Approximating Spectral invariants of Harper operators on graphs II

7 Pages·0.14 MB·English

by  Varghese Mathai

APPROXIMATING SPECTRAL INVARIANTS OF HARPER
OPERATORS ON GRAPHS II
2
0
0 VARGHESE MATHAI, THOMAS SCHICK, AND STUART YATES
2

n Abstract. We studyHarperoperators andthecloselyrelateddiscrete mag-
a netic Laplacians (DML) on agraph withafreeaction of adiscrete group, as
J definedbySunada[Sun]. ThespectraldensityfunctionoftheDMLisdefined

5 using the von Neumann trace associated with the free action of a discrete
1 group on a graph. The main resultin this paper states that when the group
isamenable,thespectraldensityfunctionisequaltotheintegrateddensityof

] states of the DML that is defined using either Dirichlet or Neumann bound-
P ary conditions. This establishes the main conjecture in [MY]. The result is
S generalizedtoother selfadjointoperatorswithfinitepropagation.
.
h
t
a
m
1. Introduction
[

In[MY]MathaiandYatesexaminedthe spectralpropertiesofthe discretemag-
1
neticLaplacian(DML)onagraphwithafreeactionofadiscretegroup,asdefined
v
7 by Sunada [Sun]. One of the results proved in [MY] is that when the group is
2 amenable and has a finite fundamental domain on the graph, the spectral density
1 function of the DML — defined using the von Neumann trace associated with the
1
discrete group action — can be approximated almost everywhere by the normal-
0
izedspectraldensitiesofaseriesoffinite approximationsofthe DML,consistingof
2
0 restrictions of the DML to a regular exhaustion of the graph. The latter approxi-
/ mation is known in the physics literature as the integrated density of states of the
h
DML,sothattheresultcanberephrasedassayingthatthevonNeumannspectral
t
a densityfunctionoftheDMLisequaltotheintegrateddensityofstatesoftheDML
m
almost everywhere. This partial result was known to the experts, cf. [Bel], [Sh].
: It wasshownin [MY] that infact the approximationholds at everypoint when-
v
i everthe DML is associatedwith a rationalweightfunction, andit was conjectured
X
therethatthe assumptiononthe weightfunctionwasirrelevent. Basedonthe idea
r of the middle named author, the conjecture is established here in complete gener-
a
ality,andatthe same time a muchmorestraightforwardargumentis presentedfor
the corresponding result in [MY]. Related arguments can be found for example in
[DLMSY], [Eck] and [El], and are known to other authors, in particular Wolfgang
Lu¨ck and Holger Reich.
Hereafter, let X be a combinatorial graph with finite fundamental domain F
under the free action of an amenable discrete group Γ. Each edge [e] of X has
associated with it two oriented edges e and e with opposite orientation. While
denoting the set of oriented edges by EdgeX, it will be convenient to work with a
1991 Mathematics Subject Classification. Primary: 58G25,39A12.
Key words and phrases. Harper operator, discrete magnetic Laplacian, DML,approximation
theorems,amenablegroups,vonNeumannalgebras,graphs,integrated densityofstates.
V.M.andS.Y.acknowledge supportfromtheAustralianResearchCouncil.
1
2 VARGHESE MATHAI, THOMAS SCHICK, AND STUART YATES
subsetE+ oftheseedgesinwhicheachcombinatorialedgehasexactlyoneoriented
representative. Unless otherwise stated, functions over X will refer to complex-
valued functions over the vertices of X.
By Følner’s characterization of amenability (see also [Ad]) there must exist a
regularexhaustion ofΓ: atoweroffinitesubsetsΛ ⊂Λ , Λ =Γsatisfying
m m+1 Sm m
#∂ Λ
(1.1) lim δ k =0 ∀δ >0,
k→∞ #Λk
where ∂ Λ denotes the δ-neighborhood of the boundary of Λ in Γ in the word
δ k k
metric (with respect to some fixed generating set.)
Let{X }beasequenceofsubgraphsofX,withX beingthelargestsubgraph
m m
of X contained within the translates by Λ of the fundamental domain. Then the
m
X form a regular exhaustion of the graph, satisfying
m
#Vert∂ X
(1.2) lim δ m =0 ∀δ >0,
m→∞ #VertXm
where ∂ X refers to the subgraph which is the δ-neighborhood of the boundary
δ m
of X in X in the simplicial metric.
m
The discrete magneticLaplacian∆ is anexample ofanoperatorwithbounded
σ
propagation, that is there exists a constant c such that for any function f with
support Y ⊂ X, the function ∆ f has support in the c-neighborhood of Y. The
σ
following theorem, provedin Section 3, provides an approximationfor the spectral
density function of such operators. This was posed (for the DML) as Conjecture
0.1 of [MY], where a proof was obtained for all points λ only when the operator
was the DML associated with a rational weight function σ.
Theorem 1.1 (Strong spectral approximation theorem). LetX beagraphonwhich
there is a free group action by an amenable group Γ, with finite fundamental do-
∞
main. Let X be the regular exhaustion of X corresponding to a regular
(cid:8) m(cid:9)m=1
exhaustion Λ of Γ. Let A be a self-adjoint operator of bounded propagation acting
m
on l2(X), which commutes with an adjoint-closed set of twisted translation opera-
tors, and has spectral density function F. Construct finite approximations A to
m
A by restricting A to the space of functions supported on X , and denote by F
m m
the normalized spectral density function of A ,
m
1
F (λ)= #{eigenvalues ≤λ of A , counting multiplicity}.
m m
#Λ
m
Then
(1.3) lim F (λ)=F(λ), ∀λ∈R.
m
m→∞
2. The discrete magnetic Laplacian and operators of bounded
propagation
TheDMLandtheHarperoperatoraredefinedbyaU(1)-valuedweightfunction
σ on the edges of the graph X which is weakly Γ-invariant. That is, for any edge
e, σ(e)=σ(e), and for anyγ ∈Γ there exists some U(1)-valued function s on the
γ
vertices of X such that
(2.1) σ(γe)=σ(e)s (t(e))s (o(e))
γ γ
APPROXIMATING SPECTRAL INVARIANTS OF HARPER OPERATORS II 3
where o(e) and t(e) are the origin and terminus of the edge e respectively. The
Harper operator H is then given by
σ
(Hσf)(v):= X σ(e)f(o(e))− X σ(e)f(t(e))
e∈E+ e∈E+
t(e)=v o(e)=v
and the DML ∆ is defined by
σ
(∆ f)(v):=O(v)f(v)−(H f)(v),
σ σ
where O(v) is the valence of the vertex v.
Forsomes satisfyingtheweakΓ-invariancepropertyforσ (2.1),onecandefine
γ
the magnetic translation operators T ,
γ
(2.2) (T f)(x)=s (γ−1x)f(γ−1x).
γ γ
Theseformanadjoint-closedsetofoperatorsoverthespacel2(X)ofL2functionson
theverticesofX,andthusdetermineavonNeumannalgebraB(l2(X))TΓ consisting
of all operators over l2(X) that commute with the T . This von Neumann algebra
γ
has a finite trace Tr , defined in (2.4). The DML is a self-adjoint operator in this
Γ
von Neumann algebra, and so one can define its spectral density function F by
(2.3) F(λ)=Tr E(λ)
Γ
where E(λ) = χ(−∞,λ](∆σ) is the spectral projection of the DML for the interval
(−∞,λ].
More generally one can start with an adjoint-closed set of twisted translation
operators T of the form (2.2), and then consider a self-adjoint bounded operator
γ
Ainthe vonNeumannalgebraB(l2(X))TΓ which—likethe DML —hasbounded
propagation.
Definition 2.1. An operator A has propagation bounded by R if for any function
f ∈ l2(X), the support of Af is a subset of the R-neighborhood of the support of
f in the simplicial metric.
TheoperatorAisweaklyΓ-equivariant ifthereexistsaΓ-indexedsetoffunctions
t : VertX →U(1) defining twisted translation operatorsT :
γ γ
(T f)(x)=t (γ−1x)f(γ−1x),
γ γ
such that A commutes with T and T∗ for all γ ∈Γ. Then A is an element of the
γ γ
associated von Neumann algebra B(l2(X))TΓ with trace
(2.4) TrΓ(A)= X <Aδv,δv >.
v∈F
For eachofthe subgraphsX one candefine afinite operatorA byrestricting
m m
A to those functions with support on the vertices of X ; explicitly, A =P Ai
m m m m
whereP istheorthogonalprojectionontol2(X )andi : l2(X )→l2(X)isthe
m m m m
canonical inclusion. We define the normalized spectral density functions of these
restricted operators by
1
(2.5) F (λ)= #{eigenvalues ≤λ of A , counting multiplicity}.
m m
#Λ
m
Part(i)ofTheorem2.6of[MY]relateslimitsoftheseF tothespectraldensity
m
functionF ofAwhenAisthediscretemagneticLaplacian. Theargumentdeployed
though extends trivially to bounded self-adjoint weakly Γ-equivariant operators of
bounded propagationgenerally.
4 VARGHESE MATHAI, THOMAS SCHICK, AND STUART YATES
Theorem 2.2 (Weak spectral approximation theorem). Take X to be a graph as
above, and let A be a bounded self-adjoint weakly Γ-equivariant operator over l2(X)
of bounded propagation, with restrictions A to the regular exhaustion X of X.
m m
Define the following limits of the normalized spectral density functions,
F(λ)=limsupF (λ)
m
m→∞
F(λ)=liminfF (λ)
m
m→∞
(2.6)
+
F (λ)= lim F(λ+δ)
δ→+0
F+(λ)= lim F(λ+δ).
δ→+0
Then
(2.7) F(λ)=F+(λ)=F+(λ), ∀λ∈R.
Proof. Refer to the proof of Theorem 2.6 of [MY].
As adirectconsequence,onehasthatateverypointλatwhichF iscontinuous,
that is, for every λ6∈spec A,
point
(2.8) lim F (λ)=F(λ).
m
m→∞
In particular, equation (2.8) holds for all but at most a countable set of points. In
the following section a new argumentis presented that extends this result to all λ.
3. The strong spectral approximation result
A consequence of Theorem 2.2 is that if the jump of F at a discontinuity λ can
be approximated by the jumps at λ of the piece-wise constant normalized spectral
density functions F , then the limit of F (λ) exists and equals F(λ).
m m
We first need a small technical lemma.
Lemma 3.1. Given two sequences {a } and {b } with a ≤ b for all i, it follows
i i i i
that sup a ≤sup b and inf a ≤inf b .
i i i i i i i i
In particular, if {f } is a sequence of monotonically increasing functions, then
i
liminfi→∞fi and limsupi→∞fi are also monotonically increasing.
Proof. From a ≤b for all i, it follows that
i i
infa ≤a ≤b ≤supb ∀k.
i k k i
i i
Then inf a ≤b for all k implies inf a ≤inf b , anda ≤sup b for allk implies
i i k i i i i k i i
sup a ≤sup b .
i i i i
Considerthe sequence {f } ofmonotonicallyincreasingfunctions; for anyδ >0,
i
f (x+δ)≥f (x) for every i. Therefore
i i
inf f (x)≤ inf f (x+δ) ∀k.
i i
i>k i>k
Taking the limit or supremum over k then gives
liminff (x)≤liminff (x+δ),
k k
k→∞ k→∞
thatis,thefunctionliminfk→∞fk ismonotonicallyincreasing. Thesameargument
with sup instead of inf gives the corresponding inequality for limsup f .
k→∞ k
The corollary then follows from Theorem 2.2.
APPROXIMATING SPECTRAL INVARIANTS OF HARPER OPERATORS II 5
Corollary 3.2. Denote by D(λ) the jump in F at λ,
D(λ)= lim F(λ)−F(λ−δ),
δ→+0
and similarly let D (λ) be the jump in the normalized spectral density functions
m
F ,
m
D (λ)= lim F (λ)−F (λ−δ).
m m m
δ→+0
Then for any λ∈R,
(3.1) lim D (λ)=D(λ) =⇒ lim F (λ)=F(λ).
m m
m→∞ m→∞
Proof. Fixsomeǫ>0andλ∈R,andsupposethatlimm→∞Dm(λ)=D(λ). Then
we will show that F(λ)−3ǫ<F (λ)<F(λ)+ǫ for all sufficiently large m.
m
By the right continuity and monotonicity of F, there exists a δ > 0 such that
F(x) < F(λ)+ ǫ for all λ ≤ x < λ+δ. The F approach F at all points of
2 m
continuityofF;asF hasatmostacountablenumberofdiscontinuitiesthereexists
some x0 ∈[λ,λ+δ) where limm→∞Fm(x0)=F(x0)<F(λ)+ 2ǫ. So for m greater
than some M , it follows that F (x ) < F(λ) + ǫ. The F are monotonically
1 m 0 m
increasing, so
(3.2) ∃M such that F (λ)<F(λ)+ǫ ∀m>M .
1 m 1
Now F(λ)=D(λ)+limδ→+0F(λ−δ), so we canchoose d suchthat F(λ−d)+
D(λ) > F(λ)−ǫ. As the F are monotonically increasing functions, F is also by
m
Lemma 3.1. By Theorem 2.2, F+(µ)=F(µ) for all µ. Therefore F(µ+δ)>F(µ)
for all δ > 0. Letting µ = λ−d and δ = d gives F(λ− d) > F(λ−d) and so for
2 2
sufficiently large m, F (λ− d)>F(λ−d)−ǫ.
m 2
From F (λ)≥F (λ− d)+D (λ) and F(λ−d)>F(λ)−D(λ)−ǫ,
m m 2 m
∃M such that F (λ)>F(λ−d)−ǫ+D (λ)
2 m m
(3.3) >F(λ)+Dm(λ)−D(λ)−2ǫ
≥F(λ)−|D (λ)−D(λ)|−2ǫ ∀m>M .
m 2
Bysupposition,limm→∞Dm(λ)=D(λ). InparticularwecanpickanM greater
than M and M such that |D (λ)−D(λ)|<ǫ for all m>M. Therefore
1 2 m
∃M such that m>M =⇒ F(λ)+ǫ>F (λ)>F(λ)−3ǫ,
m
and thus
lim D (λ)=D(λ) =⇒ lim F (λ)=F(λ).
m m
m→∞ m→∞
The following theorem establishes this approximation of the jumps. Its proof is
modelled on the argument in [El].
Theorem 3.3. ThejumpD(λ)ofF atλisthelimitofthejumpsofthenormalized
spectral density functions:
lim D (λ)=D(λ) ∀λ∈R.
m
m→∞
6 VARGHESE MATHAI, THOMAS SCHICK, AND STUART YATES
Proof. Notethatthe jumps canbe expressedintermsofthe dimensionsofkernels,
D(λ)=dim ker(A−λ),
Γ
(3.4) 1
D (λ)= dimker(A −λ).
m m
#Λ
m
Consider subsets Y of X consisting of the r-interior of X , where A has
m m m
propagationbounded by r.
For each X define dim for a subspace W of l2(X) by
m Xk
1
dimXkW = #Λ X hPWδx,δxi
k x∈Xk
where P is the orthogonal projection onto W. This has the following properties
W
for subspaces W, V of l2(X):
1. W ⊥V =⇒ dim (W ⊕V)=dim W +dim V,
Xk Xk Xk
2. W ⊂V =⇒ dim W ≤dim V,
Xk Xk
3. PW ∈B(l2(X))TΓ =⇒ dimXkW =dimΓW,
4. W ⊂l2(X ) =⇒ dim W = 1 dimW, regarding l2(X ) as a subspace of
k Xk #Λk k
l2(X).
In particular, dim l2(X ) = #F and dim l2(Y ) converges to #F as k ap-
Xk k Xk k
proaches infinity.
Leti′ : l2(Y )→l2(X )betheinclusionwith(i′ f)(x)=0forallxinX \Y .
m m m m m m
Let A′ : l2(Y ) → l2(X ) be the operator A i′ . For any function f in l2(Y ),
m m m m m m
(A′ −λi′ )f =(A −λ)(i′ f). Therefore as subspaces of l2(X),
m m m m
ker(A′ −λi′)⊂ker(A −λ),
k k k
(3.5)
im(A′ −λi′)⊂im(A −λ).
k k k
Let D′(λ)= 1 dimker(A′ −λi′). Then taking dim ,
k #Λm k k Xk
D′(λ)=dim ker(A′ −λi′)≤dim ker(A −λ)=D (λ),
k Xk k k Xk k k
dim im(A′ −λi′)≤dim im(A −λ).
Xk k k Xk k
One also has for any m,
lim dim ker(A′ −λi′)+dim im(A′ −λi′)= lim dim l2(Y )
k→∞ Xk k k Xk k k k→∞ Xk k
=#F =dim ker(A −λ)+dim im(A −λ).
Xm m Xm m
Consequently,
(3.6) lim D (λ)−D′(λ)=0.
k k
k→∞
Recall that A = P Ai , where P is the projection onto l2(X ) and i is the
k k k k k k
canonical inclusion of l2(X ) into l2(X). In terms of A then, A′ = P Ai i′. By
k k k k k
the bounded propagation of A, the support of Ai i′f is contained within X for
k k k
any f in l2(Y ) and hence as in (3.5),
k
ker(A′ −λi′)⊂ker(A−λ),
k k
im(A′ −λi′)⊂im(A−λ).
k k
APPROXIMATING SPECTRAL INVARIANTS OF HARPER OPERATORS II 7
The kernel and image of A−λ are invariant under the twisted translations T ,
γ
and so for these subspaces, dim equals dim . It follows that
Xk Γ
D′(λ)=dim ker(A′ −λi′)≤dim ker(A −λ)=D(λ),
k Xk k k Γ k
dim im(A′ −λi′)≤dim im(A −λ),
Xk k k Γ k
and
lim dim ker(A′ −λi′)+dim im(A′ −λi′)
k→∞ Xk k k Xk k k
=#F =dim ker(A−λ)+dim im(A−λ).
Γ Γ
Therefore
lim D′(λ)=D(λ),
k
k→∞
which together with Equation (3.6) gives the required limit
lim D (λ)=D(λ).
k
k→∞
Theorem 1.1 follows from Theorem 3.3 and Corollary 3.2.
References
[Ad] T.Adachi,AnoteontheFølnerconditionofamenability,NagoyaMath.J.131(1993)
67–74.
[Bel] J.Bellissard,GapLabelingTheoremsforSchro¨dinger’sOperators,Fromnumbertheory
tophysics(LesHouches,1989), 538–630, Springer,Berlin,1992.
[DLMSY] J.Dodziuk,P.Linnell,V.Mathai,T.SchickandS.Yates,ApproximatingL2-invariants,
andtheAtiyahconjecture, math.GT/0107049.
[Eck] B. Eckmann, Approximating ℓ2-Betti numbers of an amenable covering by ordinary
Bettinumbers,Comment. Math. Helv.74(1999) 150-155.
[El] G.Elek,OntheanalyticzerodivisorconjectureofLinnell,math.GR/0111180.
[MY] V. Mathai and S. Yates, Approximating spectral invariants of Harper operators on
graphs,J. Functional Analysis,inpress,math.FA/0006138.
[Sh] M. Shubin, Discrete Magnetic Schro¨dinger operators, Commun. Math. Phys., 164
(1994), no.2,259–275.
[Sun] T.Sunada,AdiscreteanalogueofperiodicmagneticSchro¨dingeroperators,Contemp.
Math.173(1994), 283–299.
Departmentof Mathematics,University ofAdelaide, Adelaide 5005,Australia
E-mail address: [email protected]
FB Mathematik,Universita¨tGo¨ttingen,Bunsenstr. 3,37073Go¨ttingen,Germany
E-mail address: [email protected]
Departmentof Mathematics,University ofAdelaide, Adelaide 5005,Australia
E-mail address: [email protected]