EIGENFUNCTIONS GROWTH OF R-LIMITS ON GRAPHS

SIEGFRIED BECKUS, LATIF ELIAZ

Abstract. We derive characterizations of the essential spectrum σess of Schrodinger operatorson infinite graphs, which involve the concept of R-limits. This concept, which was introducedpreviously for operators on N and Zd as “right-limits”, captures the “behaviour at infinity”of the operator. We first focus on such an operator H defined on a graph of sub-exponentialgrowth rate. By closely inspecting the growth rate of generalized eigenfunctions we are ableto show that each point in σess(H) corresponds to a bounded generalized eigenfunction of acorresponding R-limit of H. Additionally, if the graph is of uniform sub-exponential growth wederive such a correspondence also in the reverse direction, and show that the essential spectrumis equal to the union over the spectra of the R-limits.

1. Introduction

This work deals with Schrodinger operators H : `2(G)→ `2(G) of the form

(1.1) (Hψ)(v) :=∑u∼v

(ψ(u)− ψ(v)

)+W (v)ψ(v)

where W : V (G) → R is a bounded function and G is an infinite, connected graph with auniform bound on the vertex degree.

The essential spectrum of a linear bounded selfadjoint operator is invariant by compact pertur-bations (Weyl’s theorem). In light of this, one could naturally expect that the essential spectrumonly depends on the geometry at infinity. This relation has been exposed for a Schrodinger op-erator on N or Zn, as the essential spectrum was characterized by the union over the set ofenergies σ∞(H ′) corresponding to bounded generalized eigenfunctions of so-called right limits ofH (see [11, 41]). Here the term right limits is used in order to reflect the study of the geometryat infinity. In particular, if G is the Cayley graph of Z with the usual generators, a right limitis a strong limit point of a sequence of shifts of the original operator. This was extended to Zdin [23] and recently to general graphs with uniformly bounded vertex degree in [7]. Since thename right limit is not longer appropriate, these operators are called R-limits.

For infinite connected graphs with uniform bound on the vertex degree, the authors of [7]proved that the union over the spectra of all R-limits of H is contained in the essential spectrumof H, where H is a bounded Jacobi operator. Moreover they show that the converse inclusionholds on regular trees, and that on the contrary there exists an infinite, connected graph withuniform bound on the vertex degree such that equality does not hold. This is the starting pointof the present work. We show that under additional assumptions on the growth rate of balls inthe graph, the converse inclusion is obtained. Specifically, if the graph admits a sub-exponentialgrowth, then the essential spectrum is contained in the union of σ∞(H ′) over all R-limits ofH (see Theorem 2.2). Additionally, if the graph admits a uniform sub-exponential growth, wederive an equality between the essential spectrum and the union over the spectra of the R-limits,recovering the results known for Zn (see Theorem 2.5). In addition, we show that if the uniformsub-exponential growth is discarded, equality does not hold in general.

1

2 SIEGFRIED BECKUS, LATIF ELIAZ

Such results go back to the concept of limit operators based on Favard [13], Muhamadiev[30, 31], Lange-Rabinovich [22], Rabinovich-Roch-Silbermann [34, 35, 36] and Chandler-Wilde-Lindner [10, 11], see also [25, 26, 33, 42]. After the completion of this paper we learned about arecent interesting development [20] in this direction. There these methods were pushed forwardto metric measure spaces, relevant also to discrete graphs under some assumptions. Anotherapproach to tackle such questions comes from C∗-algebras [15, 16, 17, 18, 19, 27] which uses theconcept of localization at infinity, which coincides with the concept of R-limits. In this case theyobtain similar results for operators on locally compact, non-compact abelian groups. Recentlythese concepts were also extended to groupoid C∗-algebra [1, 8, 9]. For a more comprehensivereview and further references on the subject see [11, 23, 41].

In order to prove our second main result (Theorem 2.5), the so-called Shnol-type theoremis used. Shnol [39] proved that if a generalized eigenfunction admits at most a polynomialgrowth rate then the corresponding energy is in the spectrum of the operator. This result wasindependently discovered by Simon [40]. Since then various remarkable generalizations to theDirichlet setting were proven, see e.g. [2, 3, 4] and references therein. In the literature also theconverse question is addressed [5, 14, 24]. To be more precise, one seeks to find for µ-almostevery element in the spectrum, a generalized eigenfunction that has at most sub-exponentialgrowth where µ is the spectral measure of the operator H. Such a converse theorem is used inthe proof of Theorem 2.5.

1.1. Organisation. The main results of this work are presented in Section 2. Then the conceptof R-limits as wells as the proof of Theorem 2.2 are provided in Section 3. With this at hand,Theorem 2.5 is proven in Section 4.

1.2. Acknowledgements. L.E. acknowledges the support of the Israel Science Foundation(grants No. 399/16 and 970/15) founded by the Israel Academy of Sciences and Humanities.Parts of this work are included in the PhD thesis of L.E. [12], carried under the supervision ofJ. Breuer from the Hebrew University of Jerusalem. We are grateful to him for his significantsupport during the preparation of this work. S.B acknowledges the support of the Israel ScienceFoundation (grants No. 970/15) founded by the Israel Academy of Sciences and Humanities.

2. Setting and main results

A graph G consists of a countable vertex set V (G) and an edge set E(G) where an edge isrepresented by a tuple of vertices. Throughout this work we deal with undirected graphs andso the edge (u, v) is identified with the edge (v, u) for u, v ∈ V (G). The tuple (u, u) for someu ∈ V (G) is called loop. We only consider graphs without loops. Two vertices u and v are calledadjacent (u ∼ v) if (u, v) ∈ E(G). Then the vertex degree deg(v) of a vertex v ∈ V (G) is definedby

deg(v) := |{u ∈ V (G) : v ∼ u}|,where |S| denotes the cardinality of the set S. The tuple (G, v0) is called rooted d-boundedgraph if v0 ∈ V (G) is a fixed vertex and deg(v) ≤ d for all v ∈ V (G). Then `2(G) := `2(V (G))denotes the Hilbert space of all square summable functions ψ : V (G) → C. Furthermore,`∞(G) := `∞(V (G)) is the Banach space of bounded functions ψ : V (G) → C equipped withthe norm ‖ψ‖∞ := supu∈V (G) |ψ(u)|.

Throughout this work, we study the self-adjoint, linear and bounded operators acting onthe Hilbert space `2(G) of the form (1.1). Whenever W is chosen to be identically zero, theoperator is denoted by ∆, which is called the graph Laplacian. Furthermore, A = AG denotes

EIGENFUNCTIONS GROWTH ON GRAPHS 3

the adjacency operator on the graph G, which is a Schrodinger operator with W (v) := deg(v)for v ∈ V (G). Throughout this work a triple (H,G, v0) denotes a Schrodinger operator of theform (1.1) defined on the rooted graph (G, v0).

Let H be a Schrodinger operator on a connected infinite rooted graph (G, v0). The spectrumof H is denoted by σ(H). Then the discrete spectrum σdisc(H) ⊆ σ(H) is the set of isolatedeigenvalues of finite multiplicity and the essential spectrum is σess(H) := σ(H) \ σdisc(H).Furthermore, a function ψ : V (G)→ C is called a generalized eigenfunction of H with eigenvalueλ if ψ 6≡ 0 and Hψ(v) = λψ(v) for all v ∈ V (G). With this at hand, σ∞(H) denotes the set ofall λ such that there exists a bounded generalized eigenfunction ψ ∈ `∞(G) with eigenvalue λ.

Given a graph G, a chain of vertices (v0, v1, . . . , vn) is called a path in G if vi−1 ∼ vi for all1 ≤ i ≤ n. Clearly, such a path can be represented by n edges. A graph is connected if for everyu, v ∈ V (G), there is a path (v0, v1, . . . , vn) such that v0 = u and vn = v. Throughout this workwe deal only with connected and infinite (i.e. |V (G)| = ∞) graphs. Next the combinatorialgraph distance on G is defined by

dist (u, v) := inf{n∣∣ (v0, v1, . . . , vn) is a path with v0 = u and vn = v

}.

Note that the latter gives the smallest number of edges which are contained in a path connectingu and v. For a rooted graph (G, v0), the notation

|v| := dist (v, v0)

is used for the distance of a vertex v ∈ V (G) from the root v0. Then the sphere S(G)v0 (r) of radius

r ∈ N around the root v0 and the number of vertices Nv0(r) in such a sphere are defined by

S(G)v0 (r) := {v ∈ G |dist(v, v0) = r} , N(r) := Nv0(r) :=

∣∣∣S(G)v0 (r)

∣∣∣ .Similarly, the ball B

(G)v0 (r) of radius r ∈ N around the root v0 and the number of vertices νv0(r)

in such a ball are defined by

B(G)v0 (r) := {v ∈ G | dist(v, v0) ≤ r} , ν(r) := νv0(r) :=

∣∣∣B(G)v0 (r)

∣∣∣ .Definition 2.1. A rooted graph (G, v0) is of sub-exponential growth rate if for each γ > 1,there exists C = Cγ,v0 > 0 such that for every r ∈ N,

νv0(r) < Cγr.

Furthermore, a graph G is of uniform sub-exponential growth rate if the constant C > 0 canbe chosen independently of the choice of the root. Specifically, for each γ > 1, there exists aconstant C = Cγ > 0 such that

νu(r) < Cγr

holds for every u ∈ G and r ∈ N.

Notice that equivalently one can use N∗ instead of ν∗ in these definitions.

The concept of R-limits of a Schrodinger operator H defined on a graph G was recentlyintroduced in [7]. A precise mathematical definition is provided in Section 3.1. The main ideais to deal with the geometry at infinity of the underlying object. Specifically, (H ′, G′, v′0) is anR-limit of (H,G, v0) if there is a sequence of vertices {vn}n∈N ∈ V (G) going to infinity suchthat locally around v′0 (in each ball) the operator H ′ acts (up to a small error) like H aroundvn for n large enough. Note that a sequence of vertices goes to infinity if it leaves every finitesubset of the vertex set.

4 SIEGFRIED BECKUS, LATIF ELIAZ

The first main result of this work holds for any graph of sub-exponential (not necessarilyuniform) growth rate.

Theorem 2.2. Let (G, v0) be an infinite, connected rooted d-bounded graph of sub-exponentialgrowth rate, and H be a Schrodinger operator on `2 (G) of the form (1.1). Then

σess (H) ⊆⋃

H′∈R(H)

σ∞(H ′).

The latter inclusion is preserved also for the adjacency operator H = A on the d-regulartree G := Td, although this graph has exponential growth. In this case the only R-limit isthe same operator H ′ = A on Td. Then the essential spectrum σess(A) equals to σ(A) =[−2√d− 1, 2

√d− 1

], and furthermore, [−d, d] ⊂ σ∞(A) holds, see e.g. [29, Equations (7.6–

7.7)]. Thus,

[−d, d]∖ [−2√d− 1, 2

√d− 1

]⊂ σ∞(A)\σess(A)

follows where the set on the left hand side is non-empty.

Next we show that the inclusion of Theorem 2.2 can also be strict for graphs with sub-exponential growth rate. This is done by providing a specific example motivated by the previousconsiderations about the d-regular tree. This example is a particular case of a more general classof examples that we define next.

Definition 2.3. Let{Gk, v

(k)1 , v

(k)2

}∞k=1

be a sequence of finite graphs with two specific vertices

v(k)1 , v

(k)2 ∈ Gk, and let {k`}∞`=1 ⊆ N be an increasing sequence. The chain graph corresponding

to{Gk, v

(k)1 , v

(k)2

}∞k=1

and {k`}∞`=1 is defined iteratively as follows:

• Begin with the half line graph, with vertices V = N and edges E = {(n, n+ 1) |n ∈ N}.• For each k ∈ {k`}∞`=1, replace the vertex k with the graph Gk.

• The edges (k − 1, k), (k, k + 1) are replaced with the edges(k − 1, v

(k)1

),(v(k)2 , k + 1

).

• In case that k = k` = k`−1 + 1 the edge (k− 1, k) is replaced with the edge(v(k−1)2 , v

(k)1

).

Proposition 2.4. There exists an infinite, connected rooted d-bounded graph G of sub-exponential growth rate so that the adjacency operator H := AG satisfies that⋃

H′∈R(H)

σ∞(H ′)\ σess(H)

is nonempty.

Proof. Let Td be the d-regular tree (d > 2) with root v0, and denote by T (k) the finite subgraph

which is the restriction of Td to B(T )v0 (k). Let {k`}∞`=1 be the sequence defined by k` = d`+1.

For each k ∈ {k`}∞`=1 let v(k)1 , v

(k)2 be two vertices in S(T )v0 (k) ⊂ T (k) of maximum distance from

each other. We will consider the adjacency operator A = AT

on the tree T which is the chain

graph corresponding to the above{T (k), v

(k)1 , v

(k)2

}and {k`}. The graph T with root v0 = 1 is

of sub-exponential growth, as for each k ∈ N, νv0(k) < k. Additionally, it is a sub-graph of Td,and thus (see, e.g. [28, Corollary 4.5])

ρ(T)≤ ρ(Td) = 2

√d− 1,

EIGENFUNCTIONS GROWTH ON GRAPHS 5

where ρ(G) = sup{|λ| : λ ∈ σ(AG)} is the spectral radius of the graph G. Thus

σess(A) ⊆ σ(A) ⊂[−2√d− 1, 2

√d− 1

]= σ(AT ).

On the other hand, one of the R-limits of A is the adjacency operator on the d-regular tree, forwhich, as mentioned above,

σ∞(AT ) = [−d, d].

As a consequence[− d, d

]\[−2√d− 1, 2

√d− 1

]⊆

⋃H′∈R(H)

σ∞(H ′)\ σess(H).

�

It is worth mentioning that the last example is sub-exponentially growing but it does notadmit uniform sub-exponential growth. If we require a uniform sub-exponential growth rate ofthe graph, we get the following result:

Theorem 2.5. Let G be an infinite, connected d-bounded graph of uniform sub-exponentialgrowth and let H be a bounded Schrodinger operator on `2(G). Then,

σess(H) =⋃

H′∈R(H)

σ(H ′)(2.1)

=⋃

H′∈R(H)

σ∞(H ′).

Remark 2.6. In [7, Theorem 3] an example is given for a graph for which the essential spectrumis strictly larger than the union over the spectra of all R-limits. While this example does not obeya sub-exponential growth rate, it can be adapted. For indeed, recall that the graph given in this

example is in fact a chain graph G corresponding to a specific sequence of graphs{Gk, v

(k)1 , v

(k)2

}for which both the girth and the cardinality satisfies girth(Gk), |Gk| → ∞ as k →∞. Originallyin [7] the sequence {k`} is taken such that it satisfies just k` − k`−1 → ∞. Instead we can now

choose k` =∑`

j=1 |Gj |. Consequently the growth rate of the graph, e.g. around 1 ∈ N, satisfies

k ≤ N1(k) ≤ 2k , k ∈ N

and so it has sub-exponential growth. The change of {k`} does not affect the proof of [7, Theorem3], by which d is contained in the essential spectrum of AG (the adjacency operator on G), whileit is not in the spectra of any of the R-limits of AG.

It is worth mentioning that the graphs Gk are locally tree-like, of exponential growth rate, andthus G does not has a uniform sub-exponential growth. In light of this, the constraint of havinga uniform sub-exponential growth of the graph G in Theorem 2.5 is crucial.

The set of graphs of uniform sub-exponential growth contains many different examples. Forinstance, one class of such graphs is the set of (infinite) penny graphs, i.e. graphs whose verticescan be represented by unit circles, with no two of these circles crossing each other, and withtwo adjacent vertices if and only if they are represented by tangent circles. Similarly one canconsider any planar graph in which each vertex can be represented by a closed shape such thatits girth and the surrounded area are bounded below by a constant greater than zero.

6 SIEGFRIED BECKUS, LATIF ELIAZ

3. Characterizing σess(H) using generalized eigenfunctions

3.1. R-limits. If η : A→ B is a bijective map on two finite sets A and B, denote by Iη : `2(A) ∼=C|A| → `2(B) ∼= C|B| the isomorphism defined via Iη(δa) := δη(a). Such maps will be mainlyused for sets A and B that are balls in different graphs. Specifically, let (G, v0) be an infinite,connected, rooted d-bounded graph. Since we assume that the graph G admits a uniform boundon the vertex degree, each ball Bv(r) is finite (for any v ∈ V (G), r ∈ N). Throughout this work,Bv(r) defines a subgraph of G by restricting the edge set only to those that connect to verticesin Bv(r). For the sake of simplifying the notation, this induced subgraph is also denoted byBv(r).

Recall that a bijective map φ : V (G)→ V (G′) between two graphs G and G′ (finite or infinite)is called a graph isomorphism if the induced map φE : E(G) → E(G′), (u, v) 7→

(φ(u), φ(v)

)is also bijective. Then two graphs G and G′ are isomorphic (G ∼ G′) if there exists a graphisomorphism between them. Furthermore, two balls Bu(r) and Bu′(r) for u ∈ V (G) and u′ ∈V (G′) are isomorphic (Bu(r) ∼ Bu′(r)) if the corresponding subgraph of Bu(r) is isomorphic tothe subgraph Bu′(r). It is immediate to show that if G is a connected d-bounded graph and G′ isisomorphic to G then G′ is also a connected d-bounded graph. Furthermore, deg(v) = deg(φ(v))holds for all v ∈ V (G) where φ denotes the graph isomorphism.

Define the projection

Pv0,r : `2(G)→ `2(Bv0(r)

),(Pv0,rψ

)(v) := χBv0 (r)

(v)ψ(v) ,

where χBv0 (r)is the characteristic function of the ball Bv0(r). Note furthermore that `2

(Br(v0)

)is naturally embedded into `2(G) by extending a function with zeros. In the following, we willnot distinguish between the finite dimensional space `2

(Bv0(r)

)and its embedding in `2(G). For

a Schrodinger operator H on the rooted graph (G, v0) denote by Hv0,r : `2(Bv0(r))→ `2(Bv0(r))the operator

Hv0,r := Pv0,rHPv0,r.

It is worth pointing out that Hv0,r can be represented as a matrix acting on C|Bv0 (r)|. With thisnotion at hand, ψ : V (G)→ C is a generalized eigenfunction of H with eigenvalue λ if and onlyif Pv0,rHψ = λPv0,rψ for all r ∈ N.

Recall that a sequence of vertices {vn}n∈N goes to infinity if it leaves any finite subset ofV (G), or equivalently, if limn→∞ dist(v0, vn) = ∞. If it does so monotonically we say thatit converges to infinity. Let (G, v0) and (G′, v′0) be two rooted graphs. A sequence of mapsfr : Bv0(r)→ Bv′(r) for r ∈ N is called coherent (for v0) if for s > r, the restriction of the mapfs to Bv0(r) equals fr, namely

fs(u) = fr(u) , u ∈ Bv0(r).

Definition 3.1. Let Hn be a sequence of Schrodinger operators on the rooted d-bounded graphs(Gn, vn) for n ∈ N and let H ′ be a Schrodinger operator on a rooted graph (G′, v′0). Then thesequence {(Hn, Gn, vn)}n∈N is called convergent to (H ′, G′, v′0) if

(C1) There are coherent maps{fn,r : Bvn(r)→ Bv′0(r)

}r∈N

for each n ∈ N, such that for

every r ∈ N, there exists an Nr ∈ N satisfying that fn,r : Bvn(r) → Bv′0(r) is a graphisomorphism for all n ≥ Nr;

EIGENFUNCTIONS GROWTH ON GRAPHS 7

(C2) For each r ∈ N,

limn→∞,n≥Nr

∥∥Ifn,rPvn,rHnPvn,rI−1fn,r−H ′v′0,r

∥∥ = 0

where Nr ∈ N is the constant chosen in (C1).

Note that we require in the latter definition that the maps fn,r eventually (in n ∈ N) aregraph isomorphism between the balls Bvn(r) and Bv′0(r). However, if n < Nr these maps

are not necessarily graph isomorphisms. If only (C1) holds, we call the sequence of rooted d-bounded graphs {(Gn, vn)}n∈N convergent to the rooted d-bounded graph (G′, v′0). Additionallynote that in Definition 3.1 one can replace (C1) with the requirement that there exist mapsfn,r : Bvn(r) → Bv′0(r) that are eventually bijective. In this case by (C2) the maps fn,r areeventually graph isomorphism.

Since we have graph isomorphisms, we derive that if {Gn, vn}n∈N converges to the rootedgraph (G′, v′0) then G′ is connected and infinite.

Definition 3.2. Let H be a Schrodinger operator on a graph G. Then a Schrodinger operatorH ′ on a graph G′ is called an R-limit of H if there exist a vertex v′0 ∈ V (G′) and a sequence ofvertices vn ∈ V (G) that converges to infinity such that {(H,G, vn)}n∈N converges to (H ′, G′, v′0).

Whenever it is necessary to specify the limit graph or the coherent maps, we say (H ′, G′, v′0)

is an R-limit with respect to the coherent maps{fn,r : Bvn(r)→ Bv′0(r)

}r,n∈N

. It is worth men-

tioning that for n ≥ Nr in the latter definition Ifn,r : `2(Bvn(r))→ `2(Bv′0(r)) is an isomorphism

and so I−1fn,ris well-defined.

Notice that the above definition for R-limits is equivalent to the definition given in [7, 12].Nevertheless the definition is presented here slightly different in order to relate it to the more gen-eral notion of convergence of a sequence of Schrodinger operators, introduced in Definition 3.1,which will be useful for us in this paper.

3.2. General observations. The proof of Theorem 2.2 requires some preliminary propositionsfor which we don’t have to impose any restriction on the growth rate of the graph. The argumentis inspired by the proof of [41, Theorem 7.2.1], which concerns operators on `2 (N).

We begin with more relevant notations. Let G be a graph and ψ : V (G)→ C be a map. Thenthe support of ψ is defined by supp(ψ) := {u ∈ V (G′) : ψ(u) 6= 0}. Denote by Cc(G) the set ofall ψ : V (G)→ C such that supp(ψ) is finite. Clearly, Cc(G) ⊆ `2(G) holds.

Proposition 3.3. Let {(Gn, vn)}n∈N be a sequence of infinite, connected, rooted and d-boundedgraphs. Assume that {(Hn, Gn, vn)}n∈N converges to (H ′, G′, v′0) and satisfies supn∈N ‖Hn‖ ≤C for some C > 0. Then (G′, v′0) is an infinite, connected rooted d-bounded graph, H ′ is aSchrodinger operator on (G′, v′0) of the form (1.1) and ‖H ′‖ ≤ 2C holds.

Proof. Let{fn,r : Bvn(r)→ Bv′0(r)

}r∈N

be the coherent maps (that are eventually isomorphic)

such that {(Hn, Gn, vn)}n∈N converges with respect to these maps to (H ′, G′, v′0). That (G′, v′0) isan infinite connected d-bounded graph follows immediately from the definition and that (Gn, vn)is an infinite connected rooted d-bounded graph for each n ∈ N.

Let u ∈ V (G′) and set r := |u| + 1. By definition, there exists an Nr ∈ N such thatfn,r : Bvn(r)→ Bv′0(r) is an isomorphism for all n ≥ Nr and

limn→∞,n≥Nr

∥∥Ifn,rPvn,rHnPvn,rI−1fn,r−H ′v′0,r

∥∥ = 0.

8 SIEGFRIED BECKUS, LATIF ELIAZ

Since Bv′0(r) is isomorphic to Bvn(r) and Bu(1) ⊆ Bv′0(r), we derive deg(u) ≤ d as graph

isomorphisms preserve the vertex degree. Thus, (G′, v′0) is a rooted d-bounded graph as u ∈V (G′) was arbitrary. Furthermore, Bv′0(r) and Bvn(r) are finite sets of the same cardinality and

so Ifn,rPvn,rHnPvn,rI−1fn,rand Hv′0,r

can be represented as matrices that converge in the matrix

norm to each other. This is equivalent to the convergence of the coefficients. Since (Hnψ)(v)depends only on the values of ψ on the neighbours of v and v itself by (1.1), it is straightforwardto show that H ′ is a Schrodinger operator of the form (1.1).

Let ψ ∈ Cc(G′) be such that ‖ψ‖ ≤ 1. Then there is an r ∈ N such that supp(ψ) ⊆ Bv′0(r−1).Set ε := C. According to Definition 3.1, there is an n0 ∈ N such that∥∥Ifn0,r

Pvn0 ,rHn0Pvn0 ,r

I−1fn0,r−Hv′0,r

∥∥ < ε.

Since supp(ψ) ⊆ Bv′0(r − 1) (r − 1 is important here) and H ′ is a Schrodinger operator of the

form (1.1), we have

H ′ψ = H ′vn0 ,rψ.

Thus, the previous considerations lead to

‖H ′ψ‖ =‖H ′v′0,rψ‖

≤∥∥H ′v′0,rψ − Ifn0,r

Pvn0 ,rHn0Pvn0 ,r

I−1fn0,rψ∥∥+

∥∥Ifn0,rHvn0 ,r

I−1fn0,rψ∥∥

<ε+ C

as ‖ψ‖ ≤ 1 and Ifn0,ris an isomorphism. Hence, the desired result follows from the choice of ε

and a standard approximation of ϕ ∈ `2(G′) by a compactly supported function. �

Proposition 3.4. Let (G, v0) be a rooted graph with uniform bounded vertex degree, and H be aSchrodinger operator on `2(G) as defined in (1.1). If {(H ′m, G′m, u′m)}m is a sequence of R-limits

of H that converge in the sense of Definition 3.1 to (H, G, v0) then (H, G, v0) is an R-limit ofH. In particular, R(H ′) ⊆ R(H) holds for all R-limit H ′ of H, and thus also

R(R(H)) ⊆ R(H).

Proof. By assumption, we have the following

(1) The sequence (H ′m, G′m, u

′m) converges to

(H, G, v0

)along the sequence of coherent

maps{f ′m,r : Bu′m(r)→ Bv0(r)

}m,r∈N that are eventually graph isomorphisms according

to Definition 3.1.(2) For m ∈ N, there is a sequence

{u(m)k

}k∈N

⊆ V (G) and a sequence of coher-

ent maps{f(m)k,r : B

u(m)k

(r)→ Bu′m(r)}k,r∈N

such that {(H,G, u(m)k )}k∈N converges to

(H ′m, G′m, u

′m).

Let ε > 0 and R ∈ N. Invoking (1) there is an MR ∈ N such that f ′m,R : Bu′m(R) → Bv0(R)is a graph isomorphism for m ≥MR and∥∥If ′m,R

(H ′m)u′m,R

I−1f ′m,R

− Hv0,R

∥∥ < ε

2, m ≥MR.

EIGENFUNCTIONS GROWTH ON GRAPHS 9

Let m ≥ MR. Invoking (2) there is an K(R,m) ∈ N such that f(m)k,R : B

u(m)k

(R)→ Bu′m(R) is

a graph isomorphism for k ≥ K(R,m) and∥∥If(m)k,R

Hu(m)k ,R

I−1f(m)k,R

−(H ′m)u′m,R

∥∥ < ε

2, k ≥ K(R,m).

For m ≥MR, the map

gm,R := f ′n,R ◦ f(m)K(R,m),R : B

u(m)K(R,m)

(R)→ Bv0(R)

is a graph isomorphism and If ′m,RIf(m)K(R,m),R

= Igm,R . Thus, we derive∥∥Igm,RHu(m)K(R,m)

,RI−1gm,R

− If ′m,RH ′u′m,RI

−1f ′m,R

∥∥ =∥∥If(m)K(R,m),R

Hu(m)K(R,m)

,RI−1f(m)K(R,m),R

−H ′u′m,R∥∥ < ε

2.

With this at hand, the triangle inequality leads to∥∥Igm,RHu(m)K(R,m)

,RI−1gm,R

− Hv0,R

∥∥ ≤∥∥Igm,RHu(m)K(R,m)

,RI−1gm,R

− If ′m,RH ′u′m,RI

−1f ′m,R

∥∥+∥∥If ′m,R

H ′u′m,RI−1f ′m,R

− Hv0,R

∥∥ < ε

for all m ≥MR. Since ε > 0 was arbitrary, H is an R-limit of H.In particular we get the last claim by taking H ′m ≡ H ′ with G′m ≡ G′ and u′m →∞. �

Lemma 3.5. Let {(Gn, vn)}n∈N be a sequence of connected infinite d-bounded graphs and Hn bea sequence of Schrodinger operators of the form (1.1) on (Gn, vn) such that supn∈N ‖Hn‖ ≤ Cfor some C > 0. Then there exists a Schrodinger operator H ′ on a rooted d-bounded graph(G′, v′0) and a subsequence {Hnk

, Gnk, vnk}k∈N that converges to (H ′, G′, v′0).

Proof. By assumption deg(v) ≤ d holds for all v ∈ V (Gn) and n ∈ N. Thus, for each r ∈ N, theset

Br := {Bv(r) : v ∈ V (Gn), n ∈ N}/ ∼is finite, where ∼ is the equivalence relation induced by graph isomorphism. Then a Cantordiagonalization argument gives a convergent subsequence {(Gn, vn)}n∈N to a rooted d-boundedgraph (G′, v′0). By a similar argument and by passing to another subsequence one gets the desiredresult that {Hnk

, Gnk, vnk}k∈N converges to (H ′, G′, v′0). Since these arguments are standard,

we only provide a short sketch here.

Let r = 1. Since Br is finite, there is a subsequence {N(1, k)}k ⊆ N with N(1, k) → ∞ suchthat BvN(1,k1)

(r) ∼ BvN(1,k2)(r) for all k1, k2 ∈ N. Now let r = 2. By the same argument, there

is a subsequence {N(2, k)}k ⊆ {N(1, k)}k such that BvN(2,k1)(R) ∼ BvN(2,k2)

(R)for all k1, k2 ∈ Nand 1 ≤ R ≤ r. By recursion we get for each r ∈ N a subsequence

(3.1) {N(r, k)}k ⊆ {N(r − 1, k)}k ⊆ . . . ⊆ {N(2, k)}k ⊆ {N(1, k)}k ⊆ Nsuch that

BvN(r,k1)(R) ∼ BvN(r,k2)

(R) , k1, k2 ∈ N , 1 ≤ R ≤ r.Due to (3.1), BvN(r,k)

(R) is isomorphic to BvN(R,1)(R) for every k ∈ N and 1 ≤ R ≤ r.

In order to define the graph (G′, v′0) it suffices to define Bv′0(r) for all r ∈ N modulo graph

isomorphism. Define Bv′0(r) := BvN(r,1)(r). By construction BvN(r,1)

(r) ∼ BvN(R,k)(r) holds for

10 SIEGFRIED BECKUS, LATIF ELIAZ

all k ∈ N and R ≥ r. Thus, the rooted graph (G′, v′0) is well-defined (up to graph isomorphism).By construction, (G′, v′0) is a connected, infinite rooted d-bounded graph.

We claim that the diagonal sequence {GN(k,k), vN(k,k)}k∈N converges to (G′, v′0). This can beseen as follows. Define fk,r : BvN(k,k)

(r)→ Bv′0(r) for k ≥ r to be the graph isomorphism between

BvN(k,k)(r) and Bv′0(r) = BvN(r,1)

(r) which exists by construction as k ≥ r. If k ≤ r, define

fk,r : BvN(k,k)(r) → Bv′0(r) by fk,r(u) := v′0 for all u ∈ BvN(k,k)

(r). By construction, these mapsare eventually graph isomorphisms, namely they fulfil the constraints given in Definition 3.1.

By the latter considerations we have shown that there is a subsequence that converges to aninfinite, connected rooted d-bounded graph (G′, v′0). In order to simplify the notation, supposethat {(Gn, vn)}n∈N converges to (G′, v′0). The operators Hn are uniformly bounded in n ∈ N.

Thus, for fixed r ∈ N, there is a subsequence {nk}k∈N such that Ifnk,rPvnk,rHnk

Pvnk,rI−1fnk,r

converges in norm (using that `2(Bvnk

(r))

is a finite dimensional vector space). By a similar

argument as for the graph sequence one can construct (with a Cantor diagonalization argument)an operator H ′ on (G′, v′0) and a subsequence {nk}k∈N such that for each r ∈ N

limk→∞,nk≥Nr

∥∥Ifnk,rPvnk,rHnk

Pvnk,rI−1fnk,r

−H ′v′0,r∥∥ = 0

where Nr ∈ N is chosen such that fnk,r defines a graph isomorphism for nk ≥ Nr. According toProposition 3.3, H ′ is a bounded Schrodinger operator of the form (1.1). �

The following is a useful assertion to prove the existence of a bounded generalized eigenfunc-tion of an R-limit for some λ ∈ C.

Proposition 3.6. Let (H,G, v0) and {(Hn, Gn, vn)}n∈N be such that either

(a) each (Hn, Gn, vn) is an R-limits of (H,G, v0), or(b) Hn = H, Gn = G and {vn}n∈N converges to infinity.

Let ϕ(n) : V (Gn)→ C, λn ∈ C and Rn ∈ N be such that λn → λ′, limn→∞Rn =∞,

Pvo,rHnϕ(n) = λnPv0,rϕ

(n) , for all r ≤ Rn , n ∈ N ,

and

maxu∈Bvn (n)

∣∣∣ϕ(n)(u)∣∣∣ ≤ C ∣∣∣ϕ(n) (vn)

∣∣∣ 6= 0,

for some constant C > 0. Then there exists an R-limit (H ′, G′, v′0) ∈ R(H) that is a limit of{(Hn, Gn, vn)}n∈N and a generalized eigenfunction 0 6= ϕ′ ∈ `∞(G′) of H ′ with eigenvalue λ′.

Proof. Due to Lemma 3.5, there is no loss of generality (by passing to a subsequence) in assumingthat the sequence {(Hn, Gn, vn)}n∈N converges to (H ′, G′, v′0) with respect to the coherent maps{fn,r}n,r∈N in the sense of Definition 3.1. In addition there is no loss of generality (again, by

passing to a subsequence) to assume that fk,n : Bvk(n)→ Bv′0(n) is a graph isomorphism for all

k ≥ n. Define ψ(n) : V (G′)→ C by

ψ(n)(u) :=1

ϕ(n)(vn)

(Ifn,nPvn,nϕ

(n))

(u)

Since f−1n,n(v′0) = vn, we have ψ(n) (v′0) = 1 and∥∥∥ψ(n)∥∥∥ = sup

u∈G

∣∣∣ψ(n) (u)∣∣∣ = sup

u∈Bv′0(n)

∣∣∣ψ(n) (u)∣∣∣ ≤ C

EIGENFUNCTIONS GROWTH ON GRAPHS 11

by the assumption on the sequence of generalized eigenfunctions. Thus, ψ(n) ∈ `∞(G).

By construction, the sequence ψ(n)(u) is uniformly bounded for every u ∈ V (G′). Hence,the Bolzano-Weierstrass theorem and a Cantor diagonalization argument yield that there is asubsequence {ψ(n(`))}`∈N ⊆ N and a ψ′ : V (G′)→ C such that for all r ∈ N,

lim`→∞

∥∥∥Pv′0,r (ψ(n(`)) − ψ′)∥∥∥ = 0.

Furthermore, |ψ′(u)| ≤ C holds for all u ∈ V (G′), namely ψ′ ∈ `∞(G′). Furthermore, ψ′(v′0) = 1

follows from ψ(n) (v′0) = 1.

In case (a), each Hn is an R-limit of H, and Proposition 3.4 asserts that H ′ is also an R-limitof H and ‖H ′‖ ≤ 2‖H‖. If (b) holds, H ′ is also an R-limit of H as {vn}n∈N goes to infinity.Thus, it is left to show that ψ′ ∈ `∞(G′) defines a generalized eigenfunction of H ′ with eigenvalueλ′.

Let ε > 0 and ϕ ∈ Cc(G′) be such that ‖ϕ‖ = 1, then there exists an r ∈ N such thatsupp(ϕ) ⊆ Bv′0(r − 1). Since H ′ is an R-limit of H, it is of the form (1.1), see Proposition 3.3.

Thus, Pv′0,rH′ψ′ = Pv′0,rPv′0,r+1H

′Pv′0,r+1ψ′ follows implying∥∥Pv′0,r (H ′ψ′ − λψ′) ∥∥ ≤‖Pv′0,r‖∥∥H ′v′0,r+1ψ

′ − λ′Pv′0,r+1ψ′∥∥

≤∥∥H ′v′0,r+1ψ

′ − λ′Pv′0,r+1ψ′∥∥.

In order to simplify the notation, set

Ln,r := Ifn,rPvn,rHnPvn,rI−1fn,r

acting on `2(Bv′0(r + 1)

). Since λn(`) → λ′ there is a C0 > 0 such that |λn(`)| ≤ C0. Choose

` ∈ N such that n(`), Rn(`) > r and∥∥∥Ln(`),r+1 −H ′v′0,r+1

∥∥∥ < ε

3‖Pv′0,rψ′‖,∥∥∥Pv′0,r+1

(ψ′ − ψ(n(`))

)∥∥∥ < ε

6 max{‖H‖, C0},

|λn(`) − λ′| <ε

6‖Pv′0,rψ′‖.

Note that ‖Pv′0,rψ′‖ 6= 0 as ψ′(v′0) = 1 by construction. Furthermore,

∥∥Pv′0,rLn(`),r+1

∥∥ ≤ 2‖H‖follows as Hn is an R-limit and so ‖Hn‖ ≤ 2‖H‖ holds by Proposition 3.3. Hence,∥∥Pv′0,r (H ′ψ′ − λψ′) ∥∥

≤∥∥Pv′0,rH ′v′0,r+1ψ

′ − λ′Pv′0,r+1ψ′∥∥

≤∥∥Pv′0,rH ′v′0,r+1ψ

′ − Pv′0,rLn(`),r+1ψ′∥∥+

∥∥Pv′0,rLn(`),r+1ψ′ − Pv′0,rLn(`),r+1ψ

(n(`))∥∥

+∥∥Pv′0,rLn(`),r+1ψ

(n(`)) − λn(`)Pv′0,rψ′∥∥+

∥∥λn(`)Pv′0,rψ′ − λ′Pv′0,r+1ψ′∥∥

=:(1) + (2) + (3) + (4)

follows by using the triangle inequality. We estimate each of the summands (1), (2), (3) and (4)separately. Specifically, the previous considerations and the choice of l lead to

(1) ≤∥∥∥Ln(`),r+1 −H ′v′0,r+1

∥∥∥ ‖Pv′0,r+1ψ′‖ ≤ ε

3

12 SIEGFRIED BECKUS, LATIF ELIAZ

and(2) ≤

∥∥Pv′0,rLn(`),r+1

∥∥∥∥Pv′0,r+1

(ψ′ − ψ(n(`))

)∥∥ ≤ ε

3.

Since Rn(`) ≥ r, we have Pv0,rHnϕ(n(`)) = λnPv0,rϕ

(n(`)). Hence, using the choice n(`) > r,

Pv′0,rLn(`),r+1ψ(n(`)) = λn(`)Pv′0,rψ

(n(`))

follows. Thus,

(3) =∥∥λn(`)Pv′0,rψ(n(`)) − λn(`)Pv′0,rψ

′∥∥ < ε

6is derived. Finally, using once more the choice of `, we deduce

(4) ≤ |λn(`) − λ′| ‖Pv′0,rψ′‖ < ε

6.

Combining the latter estimates, we conclude∥∥Pv′0,r (H ′ψ′ − λψ′) ∥∥ < (1) + (2) + (3) + (4) < ε

Since ε > 0 was arbitrary, we derive∣∣Pv′0,r(H ′ψ′ − λ′ψ′

)〉∣∣ = 0, namely λ′ is a generalized

eigenvalue of H ′ with generalized eigenfunction ψ′ ∈ `∞(G′). �

Corollary 3.7. The union⋃H′∈R(H) σ∞(H ′) is closed.

Proof. Assume {λn}n∈N ⊂⋃H′∈R(H) σ∞(H ′) and λn → λ′. For n ∈ N, let Hn be an R-limit

on the d-bounded graph Gn and 0 6= ϕn ∈ `∞(Gn) be such that (Hn − λn)ϕn = 0. Choose

un ∈ V (Gn) such that ϕn(un) ≥ ‖ϕn‖∞2 . Then the conditions of Proposition 3.6 are satisfied (for

C = 2) and so we derive that λ′ ∈⋃H′∈R(H) σ∞(H ′). �

Another consequence of Proposition 3.6 is the following statement. Let {ϕi}i∈N be a sequenceof functions ϕi : V (G)→ C. We will use the notation

q(i)`,k := max

`(k−1)≤|u|<`k|ϕi(u)| , i ∈ N ,

and, denote by u(i)`,k ∈ V (G) a vertex satisfying `(k − 1) ≤ |u(i)`,k| < `k and q

(i)`,k = |ϕi(u(i)`,k)|.

Lemma 3.8. Let (H,G, v0) be given where H is a Schrodinger operator on the infinite con-nected rooted d-bounded graph (G, v0). Suppose we are given a sequence {ϕj}j∈N of generalizedeigenfunctions of H with eigenvalue λj satisfying

(a) limλj = λ;(b) There is an s > 1 such that for each k ∈ N, there are subsequences ni →∞ and `i →∞

satisfying

q(ni)`i,k

= |ϕni(u(ni)`i,k

)| ≥ 1

smax

{q(ni)`i+1,k, q

(ni)`i−1,k

}.

Then there is an R-limit (H ′, G′, v′0) of H, and a generalized eigenfunction 0 6= ϕ′ ∈ `∞(G′) ofH ′ with eigenvalue λ.

Proof. We will proof in the following that for each k ∈ N, there is an R-limit(H(k), G(k), v

(k)0

)of H together with a ϕ(k) : V (G(k))→ C and λ(k) := λ satisfying all the constraints of Proposi-tion 3.6. This leads to the desired result.

Let k ∈ N. For the sake of simplifying the notation and since k stays fixed for the rest of theproof, there is no loss of generality (by passing to a subsequence) in assuming that ni = i and

EIGENFUNCTIONS GROWTH ON GRAPHS 13

`i ≥ k for all i ∈ N. Since `i → ∞, u(i)`i,k

goes to infinity. Then there is no loss of generality

in assuming that {(H,G, u(i)li,k)}i∈N converges to an R-limit(H(k), G(k), v

(k)0

), otherwise we can

pass to a convergent subsequence by Lemma 3.5. Since li ≥ k > k − 1, the requirement (b)yields

|ϕi(u)| ≤ s|ϕi(u(i)`i,k)|

for all u ∈ V (G) satisfying (`i − 1)(k − 1) ≤ |u| < (`i + 1)k. Let u ∈ Bk−1(u(i)`i,k

). Since

`i(k − 1) ≤ |u(i)`i,k| < `ik holds by definition, we conclude

(`i − 1)(k − 1) ≤ |u(i)`i,k| − (k − 1) ≤ |u| ≤ |u(i)`i,k|+ (k − 1) < (`i + 1)k.

Hence,

maxu∈Bk−1(u

(i)`i,k

)

|ϕi(u)| ≤ s|ϕi(u(i)`i,k)|

follows. Define ψ(k)i : V (G)→ C by

ψ(k)i (u) :=

ϕi(u)

|ϕi(u(i)`i,k

)|, u ∈ Bk−1(u

(i)`i,k

),

0 , otherwise.

Then ψ(k)i (u

(i)`i,k

) = 1 and ‖ψ(k)i ‖∞ ≤ s. Since {(H,G, u(i)`i,k)}i∈N converges to

(H(k), G(k), v

(k)0

)with respect to the coherent maps {fi,r : B

u(i)`i,k

(r) → Bv(k)0

(r)}i,r∈N, there is an i0 ∈ N such

that fi,k is a graph isomorphism for all i ≥ i0. Define for i ≥ i0, ϕ(k)i : V (G(k)) → C by

ϕ(k)i := Ifi,kψ

(k)i . We remind the reader that this formally just defines a function on `2

(Bv(k)0

(k))

that we embed into `2(G(k)) by extending it by zero. Then we deduce for i ≥ i0

supp(ϕ(k)i ) ⊆ B

v(k)0

(k − 1) , ϕ(k)i

(v(k)0

)= 1 and ‖ϕ(k)

i ‖∞ ≤ s.

Since Bv(k)0

(k−1) is finite, there is no loss of generality (by passing to another subsequence) that

{ϕ(k)i }i∈N converges pointwise (and so in `2-norm as all functions are supported on B

v(k)0

(k− 1))

to a map ϕ(k) : V (G(k)) → C such that supp(ϕ(k)) ⊆ Bv(k)0

(k − 1), ϕ(k)(v(k)0

)= 1 and 0 <

‖ϕ(k)‖ ≤ s. Thus,

supv∈B

v(k)0

(k)|ϕ(k)(v)| ≤ s = s

∣∣∣ϕ(k)(v(k)0

)∣∣∣Now we are almost in the setting of Proposition 3.6. More precisely, we need to show that ϕ(k)

are “approximate generalized eigenfunctions”.

Let r ≤ k − 2. Then the convergence of {(H,G, u(i)`i,k)}i∈N to(H(k), G(k), v

(k)0

)and the `2-

convergence of {ϕ(k)i }i∈N to ϕ(k) imply the following: For all ε > 0, there is an i1 ∈ N such that

14 SIEGFRIED BECKUS, LATIF ELIAZ

for i ≥ i1, ∥∥∥∥Ifi,r+1Hu(i)`i,k

,r+1Ifi,r+1

−H(k)

v(k)0 ,r+1

∥∥∥∥ < ε

2s‖ϕ(k)‖2,∥∥∥ϕ(k)

i − ϕ(k)∥∥∥2<

ε

8 max{‖H‖, |λi|, 1},

|λi − λ| <ε

8‖ϕ(k)‖2.

Note that ϕ(k) is supported on Bv(k)0

(k − 1) and so ‖ϕ(k)‖2 is finite. Furthermore λi → λ holds

implying |λi| is uniformly bounded in i ∈ N. According to Proposition 3.3, H(k) is a Schrodinger

operator of the form (1.1) and ‖H(k)‖ ≤ 2‖H‖. Using (1.1), we derive

Pv(k)0 ,r

H(k)ϕ(k) = Pv(k)0 ,r

H(k)

v(k)0 ,r+1

Pv(k)0 ,r+1

ϕ(k).

Let v ∈ Bv(k)0

(r). Then(Ifi,r+1

Pu(i)`i,k

,r+1I−1fi,kζ

)(v) = ζ(v), ζ ∈ `2(G(k)),

holds as the maps are coherent and r+ 1 ≤ k − 1. Denote by χk,r the characteristic function ofBv(k)0

(r). Then, a short computation gives(Pv(k)0 ,rIfi,r+1

Hu(i)`i,k

,r+1I−1fi,r+1

Pv(k)0 ,r+1

ϕ(k)i

)(v) =χk,r(v)

(Ifi,r+1

Hu(i)`i,k

,r+1ψ(k)i

)(v)

=χk,r(v)(Hψ

(k)i

)(f−1i,r+1(v)

)=λi

(Pv(k)0 ,rIfi,r+1

ψ(k)i

)(v)

=λi

(Pv(k)0 ,r

ϕ(k)i

)(v)

invoking the definition of ψ(k)i and that ϕi is a generalized eigenfunction of H with eigenvalue

λi. With this at hand, the triangle inequality leads to∥∥∥Pv(k)0 ,r

H(k)ϕ(k) − λPv(k)0 ,r

ϕ(k)∥∥∥ ≤ T1 + T2 + T3 + T4

where

T1 :=

∥∥∥∥Pv(k)0 ,rH

(k)

v(k)0 ,r+1

Pv(k)0 ,r+1

ϕ(k) − Pv(k)0 ,rIfi,r+1

Hu(i)li,k

,r+1I−1fi,r+1

Pv(k)0 ,r+1

ϕ(k)

∥∥∥∥ ,T2 :=

∥∥∥∥Pv(k)0 ,rIfi,r+1

Hu(i)li,k

,r+1I−1fi,r+1

Pv(k)0 ,r+1

ϕ(k) − Pv(k)0 ,rIfi,r+1

Hu(i)li,k

,r+1I−1fi,r+1

Pv(k)0 ,r+1

ϕ(k)i

∥∥∥∥ ,T3 :=

∥∥∥∥Pv(k)0 ,rIfi,r+1

Hu(i)li,k

,r+1I−1fi,r+1

Pv(k)0 ,r+1

ϕ(k)i − λiPv(k)0 ,r

ϕ(k)

∥∥∥∥ ,T4 :=

∥∥∥λiPv(k)0 ,rϕ(k) − λP

v(k)0 ,r

ϕ(k)∥∥∥ .

EIGENFUNCTIONS GROWTH ON GRAPHS 15

Then the previous considerations yield that

T1 <ε

2, T2 ≤ 2‖H‖‖ϕ(k) − ϕ(k)

i ‖ <ε

4,

T3 = |λi|∥∥P

v(k)0 ,r

ϕ(k)i − Pv(k)0 ,r

ϕ(k)∥∥ < ε

8, T4 = |λi − λ|

∥∥Pv(k)0 ,r

ϕ(k)∥∥ < ε

8

Thus,

Pv(k)0 ,r

H(k)ϕ(k) = λPv(k)0 ,r

ϕ(k) , r ≤ k − 2 ,

follows as ε > 0 was arbitrary.

Combining all the previous considerations, Proposition 3.6 applies for the sequence{(H(k), G(k), u(k))}k∈N, the sequence {ϕ(k)}k∈N, Rk := k − 2 and λk := λ. Thus, there is anR-limit (H ′, G′, v′0) of H and a generalized eigenfunction 0 6= ϕ′ ∈ `∞(G′) of H ′ with eigenvalueλ finishing the proof. �

The next two propositions rely on Proposition 3.6. We show that under certain conditions theexistence of a generalized eigenfunction of H results in the existence of a bounded generalizedeigenfunction of some R-limit of H. The first proposition treats bounded generalized eigenfunc-tions, while in the second proposition they are unbounded. For the proof, an auxiliary lemmais needed for each proposition.

Lemma 3.9. Let s > 1 and {an}n∈N be a sequence of positive numbers such that limn→∞ an = 0.Then either there exists C > 0 such that

(3.2) an < Cs−n, n ∈ N.

or there exists a subsequence {ank}∞k=1 such that for any k ∈ N

(3.3) ank> ank+1 and sank

≥ ank−1.

Proof. Define the set

N1 :=

{n ∈ N | an ≥ max

k>nak

}.

Since an → 0, N1 is infinite. Define further,

N2 := N1 ∩ {n ∈ N | an−1 ≤ san} .If N2 is infinite we get a subsequence satisfying (3.3). Otherwise, N2 is finite. Since N1 is infiniteand N2 is finite, there is an n0 ∈ N1 such that an−1 < san for all n ≥ n0. Let n ∈ N1 be suchthat n ≥ n0. Thus,

an < san < an−1

follows implying n − 1 ∈ N1. If n − 1 ≥ n0, one derives similarly an−2 > san−1 > r2an.Recursively, we get an0 > sn−n0an implying

an < an0sn0s−n , n ≥ n0.

Thus, (3.2) follows proving the statement. �

Proposition 3.10. Let H be Schrodinger operator on an infinite, connected rooted d-boundedgraph (G, v0) and 0 6= ϕ ∈ `∞(G) be a bounded generalized eigenfunction of H with eigenvalueλ. Then one of the following assertions holds.

(a) There exists an R-limit (H ′, G′, v′0) of H, and a bounded generalized eigenfunction 0 6=ϕ′ ∈ `∞(G′) of H ′ with eigenvalue λ.

16 SIEGFRIED BECKUS, LATIF ELIAZ

(b) There are constants γ > 1 and C > 0 such that

|ϕ (u) | ≤ C · γ−|u| , u ∈ G .

Proof. Let k ∈ N. Then there is a um,k ∈ V (G) such that m(k − 1) ≤ |um,k| < mk and

|ϕ(um,k)| = maxm(k−1)≤|u|<mk

|ϕ (u)| =: qm,k

Note that the latter is a maximum as Bv0(r) is finite for every r ∈ N. We will treat the twocases where (A) {qm,k}m∈N does not tend to zero for some k ∈ N and (B) {qm,k}m∈N does tendto zero for all k ∈ N.

(A) Suppose k ∈ N is chosen such that {qm,k}m∈N does not tend to zero. Let {qm`,k}`∈Nbe a subsequence such that lim`→∞ qm`,k = q > 0. Such a subsequence exists as {qm,k}m∈N isuniformly bounded as ϕ ∈ `∞(G). Choose `0 ∈ N such that qm`,k > q/2 for every ` > `0. Then,

maxu∈Bum`,k

(`)|ϕ(u)| ≤ ‖ϕ‖∞ ≤

2‖ϕ‖∞q

|ϕ (um`,k)| 6= 0.

Thus, all the requirements of Proposition 3.6 are satisfied for the sequence {(H,G, um`,k)}`∈Nwith λn = λ and ϕ(n) = ϕ. Hence, the statement (a) of Proposition 3.10 follows from Proposi-tion 3.6.

(B) Suppose now that limm→∞ qm,k = 0 for all k ∈ N and let s > 1. Due to Lemma 3.9,either (B.1) there is a k ∈ N such that (3.2) holds for a suitable constant and am := qm,k or(B.2) for all k ∈ N, there is a subsequence {ami}i∈N satisfying (3.3).

(B.1) There exists a k ∈ N and a constant C > 0 such that

qm,k < Cs−m , m ∈ N.

If u ∈ V (G) then there is an m ∈ N such that m(k − 1) ≤ |u| < mk and so −m < − |u|k .Consequently, the latter considerations yield

qm,k < Cs−|u|k .

Specifically, ϕ is exponentially decaying as claimed in (b) with γ := s1k > 1.

(B.2) For all k ∈ N, there is a subsequence {qmi,k}i∈N satisfying

(3.4) qmi,k > qmi+1,k and s qmi,k ≥ qmi−1,k.

Hence,

qmi,k ≥1

smax

{qmi−1,k, qmi+1,k

}follows and ϕ is a generalized eigenfunction of H with eigenvalue λ. Thus, Lemma 3.8 appliedto ϕj := ϕ and λj := λ implies (a). �

Next we prove Proposition 3.12, and again begin by presenting a useful lemma.

Lemma 3.11. Let {an}n∈N be an unbounded sequence of real numbers such that there is ans > 1 and a C > 0 satisfying

an < Csn , n ∈ N.Then, for each γ > s, there exists a subsequence {ank

}∞k=1 such that for all k ∈ N,

ank> ank−1 and γank

≥ ank+1.

EIGENFUNCTIONS GROWTH ON GRAPHS 17

Proof. Define,

N1 =

{n ∈ N | an ≥ max

m≤nam

}.

Since an is unbounded, |N1| =∞. Define further,

N2 = N1 ∩ {n ∈ N | an+1 ≤ γan} .Clearly, the statement is proven if N2 is infinite. Assume by contradiction that N2 is finite.Then there exists an n0 so that for every n ≥ n0, an+1 > γan. Since N1 is infinite, there is ann1 ∈ N1 such that n1 ≥ n0. Then a short induction yields that for n ≥ n1, we have

n ∈ N1 and an > γn−n1an1 .

Since there is an n1 ≥ n0, we get

an1+1 > γan1 > an1 ≥ maxm≤n1

am.

Thus, n1 + 1 ∈ N1 and so the base case is proven. Analogously, the induction step is proven.

Thus, there is a constant C1 > 0 such that an > C1γn for all n ∈ N. Since γ > s, this

contradicts the assumption an < Csn for some constant C > 0. Hence, N2 is infinite. �

Proposition 3.12. Let ϕ be a unbounded generalized eigenfunction of H with eigenvalue λ.Then one of the following assertions holds.

(a) There exists an R-limit (H ′, G′, v′0) of H, and a bounded generalized eigenfunction 0 6=ϕ′ ∈ `∞(G′) of H ′ with eigenvalue λ.

(b) There exists a constant γ > 1, such that for all C > 0, there is a u = u(C) ∈ V (G)

satisfying |ϕ (u) | ≥ C γ|u|.

Proof. Suppose (b) is not satisfied. We will prove that then (a) holds. Let k ∈ N. Define

qm,k := maxm(k−1)≤|u|<mk

|ϕ (u)| , m ∈ N.

Since (b) does not hold, for all γ > 1, there is a constant Cγ,k > 0 such that

|ϕ(u)| ≤ Cγ,kγ|u| , u ∈ V (G).

Let s > 1 and set γ := s1k > 1. Thus, there is a constant Ck = C(k, r) > 0 such that

qm,k < Ck γmk = Ck s

m.

Since ϕ is unbounded, limm→∞ qm,k =∞. Then Lemma 3.11 implies that there is a subsequence{qmi,k}i∈N such that for every γ′ > s,

qmi,k > qmi−1,k and γ′qmi,k ≥ qmi+1,k.

Thus, Lemma 3.8 applied to ϕj := ϕ and λj := λ yields (a). �

3.3. Generalized eigenfunctions and Shnol type theorems. A function ϕ : V (G) → Con a graph G has sub-exponential growth (with respect to the graph metric dist) if for one (any)vertex v0 ∈ V (G),

V (G) 3 v 7→ e−α dist(v,v0) ϕ(v)

is an element of `2(G) for all α > 0. With this at hand, the following Shnol-type theorem holds,proven in [21] in more general setting of bounded Jacobi operators on a graph. This theorem isused for the proof of Theorem 2.5.

18 SIEGFRIED BECKUS, LATIF ELIAZ

Proposition 3.13 ([21, Theorem 4.8]). Let H be a Schrodinger operator of the form (1.1)on an infinite, connected, rooted graph (G, v0) with uniformly bounded vertex degree. Supposeϕ : V (G) → C is a generalized eigenfunction of H with eigenvalue λ. If ϕ is sub-exponentiallybounded, then λ ∈ σ(H).

The proof of Theorem 2.2 depends on the existence of a generalized eigenfunction for eachpoint in the spectrum admitting a suitable growth rate. The following statement provides asufficient condition to get such generalized eigenfunction which can be found in [24, Theorem 3]in a more general setting.

Theorem 3.14 (reverse Shnol’s Theorem, [24, Theorem 3]). Let (G, v0) be a connected, infinite,rooted graph and H be a Schrodinger operator on `2(G) of the form (1.1) with spectral measureµ. Suppose ω ∈ `2(G) is real-valued and positive (i.e. ω(v) > 0 for all v ∈ G). Then for µ-a.e.λ ∈ σ (H), there exists a generalized eigenfunction ϕ of H with eigenvalue λ and ϕ · ω ∈ `2(G).

Corollary 3.15. Let (G, v0) be a connected, infinite, rooted graph and H be a Schrodingeroperator on `2(G) of the form (1.1). If ω ∈ `2(G) is real-valued and positive, then{

λ ∈ σ(H)∣∣∃ϕ : G→ C, Hϕ = λϕ and ϕω ∈ `2(G)

}= σ(H).

Lemma 3.16. Let (G, v0) be an infinite, connected, rooted graph. Then ωG : V (G)→ R definedby

ωG(v) :=1

(|v|+ 1)√Nv0 (|v|)

satisfies ωG ∈ `2(G).

Proof. A short computation gives

‖ωG‖22 =∑v∈G|ωG(v)|2 =

∞∑k=0

Nv0(k) · 1

(k + 1)2Nv0(k)=∞∑k=0

1

(k + 1)2<∞.

�

As a consequence of Lemma 3.16 and Theorem 3.14 we derive

Corollary 3.17. The set

Λ1 =

{λ ∈ σ(H)

∣∣∣∣∣ ∃ a generalized eigenfunction ϕλ with eigenvalue λ such that

|ϕ (v)| ≤ (|v|+ 1)√Nv0 (|v|)

}is dense in σ(H). In particular, if G has sub-exponential growth, then the set

(3.5) Λ2 =

{λ ∈ σ(H)

∣∣∣∣ ∃ a generalized eigenfunction ϕλ with eigenvalue λ such that

ϕλ is sub-exponentially growing

}is dense in σ(H).

Proof. This follows immediately from Lemma 3.16 and Corollary 3.15. The second part followsfrom the fact that if G is sub-exponentially growing, then for each γ > 1, there is a C > 0 suchthat

(|v|+ 1)√Nv0 (|v|) ≤ C · γ|v|/2 (|v|+ 1) .

Then the statement follows from the first part. �

EIGENFUNCTIONS GROWTH ON GRAPHS 19

3.4. Proof of Theorem 2.2. The proof strategy of Theorem 2.2 is as follows. Givenλ ∈ σess(H) we use the reverse Sch’nol’s property (Theorem 3.14) to obtain a sequence of

generalized eigenfunctions{ϕ(n)

}∞n=1

of H with eigenvalue λn such that λn → λ and each ϕn issub-exponentially growing. With this at hand, Proposition 3.6, Propositions 3.10 and Proposi-tion 3.12 complete the proof of the theorem.

Recall that the discrete spectrum σdisc(H) of an operator H is defined by the set of isolatedeigenvalues of finite multiplicity. Furthermore, the essential spectrum σess(H) is defined byσ(H)\σdisc(H). Thus, if λ is an element of the essential spectrum, then either λ is an eigenvalueof infinite multiplicity or in each neighbourhood of λ there are elements of the spectrum σ(H)that are not equal to λ.

Proof of Theorem 2.2. Let λ ∈ σess(H). Then one of the following cases holds

(a) For every ε > 0 there is a λε ∈ σ(H) such that 0 < |λ− λε| < ε.(b) There exists an infinite sequence ϕn ∈ `2(G) satisfying Hϕn = λϕn and they are pairwise

orthogonal.

Begin with case (a), due to Corollary 3.17 and since (G, v0) has sub-exponential growth, theset Λ2 (of (3.5)) is dense in σ(H). Then by the assumption (a) there is a sequence of eigenvalues{λn}n∈N corresponding to generalized eigenfunctions ϕn : V (G) → C such that λn 6= λm form 6= n, λn → λ and ϕn has sub-exponential growth. Then either infinitely many of the ϕn areunbounded or only finitely many of them are unbounded

If infinitely many of the ϕn are unbounded, there is no loss of generality to assume thateach ϕn is unbounded (otherwise pass to a subsequence). Then either λn ∈ σ∞(Hn) for someR-limit Hn of H or ϕn is exponentially growing by Proposition 3.12. However, they cannotgrow exponentially, as by construction ϕn is sub-exponentially bounded and so λn ∈ σ∞(Hn)holds for some Hn ∈ R(H). Since λn → λ, Corollary 3.7 yields λ ∈

⋃H′∈R(H) σ∞(H ′) finishing

the proof.

If only finitely many of the ϕn are unbounded, there is no loss of generality in assuming thatϕn is bounded for each n ∈ N (otherwise pass to a subsequence). By Proposition 3.10 eitherinfinitely many of the λn satisfy λn ∈ σ∞(Hn) for some Hn ∈ R(H) or only finitely many. Inthe first case Corollary 3.7 finishes the proof as before. In the second case, Proposition 3.10asserts that for n large enough

|ϕn (u) | ≤ Cn · γ−|u|n , u ∈ G .Thus, ϕn ∈ `2(G) and in particular, they are orthogonal as H is self-adjoint and λn 6= λm form 6= n. This is the same situation as in (b) (where λn ≡ λ for all n ∈ N). We proceed provingthis last case which will prove also (b).

Set ψn := ϕn

‖ϕn‖2 for each n ∈ N. Then ψn is still an eigenfunction of H with eigenvalue λn(λn = λ in (b)) and the functions {ψn}n∈N are pairwise orthogonal. Let un ∈ V (G) be thevertex such that ‖ψn‖∞ = |ψn(un)|. We have two cases, either {un}n∈N converges to infinity or{un}n∈N ⊆ Bv(r) for some r > 0, v ∈ V (G) and every n ∈ N. In the first case, when {un}n∈Nconverges to infinity, then

maxu∈Bn

∣∣ψn(u)∣∣ ≤ ‖ψn‖∞ = |ψn(un)|

holds by construction, with the notation Bn = Bun(n). Thus, Proposition 3.6 yields that thereexists an R-limit (H ′, G′, v′0) being a limit point of (H,G, un) and a generalized eigenfunction0 6= ψ ∈ `∞(G′) of H ′ with eigenvalue λ.

20 SIEGFRIED BECKUS, LATIF ELIAZ

It is left to treat the case that {un}n∈N ⊆ Bv(r) for some r > 0, v ∈ V (G) and every n ∈ N.Since {ψn}n∈N are pairwise orthogonal, they converge weakly to zero (see e.g. [37, TheoremII.6]). Since ‖ψn‖∞ = |ψn(un)| and un ∈ Bv(r) for all n ∈ N, the weak convergence to zeroyields limn→∞ ‖ψn‖∞ = 0. For k ∈ N define

q(n)`,k := max

`(k−1)≤|u|<`k|ψn(u)|.

Since ψn ∈ `2(G), we derive lim`→∞ q(n)`,k = 0. Define

Ln,k :={` ∈ N : q

(n)`,k ≥ max

i≥`q(n)i,k

}.

Since lim`→∞ q(n)`,k = 0, we derive that Ln,k is infinite for all n ∈ N. Next, we will prove the

following claim that together with Lemma 3.8 will conclude the proof.

Claim 1. For each k ∈ N there are two sequences {nj}j∈N (with nj →∞) and {`j}j∈N (with

`j →∞) such that for each j ∈ N, `j ∈ Lnj ,k and q(nj)`j−1,k ≤ 2q

(nj)`j ,k

.

Before proving the claim, let us explain why it finishes the proof. By construction ψn is ageneralized eigenfunction of H with eigenvalue λn satisfying 0 6= ψn ∈ `∞(G), λn → λ. Claim1 implies for each k ∈ N, there are subsequences nj → ∞ and `j → ∞ satisfying `j ∈ Lnj ,k

and q(nj)`j−1,k ≤ 2q

(nj)`j ,k

. The condition `j ∈ Lnj ,k leads to G(nj)`j ,k≥ q

(nj)`j+1,k. Thus, the constraint

(b) of Lemma 3.8 is satisfied with s = 2. Hence, the latter lemma implies that there is anR-limit (H ′, G′, v′0) of H and a generalized eigenfunction 0 6= ϕ′ ∈ `∞(G′) of H ′ with eigenvalueλ proving Theorem 2.2.

It is left to chow Claim 1. Let us fix k ∈ N and introduce the following notation q(n)` := q

(n)`,k

and Ln := Ln,k. We will make the proof by contradiction. Thus assume by contradiction that

(3.6) ∀{nj}j(with nj →∞), ∀{`j}j(with `j →∞),∃j ∈ N : `j 6∈ Lnj or q(nj)`j−1 > 2q

(nj)lj

We first prove that this is equivalent to

(3.7) ∃n0, `0 ∈ N, ∀n ≥ n0, ∀` ≥ `0 : q(n)` < max

i≥`q(n)i or q

(n)`−1 > 2q

(n)` .

That (3.7) implies (3.6) is straightforward. In order to show the implication (3.6)⇒(3.7), assumeby contradiction that (3.6) holds but (3.7) does not hold. First note that ` ∈ Ln is equivalent

to q(n)` ≥ maxi≥` q

(n)i . Lets denote by An,` the statement q

(n)` ≥ maxi≥` q

(n)i and q

(n)`−1 ≤ 2q

(n)` .

Thus,

(3.8) ∀n0, `0 ∈ N, ∃n ≥ n0, ∃` ≥ `0 : An,l.

Then we can iteratively define a sequence (nk, `k) with nk → ∞ and `k → ∞ such that Ank,`kis true for each k ∈ N. More precisely, let (n1, `1) be such that 1 ≤ `1, 1 ≤ n1 and A`1,n1 is true,which is possible by (3.8). Suppose now we have (nk, `k) for ` ≤ k ≤ m such that Ank,`k is true.By (3.8), there is an n > nk and ` > `k such that An,` is true. Set nk+1 := n and `k+1 := `.By construction nk → ∞ and `k → ∞ and Ank,`k is true for all k ∈ N. This contradicts (3.6).Thus, we have proven that (3.6) and (3.7) are equivalent. In particular, by our assumption (3.7)holds.

EIGENFUNCTIONS GROWTH ON GRAPHS 21

In the following let `0, n0 ∈ N be chosen according to (3.7). Let n ∈ N be such that n ≥ n0,

and let ` ∈ Ln be such that ` ≥ `0 (exists since Ln is infinite). Then (3.7) leads to q(n)`−1 > 2q

(n)` .

Thus,

q(n)`−1 > 2q

(n)` ≥ max

i≥`q(n)i

follows implying that also ` − 1 ∈ Ln. Since |Ln| = ∞, the latter considerations imply that

[`0,∞) ∩ N ⊆ Ln. Thus, q(n)`−1 > 2q

(n)` holds for all ` ≥ `0 by (3.7). Altogether, we derive that

q(n)` <

1

2q(n)`−1 <

1

22q(n)`−2 < . . . <

1

2`−`0q(n)`0≤ 1

2`−l0‖ψn‖∞

holds for all n ≥ n0 and ` ≥ `0. Hence, for v ∈ S(G)v0 (r) with r ≥ `0k, the estimates

|ψn(v)| ≤ q(n)br/kc ≤1

2br/kc−`0‖ψn‖∞ ≤ ‖ψn‖∞2`0e−

ln(2)k

r , n ≥ n0 ,

are deduced where br/kc denotes the largest integer j satisfying j ≤ rk . Recall that N(r) denotes

the cardinality of the sphere S(G)v0 (r). Since (G, v0) has sub-exponential growth rate, there is a

constant Ck > 0 such that N(r) < Ckγrk for all r ∈ N where γk := e

ln(2)2k > 1 (see Definition 2.1).

Then a short computation yields∑v∈V (G) : |v|≥`0k

|ψn(v)|2 ≤ ‖ψn‖∞2`0∑r≥`0k

N(r)e−ln(2)

kr ≤ ‖ψn‖∞2`0Ck

∑r≥`0k

γ−rk .

The latter sum is convergent by the root test as γ−1k < 1. Furthermore, the cardinality of allvertices v ∈ V (G) with |v| < `0k is finite as G is d-graph. Hence,

‖ψn‖22 =∑

v∈V (G)

|ψn(v)|2 =∑

v∈V (G) : |v|<`0k

|ψn(v)|2 +∑

v∈V (G) : |v|≥`0k

|ψn(v)|2 ≤ Ck‖ψn‖∞

follows for a suitable constant Ck > 0. Since limn→∞ ‖ψn‖∞ = 0, we derive limn→∞ ‖ψn‖22 = 0,a contradiction as ‖ψn‖2 = 1. Thus, Claim 1 is proven. �

4. Proof of Theorem 2.5

The proof follows from the results mentioned above, and the following theorem from [7].

Theorem 4.1 ([7, Theorem 2]). Let G be an infinite, connected d-bounded graph and H aSchrodinger operator on G, then ⋃

H′∈R(H)

σ(H ′) ⊆ σess(H).

With this at hand, we can prove Theorem 2.5.

Proof of Theorem 2.5. We already know from Theorems 4.1 and Theorem 2.2 that⋃H′∈R(H)

σ(H ′) ⊆ σess(H) ⊆⋃

H′∈R(H)

σ∞(H ′).

Thus, it suffices to prove for each H ′ ∈ R(H),

σ∞(H ′) ⊆ σ(H ′).

Since G admits a uniform sub-exponential growth also any R-limit (H ′, G′, v′0) of (H,G, v0)is of uniform sub-exponential growth. Indeed, for each γ > 1 there exists a constant C > 0

22 SIEGFRIED BECKUS, LATIF ELIAZ

(independent of the root) such that Nu(r) < Cγr for all u ∈ V (G) and r ∈ N (see Definition 2.1).Let γ > 1 be arbitrary and r ∈ N. Since (H ′, G′, v′0) is an R-limit of (H,G, v0), there is a vertexu ∈ V (G) such that the subgraphs Bv′0(r) and Bu(r) are isomorphic (see Definition 3.1). In

particular, the spheres contained in these balls have the same cardinality, namely Nu(r) =Nv′0

(r). Since Nu(r) < Cγr where the corresponding constant C is independent of u ∈ V (G),

we derive Nv′0(r) < Cγr. Thus, the map V (G′) 3 v′ 7→ e−α dist(v′,v′0) is an element `2(G′) for any

α > 0.

Let λ ∈ σ∞(H ′), then by definition there is a bounded generalized eigenfunction ϕ of H ′ witheigenvalue λ. By the previous considerations

V (G′) 3 v′ 7→ e−α dist(v′,v′0)ϕ(v′)

is an element of `2(G′) as ϕ is uniformly bounded. Thus, ϕ is sub-exponentially boundedimplying λ ∈ σ(H ′) by Proposition 3.13. �

References

[1] K. Austin and J. Zhang, Limit operator theory for groupoids, preprint, arXiv:1903.08442, (2019).[2] S. Beckus and Y. Pinchover, Shnol-type theorem for the Agmon ground state, to appear in J. Spectr. Theory,

arXiv:1904.07176, (2018).[3] S. Beckus and B. Devyver, Generalized eigenfunctions and eigenvalues: a unifying framework for Shnol-type

theorems, arXiv:1904.07176, (2019).[4] A. Boutet de Monvel, D. Lenz and P. Stollmann, Sch’nol’s theorem for strongly local forms, Israel J. Math.

173 (2009), 189–211.[5] A. Boutet de Monvel and P. Stollmann, Eigenfunction expansions for generators of Dirichlet forms, J. Reine

Angew. Math. 561 (2003), 131–144.[6] J. Breuer, Singular continuous spectrum for the Laplacian on certain sparse trees, Commun. Math. Phys.

269 (2007), no. 3, 851–857.[7] J. Breuer, S. Denisov and L. Eliaz, On the essential spectrum of Schrodinger operators on trees, Math. Phys.

Anal. Geom. 21 (2018), no. 4, Art. 33.[8] C. Carvalho, V. Nistor and Y. Qiao, Fredholm criteria for pseudodifferential operators and induced represen-

tations of groupoid algebras, Electron. Res. Announc. Math. Sci. 24 (2017), 68–77.[9] C. Carvalho, V. Nistor and Y. Qiao, Fredholm conditions on non-compact manifolds: theory and examples,

Operator theory, operator algebras, and matrix theory 267 (2018), 79–122.[10] S.N. Chandler-Wilde, M. Lindner, Sufficiency of Favard’s condition for a class of band-dominated operators

on the axis, Journal of Functional Analysis 254 (2008), 1146–1159.[11] S. N. Chandler-Wilde and M. Lindner, Limit Operators, Collective Compactness, and the Spectral Theory of

Infinite Matrices, Mem. Amer. Math. Soc. 210 (2011), no. 989.[12] L. Eliaz, On the essential spectrum of Schrodinger operators on graphs, Ph.D thesis, Dept. of Math., Hebrew

Univ. of Jerusalem, Israel, 2019 (arXiv:1909.10059).[13] J. Favard, Sur les equations differentielles lineaires a coefficients presque-periodiques, Acta Math. 51 (1928),

no. 1, 31–81.[14] R. L. Frank, D. Lenz, D. Wingert, Intrinsic metrics for non-local symmetric Dirichlet forms and applications

to spectral theory, J. Funct. Anal. 266 (2014). no. 8, 4765–4808.[15] V. Georgescu, On the structure of the essential spectrum of elliptic operators on metric spaces, J. Funct.

Anal. 260 (2011), no. 6, 1734–1765.[16] V. Georgescu, On the essential spectrum of elliptic differential operators, J. Math. Anal. Appl. 468 (2018),

no. 2, 839–864.[17] V. Georgescu and A. Iftimovici, C∗-algebras of quantum Hamiltonians, Operator Algebras and Mathematical

Physics (Constanta, 2001), pp. 123–167, Theta, Bucharest, 2003.[18] V. Georgescu and A. Iftimovici, Crossed products of C∗-algebras and spectral analysis of quantum Hamilto-

nians, Commun. Math. Phys. 228 (2002), no. 3, 519–560.[19] V. Georgescu and A. Iftimovici, Riesz-Kolmogorov compactness criterion, Lorentz convergence and Ruelle

theorem on locally compact abelian groups, Potential Anal. 20 (2004), no. 3, 265–284.

EIGENFUNCTIONS GROWTH ON GRAPHS 23

[20] R. Hagger and C. Seifert Limit operators techniques on general metric measure spaces of bounded geometry,preprint, arXiv:1908.01985, (2019).

[21] S. Haeseler and M. Keller, Generalized solutions and spectrum for Dirichlet forms on graphs, Random walks,boundaries and spectra, Progr. Probab. 64, (2011), Birkhauser, 181–199.

[22] B. V. Lange and V. S. Rabinovich, Pseudodifferential operators in Rn and limit operators, Math. Mat. Sb.(N.S.) 129 (1986), no. 2, 175–185 (Russian, English transl. Math. USSR Sb. 57 (1987), 183–194).

[23] Y. Last and B. Simon, The Essential Spectrum of Schrodinger, Jacobi, and CMV Operators, J. Anal. Math.98 (2006), 183–220.

[24] D. Lenz and A. Teplyaev, Expansion in generalized eigenfunctions for Laplacians on graphs and metricmeasure spaces, Trans. Amer. Math. Soc. 368 (2016), no. 7, 4933–4956.

[25] M. Lindner, Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method,Birkhauser, Basel - Boston - Berlin, (2006).

[26] M. Lindner, M. Seidel, An affirmative answer to a core issue on limit operators, Journal of FunctionalAnalysis 267 (2014), 901–917.

[27] M. Mantoiu, C∗-algebras, dynamical systems at infinity and the essential spectrum of generalized Schrodingeroperators, J. Reine Angew. Math. 550 (2002), 211–229.

[28] B. Mohar, The spectrum of an infinite graph, Linear Algebra Appl. 48(1982), 245–256.[29] B. Mohar and W. Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), no. 3,

209–234.[30] E. M. Muhamadiev, On invertibility of differential operators in the space of continuous functions bounded on

the real axis, Dokl. Akad. Nauk SSSR 196 (1971), 47–49 (Russian, English transl. Soviet Math. Dokl. 12(1971), 49–52).

[31] E. M. Muhamadiev, On the invertibility of elliptic partial differential operators, Dokl. Akad. Nauk SSSR 205(1972), 1292–1295 (Russian, English transl. Soviet Math. Dokl. 13 (1972), 1122–1126).

[32] T. Poerschke, G. Stolz and J. Weidmann, Expansions in generalized eigenfunctions of selfadjoint operators,Math. Z. 202 (1989), no. 3, 397–408.

[33] V. S. Rabinovich, S. Roch, J. Roe, Fredholm indices of band-dominated operators, Integral Equations Oper.Theory 49 (2004), no. 2, 221-238.

[34] V. S. Rabinovich, S. Roch and B. Silbermann, Fredholm theory and finite section method for band-dominatedoperators, Integral Equations Operator Theory 30 (1998), 452–495.

[35] V. S. Rabinovich, S. Roch and B. Silbermann, Band-dominated operators with operator-valued coefficients,their Fredholm properties and finite sections, Integral Equations Operator Theory 40 (2001), no. 3, 342–381.

[36] V.S. Rabinovich, S. Roch, B. Silbermann, Limit Operators and Their Applications in Operator Theory,Birkhauser, (2004).

[37] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I. Functional Analysis, Academic Press,New York, (1980).

[38] M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, AcademicPress, New York, (1978).

[39] I. E. Shnol, On the behavior of the eigenfunctions of Schrodinger’s equation, Mat. Sb. (N.S.) 42 (84) (1957),273–286. erratum 46 (88) (1957), 259.

[40] B. Simon, Spectrum and continuum eigenfunctions of Schrodinger operators, J. Funct. Anal. 42 (1981), no.3, 347–355.

[41] B. Simon, Szego’s Theorem and its Descendants, M. B. Porter Lectures, Princeton University Press, Prince-ton, NJ, (2011). Spectral Theory for L2 Perturbations of Orthogonal Polynomials.

[42] J. Spakula, R. Willett, A metric approach to limit operators, Trans. Amer. Math. Soc. 369 (2017), no. 1,263–308.

[43] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space,Invent. Math. 139 (2000), no. 1, 201–240.

Institut fur Mathematik, Universitat Potsdam, Potsdam, GermanyEmail address: beckus@uni-potsdam.de

Department of Mathematics, Technion - Israel Institute of Technology, 32000 Haifa, IsraelEmail address: latif@campus.technion.ac.il