ ϕ,ϕ ∆ D2 ϕ,D2 ϕ h K i h K K K i inf deg(v). ϕ,ϕ ≥ D21ϕ,D21ϕ v∈suppϕ h i bh K K i 1 For every ψ c (Kc) there is an ϕ c (Kc) such that ψ = D2 ϕ. ∈ c ∈ c K Furthermore c (Kc) is dense in the domain of ∆ and so we conclude c K infσ (∆) = infσ (∆ ) m infσ(∆ ). ess ess K ∞ K ≥ By Proposition 1 this yields the lower bound. If M = the ∞ b ∞ upper bound is infinity. Otherwise by equation (3) supσ(∆ ) K ≤ M supσ(∆ ) and again by Proposition 1 we the upper bound. (cid:3) ∞ K Propositiobn 5. Let G be infinite. Then infσ (∆) min m ,M infσ (∆) . ess ∞ ∞ ess ≤ { } Proof Letv V,n Nbepairwisedistinctsuchthatdeg(v ) m . n n ∞ ∈ ∈ b ≤ Moreover let χ the characteristic function of v . For K compact such n n that v Kc it holds n ∈ ∆ ϕ,ϕ K inf h i ∆ χ ,χ = deg(v ) m . K n n n ∞ ϕ∈cc(Kc) ϕ,ϕ ≤ h i ≤ h i By Proposition 1 we have infσ (∆) m . On the other hand we ess ∞ ≤ have by equation (3) infσ(∆ ) M infσ(∆ ). By Proposition 1 we K K K ≤ get infσ (∆) M infσ (∆). (cid:3) ess ∞ ess ≤ b b Proof of Theorem 1. Remember the operators ∆ and ∆ are unitary equivalent. Thus σ (∆) = σ (∆). The Theorem follows from Propo- ess ess (cid:3) sition 4 and 5. e b e b Proof of Theorem 2. By Theorem 4 we have infσ(∆ ) 1 1 α2 0. K ≥ − − K ≥ q ThusbytakingthelimitsProposition1yields infσ (∆) > 0ifα > 0. b ess ∞ (cid:3) Propositions 4 and 5 give the desired result. b Remark. Define H1(V) l2(V) as the subspace consisting of all f ⊆ with 1 f = f + df,df 2 < , k kH1 k k h i ∞ where the second term in the sum is the form of ∆ which was defined in Section 1. Let j : H1(V) l2(V) be the canonical inclusion. Then →