Heat kernels on metric graphs and a trace formula

0.27 MB·English

by  Vadim Kostrykin

HEAT KERNELS ON METRIC GRAPHS AND A TRACE FORMULA
VADIMKOSTRYKIN,JU¨RGENPOTTHOFF,ANDROBERTSCHRADER
DedicatedtoJean-MichelCombesontheoccasionofhis65-thbirthday
7
0
0 ABSTRACT. Westudyheat semigroups generated byself-adjoint Laplace operators on
2 metricgraphscharacterized bythepropertythatthelocalscattering matrices associated
witheachvertexofthegraphareindependentfromthespectralparameter. Forsuchop-
n
eratorsweprovearepresentation fortheheatkernelasasumoverallwalkswithgiven
a
J initialandterminaledges.Usingthisrepresentationatraceformulaforheatsemigroupsis
proven. Applicationsofthetraceformulatoinversespectralandscatteringproblemsare
4
alsodiscussed.

1
v
9
0 1. INTRODUCTION
0
1 Metricgraphsornetworksareone-dimensionalpiecewiselinearspaceswithsingular-
0 ities atthe vertices. Alternatively,a metric graphis a metric space which can be written
7
asaunionoffinitelymanyintervals,whichareeithercompactor[0, );anytwoofthese
0 ∞
intervalsareeitherdisjointorintersectonlyinoneorbothoftheirendpoints. Itisnatural
/
h
tocallthemetricgraphcompactifallitsedgeshavefinitelength.
p
Theincreasinginterestinthetheoryofdifferentialoperatorsonmetricgraphsismoti-
-
h vatedmainly by two reasons. The first reasonis thatsuch operatorsarise in a varietyof
t applications. We refer the reader to the review [34], where a number of models arising
a
m in physics, chemistry, and engineeringis discussed. The secondreason is purelymathe-
matical: It is intriguing to study the interrelation between the spectra of these operators
:
v andtopologicalorcombinatorialpropertiesoftheunderlyinggraph.Similarinterrelations
i
X arestudiedinspectralgeometryfordifferentialoperatorsonRiemannianmanifolds(see,
e.g.[7],[15])andinspectralgraphtheoryfordifferenceoperatorsoncombinatorialgraphs
r
a (see,e.g.[9]). Metricgraphstakeanintermediatepositionbetweenmanifoldsandcombi-
natorialgraphs.
Inthepresentworkwecontinuethestudyofheatsemigroupsonmetricgraphsinitiated
in [28]. There we provided sufficient conditions for a self-adjoint Laplace operator to
generateacontractivesemigroup. Moreover,weprovedacriterionguaranteeingthatthis
semigroup is positivity preserving. For earlier work on heat semigroups generated by
Laplace operatorson metric graphsand their application to spectral analysis we refer to
[3],[13],[14],[40],[41],[46],[47].
In this article we study heat semigroups generated by self-adjoint Laplace operators
whicharecharacterizedbythepropertythatthelocalscatteringmatricesassociatedwith
eachvertexofthegraphareindependentofthespectralparameter.Allboundaryconditions
leadingtosuchoperatorsaredescribedinProposition2.4below. Inparticular,Neumann,
Dirichlet,andthesocalledstandardboundaryconditionsareinthisclass.
2000MathematicsSubjectClassification. Primary34B45,81U40;Secondary47D06.
Keywordsandphrases. Metricgraphs,heatsemigroups,traceformulas,inverseproblems.
1
2 V.KOSTRYKIN,J.POTTHOFF,ANDR.SCHRADER
Ourmaintechnicaltooltostudyheatsemigroupsonmetricgraphsarewalksonedges
ofthegraph,a conceptdevelopedin [28], [29]. We willrevisitthisconceptin Section3
below. Furthermore,wewillprovidearepresentationfortheheatkernelasasumoverall
walkswithgiveninitialandterminaledges. Thisrepresentationrelatesthetopologyofthe
graphtoanalyticpropertiesoftheheatsemigroup.
InSection4weproveatraceformulaforheatsemigroupsonarbitrary(compactaswell
asnoncompact)metricgraphs,ananalogofthecelebratedSelbergformulafordifferential
operatorsonRiemannianmanifolds(see[38],[48]forthecaseofcompactmanifoldsand
[22], [37] for the noncompactcase). A discrete analog of the Selberg trace formula on
k-regulartreesisdiscussedin[49].
The trace formulaexpressesthe trace of the semigroupdifferenceas the sum overall
cycles on the graph, that is, equivalence classes of closed walks. In the particular case
of compactgraphsandstandard boundaryconditionsour result recoversthe well-known
traceformulaobtainedbyRoth[46],[47]. Relatedresultscanbefoundin[35],[40],[41],
[50]. InthephysicalliteraturetraceformulasforLaplaceoperatorsonmetricgraphshave
beendiscussed in [2], [31], [32], [33]. Their applicationsto quantumchaosandspectral
statisticsarereviewedintherecentarticle[16].
Asanapplicationofthetraceformula,inSection5wediscussinversespectralandscat-
teringproblems. The inverseproblemsconsideredhere consistof determiningthe graph
anditsmetricstructure(i.e.thelengthsofitsedges)fromthespectrumoftheLaplaceop-
eratorandthescatteringphase(thatis,halfthephaseofthedeterminantofthescattering
matrix), underthe condition that the boundaryconditionsat all vertices of the graph are
supposed to be known. Another kind of the inverse scattering problem, the reconstruc-
tionofthegraphandtheboundaryconditionsfromthescatteringmatrix,hasbeensolved
recentlyin[29].
TheresultsofSection5provideamathematicallyrigoroussolutionoftheinversescat-
tering problem as proposed by Gutkin and Smilansky in [19]. Also these result extend
the solution of the inverse spectral problem on compact graphs given by Kurasov and
Nowaszykin[35]tomoregeneralboundaryconditions.
Acknowledgments. Itisapleasuretothanktheorganizersoftheconference“Transport
andSpectralProblemsin QuantumMechanics”heldatthe UniversityofCergy-Pontoise
in September 2006 for a very interesting and enjoyable meeting, both scientifically and
socially.TheauthorswouldliketothankM.Karowskiforhelpfuldiscussions.
2. BACKGROUND
A finite graphis a 4-tuple = (V, , ,∂), where V is a finite set of vertices, is a
G I E I
finitesetofinternaledges, isafinitesetofexternaledges. Elementsin arecalled
E I∪E
edges. Themap∂ assignstoeachinternaledgei anorderedpairof(possiblyequal)
∈ I
vertices∂(i) := v ,v andtoeachexternaledgee asinglevertexv. Thevertices
1 2
{ } ∈ E
v =: ∂−(i) and v =: ∂+(i) are called the initial and terminal vertex of the internal
1 2
edge i, respectively. The vertex v = ∂(e) is the initial vertex of the externaledge e. If
∂(i)= v,v ,thatis,∂−(i)=∂+(i)theniiscalledatadpole.Agraphiscalledcompact
if =˘{,oth}erwiseitisnoncompact.
E
Two vertices v and v′ are called adjacent if there is an internaledge i such that
∈ I
v ∂(i) and v′ ∂(i). A vertex v and the (internal or external) edge j are
∈ ∈ ∈ I ∪E
incidentifv ∂(j).
∈
Wedonotrequirethemap∂ tobeinjective. Inparticular,anytwoverticesareallowed
to be adjacent to more than one internal edge and two different external edges may be
HEATKERNELSONMETRICGRAPHS 3
incidentwiththesamevertex. If∂ isinjectiveand∂−(i)=∂+(i)foralli ,thegraph
6 ∈I
iscalledsimple.
G
Thedegreedeg(v)ofthevertexvisdefinedas
deg(v)= e ∂(e)=v + i ∂−(i)=v + i ∂+(i)=v ,
|{ ∈E | }| |{ ∈I | }| |{ ∈I | }|
thatis,itisthenumberof(internalorexternal)edgesincidentwiththegivenvertexv by
whicheverytadpoleiscountedtwice.
Itis easy to extendthe First TheoremofGraphTheory(see, e.g. [11]) to the case of
noncompactgraphs:
(2.1) deg(v)= +2 .
|E| |I|
v∈V
X
Avertexiscalledaboundaryvertexifitisincidentwithsomeexternaledge.Thesetof
allboundaryverticeswillbedenotedby∂V. Theverticesnotin∂V,thatisinV ∂V are
\
calledinternalvertices.
Thecompactgraph = (V, ,˘ ,∂ )willbecalledtheinterior ofthegraph =
int I
G I | G
(V, , ,∂). Itisobtainedfrom byeliminatingtheexternaledges.
I E G
Thestar (v) ofthevertexv V isthesetoftheedgesadjacenttov.
S ⊆E ∪I ∈
Throughoutthewholeworkwe willassumethatthegraph isconnected,thatis, for
G
anyv,v′ V thereis an orderedsequenceof vertices v = v,v ,...,v ,v = v′
1 2 n−1 n
∈ { }
such that any two successive vertices in this sequence are adjacent. In particular, this
assumptionimpliesthatanyvertexofthegraph hasnonzerodegree,i.e.,foranyvertex
G
thereisatleastoneedgewithwhichitisincident.
Wewillendowthegraphwiththefollowingmetricstructure. Anyinternaledgei
∈ I
will be associated with an interval [0,a ] with a > 0 such that the initial vertex of i
i i
corresponds to x = 0 and the terminal one to x = a . Any external edge e will
i
∈ E
be associated with a semiline [0,+ ). We call the numbera the length of the internal
i
edgei. Thesetoflengths a ,∞whichwillalsobetreatedasanelementofR|I|,will
i i∈I
{ }
be denotedby a. A compactor noncompactgraph endowedwith a metric structureis
G
calledametricgraphandiswrittenas( ,a).
G
Givenafinitegraph = (V, , ,∂)withametricstructurea = a considerthe
i i∈I
G I E { }
Hilbertspace
(2.2) ( , ,a)= , = , = ,
E I E e I i
H≡H E I H ⊕H H H H H
e∈E i∈I
M M
where =L2(I )with
j j
H
[0,a ] if j ,
j
I = ∈I
j
([0, ) if j .
∞ ∈E
o o o
LetI betheinteriorofI ,thatis,I =(0,a )ifj andI =(0, )ifj .
j j j j j
∈I ∞ ∈E
In the sequel the letters x and y will denote arbitrary elements of the product set
· I .
j
j∈E∪I
By withj denotethesetofallψ suchthatψ (x)anditsderivative
j j j j
D ∈ E ∪I ∈ H
ψ′(x)areabsolutelycontinuousandψ′′(x)issquareintegrable. Let 0 denotethesetof
j j Dj
thoseelementsψ whichsatisfy
j j
∈D
ψ (0)=0 ψ (0)=ψ (a )=0
j for j and j j j for j .
ψj′(0)=0 ∈E ψj′(0)=ψj′(aj)=0 ∈I
4 V.KOSTRYKIN,J.POTTHOFF,ANDR.SCHRADER
Let∆0bethedifferentialoperator
d2
(2.3) ∆0ψ (x)= ψ (x), j
j dx2 j ∈I∪E
withdomain (cid:0) (cid:1)
0 = 0 .
D Dj ⊂H
j∈E∪I
M
Itisstraightforwardtoverifythat∆0isaclosedsymmetricoperatorwithdeficiencyindices
equalto +2 .
|E| |I|
Weintroduceanauxiliaryfinite-dimensionalHilbertspace
(2.4) ( , )= (−) (+)
K≡K E I KE ⊕KI ⊕KI
with =C|E|and (±) =C|I|. Letd denotethe“double”of ,thatis,d = .
KE ∼ KI ∼ K K K K⊕K
Foranyψ := weset
j
∈D D
j∈E∪I
M
(2.5) [ψ]:=ψ ψ′ d ,
⊕ ∈ K
withψandψ′ definedby
ψ (0) ψ′(0)
(2.6) ψ = {ψe(0)}e∈E , ψ′ = {ψe′(0)}e∈E .
{ i }i∈I  { i }i∈I 
ψ (a ) ψ′(a )
{ i i }i∈I {− i i }i∈I
   
LetJ bethecanonicalsymplecticmatrixond ,
K
0 I
(2.7) J =
I 0
(cid:18)− (cid:19)
withIbeingtheidentityoperatoron . Considerthenon-degenerateHermitiansymplectic
K
form
(2.8) ω([φ],[ψ]):= [φ],J[ψ] ,
h i
where , denotesthescalarproductind =C2(|E|+2|I|).
h· ·i K∼
Alinearsubspace ofd iscalledisotropiciftheformωvanishesidenticallyon .
M K M
Anisotropicsubspaceiscalledmaximalifitisnotapropersubspaceofalargerisotropic
subspace.Everymaximalisotropicsubspacehascomplexdimensionequalto +2 .
|E| |I|
LetAandBbelinearmapsof ontoitself. By(A,B)wedenotethelinearmapfrom
K
d = to definedbytherelation
K K⊕K K
(A,B)(χ χ ):=Aχ +Bχ ,
1 2 1 2
⊕
whereχ ,χ . Set
1 2
∈K
(2.9) (A,B):=Ker(A,B).
M
Theorem2.1([24]). Asubspace d ismaximalisotropicifandonlyifthereexist
M ⊂ K
linearmapsA, B : suchthat = (A,B)and
K→K M M
(i) themap(A,B): d hasmaximalrankequalto +2 ,
(2.10) K→K |E| |I|
(ii) AB†isself-adjoint,AB† =BA†.
Undertheconditions(2.10)bothA ikBareinvertibleforallk>0.
±
HEATKERNELSONMETRICGRAPHS 5
Definition2.2. Twoboundaryconditions(A,B)and(A′,B′)satisfying(2.10)areequiv-
alent if the corresponding maximal isotropic subspaces coincide, that is, (A,B) =
M
(A′,B′).
M
The boundary conditions (A,B) and (A′,B′) satisfying (2.10) are equivalent if and
only if there is an invertible map C : such that A′ = CA and B′ = CB (see
K → K
Proposition3.6in[29]).
ByLemma3.3in[29],asubspace (A,B) d ismaximalisotropicifandonlyif
M ⊂ K
(2.11) (A,B)⊥ = (B, A).
M M −
Wementionalsotheequalities
(A,B)⊥ = Ker(A,B) ⊥ =Ran(A,B)†,
M
(A,B)=(cid:2)Ran( B,A)(cid:3)†.
M −
Thereisanalternativeparametrizationofmaximalisotropicsubspacesofd byunitary
K
transformationsin (see[28]andProposition3.6in[29]). Asubspace (A,B) d
is maximalisotropiKcif andonly if foran arbitraryk R 0 the operMatorA+i⊂kBKis
∈ \{ }
invertibleand
(2.12) S(k;A,B):= (A+ikB)−1(A ikB)
− −
isunitary.Moreover,givenanyk R 0 thecorrespondencebetweenmaximalisotropic
subspaces d andunitaryo∈pera\to{rs}S(k;A,B) U( +2 )on isone-to-one,
M⊂ K ∈ |E| |I| K
aresultdatingbacktoBott[6]andrediscoveredin[4],[21],and[25]. Therefore,wewill
usethenotationS(k; )forS(k;A,B)with (A,B)= .
M M M
Under the duality transformation ⊥, as a direct consequence of (2.11) and
M 7→ M
(2.12),theoperators(2.12)transformasfollows(seeCorollary2.2in[24]):
(2.13) S(k; ⊥)= S(k−1; ).
M − M
There is a one-to-one correspondence between all self-adjoint extensions of ∆0 and
maximal isotropic subspaces of d (see [24], [29]). In explicit terms, any self-adjoint
K
extensionof∆0isthedifferentialoperatordefinedby(2.3)withdomain
(2.14) Dom(∆)= ψ [ψ] ,
{ ∈D| ∈M}
where is a maximal isotropic subspace of d . Conversely, any maximal isotropic
subspacMe of d defines through (2.14) a selfK-adjoint operator ∆( ,a). If = ˘ ,
M K M I
we willsimplywrite ∆( ). Inthe sequelwe will calltheoperator∆( ,a) a Laplace
M M
operator on the metric graph ( ,a). From the discussion above it follows immediately
G
thatanyself-adjointLaplaceoperatoron equals∆( ,a)forsomemaximalisotropic
H M
subspace . Moreover,∆( ,a)=∆( ′,a)ifandonlyif = ′.
M M M M M
FromTheorem2.1itfollowsthatthedomainoftheLaplaceoperator∆( ,a)consists
M
offunctionsψ satisfyingtheboundaryconditions
∈D
(2.15) Aψ+Bψ′ =0,
with(A,B)subjectto(2.9)and(2.10). Hereψandψ′aredefinedby(2.6).
Withrespecttotheorthogonaldecomposition = (−) (+) anyelementχ
K KE ⊕KI ⊕KI
of canberepresentedasavector
K
χ
e e∈E
{ }
(2.16) χ= χ(−) .
{ i }i∈I
χ(+)
{ i }i∈I
 
6 V.KOSTRYKIN,J.POTTHOFF,ANDR.SCHRADER
Considertheorthogonaldecomposition
(2.17) =
v
K L
v∈V
M
with the linearsubspaceofdimensiondeg(v)spannedbythoseelements(2.16)of
v
L K
whichsatisfy
χ =0 if e isnotincidentwiththevertex v,
e
∈E
(2.18) χ(−) =0 if v isnotaninitialvertexof i ,
i ∈I
χ(+) =0 if v isnotaterminalvertexof i .
i ∈I
Obviously,thesubspaces and areorthogonalifv =v .
Setd := =LvC12deg(Lv)v.2Obviously,eachd 1 6inhe2ritsasymplecticstructure
Lv Lv ⊕Lv ∼ Lv
fromd inacanonicalway,suchthattheorthogonaldecomposition
K
d =d
v
L K
v∈V
M
holds.
Definition2.3. Giventhegraph = (V, , ,∂),boundaryconditions(A,B)satisfying
G G I E
(2.10) are called local on if the maximal isotropic subspace (A,B) of d has an
G M K
orthogonalsymplecticdecomposition
(2.19) (A,B)= ,
v
M M
v∈V
M
with maximal isotropic subspaces of d . Otherwise the boundary conditions are
v v
M L
callednon-local.
ByProposition4.2in[29],giventhegraph = (V, , ,∂),theboundaryconditions
G G I E
(A,B)satisfying(2.10)arelocalon ifandonlyifthereisaninvertiblemapC :
G K→K
and linear transformations A(v) and B(v) in such that the simultaneous orthogonal
v
L
decompositions
(2.20) CA= A(v) and CB = B(v)
v∈V v∈V
M M
arevalid. Fromtheequality (A,B) = (CA,CB)itfollowsthatthesubspaces
v
M M M
in(2.19)areequalto (A(v),B(v)).
M
Boundary conditions (A(v),B(v)) induce local boundary conditions (A,B) on the
graph with
G
(2.21) A= A(v) and B = B(v).
v∈V v∈V
M M
From(2.20)wegetthat
(2.22) S(k;A,B)=S(k;CA,CB)= S(k;A(v),B(v))
v∈V
M
holdswithrespecttotheorthogonaldecomposition(2.17).
Thefollowingpropositionistakenfrom[29].
Proposition 2.4. Let = (A,B) be a maximal isotropic subspace. The following
M M
conditionsareequivalent:
(i) S(k; )isk-independent,
M
HEATKERNELSONMETRICGRAPHS 7
(ii) S(k; )isself-adjointforsomek>0,
(iii) forsoMmek>0thereisanorthogonalprojectionP suchthatS(k; )=I 2P,
M −
(iv) AB† =0.
Sincethispropositionwillbecrucialinwhatfollows,werecallthe
Proof. (i) (ii). AssumethatS(k; )isk-independent.Then,by(2.12),foranyeigen-
⇔ M
vectorχ witheigenvalueλtheequality
∈K
(λ+1)Aχ+ik(λ 1)Bχ=0
−
holdsforallk>0. Undertheconditions(2.10)wehaveKerA KerB(seeLemma3.4
in[29]).Hence,λ 1,1 .Thus,S(k; )isself-adjointfor⊥allk>0.
Conversely,assu∈m{e−thatS} (k; )isseMlf-adjointforsomek > 0. Duetotheobvious
0
M
equality
(2.23) S(k; )= (k k )S(k ; )+(k+k ) −1 (k+k )S(k ; )+(k k ) ,
0 0 0 0 0 0
M − M M −
it is self-adjoint for(cid:0)all k > 0. Let χ be a(cid:1)n a(cid:0)rbitrary eigenvector of S(k ; (cid:1) )
0
∈ K M
correspondingtotheeigenvalueλ 1,1 .Observingthat
∈{− }
(k+k )λ+k k
0 0
− =λ,
(k k )λ+k+k
0 0
−
again by (2.23), we conclude that χ is an eigenvectorof S(k; ) correspondingto the
sameeigenvalueλforallk>0. Thus,S(k; )doesnotdepenMdonk>0.
M
Theequivalence(ii) (iii)isobvious.
⇔
Theequivalence(iv) (ii)followsdirectlyfromtheidentity
⇔
S(k; ) S(k; )†
M − M
=2ik(A+ikB)−1 B(A† ikB†)+(A+ikB)B† (A† ikB†)−1
− −
=4ik(A+ikB)−1(cid:2)AB†(A† ikB†)−1. (cid:3)
−
(cid:3)
WewillwriteS( )insteadofS(k; ),wheneveranyoftheequivalentconditionsof
Proposition2.4ismMet. AnalogouslyweMwilldropthek-dependencein(2.22):
S( )= S(A(v),B(v))= S( ).
v
M M
v∈V v∈V
M M
From Proposition 3.5 in [28] it follows that for any maximal isotropic subspace
M
satisfying any of the conditions of Proposition 2.4, the Laplace operator ∆( ,a) is
− M
nonnegative.
Remark 2.5. Assume that the maximal isotropic subspace d satisfies any of
M ⊂ K
the conditions of Proposition 2.4. By (2.11) the orthogonalmaximal isotropic subspace
⊥ d thenalsosatisfiestheconditionsofProposition2.4. From(2.13)itfollowsthat
MS( ⊂⊥)K= S( ).
M − M
Obviously,DirichletA = I,B = 0andNeumannA = 0,B = Iboundaryconditions
satisfytheconditionsofProposition2.4withS(I,0)= IandS(0,I)= I,respectively.
−
Wenowprovidetwoimportantexamplesofboundaryconditionssatisfyingtheconditions
referredtoinProposition2.4.
8 V.KOSTRYKIN,J.POTTHOFF,ANDR.SCHRADER
Example2.6(Standardboundaryconditions). Givenagraph = (V, , ,∂)foreach
G G I E
vertexv V withdeg(v) 2definetheboundaryconditions(A(v),B(v))thedeg(v)
∈ ≥ ×
deg(v)matrices
1 1 0 ... 0 0 0 0 0 ... 0 0
−
0 1 1 ... 0 0 0 0 0 ... 0 0
 −   
0 0 1 ... 0 0 0 0 0 ... 0 0
A(v)=.. .. .. .. .. , B(v)=.. .. .. .. ...
. . . . .  . . . . .
   
0 0 0 ... 1 1 0 0 0 ... 0 0
 −   
0 0 0 ... 0 0  1 1 1 ... 1 1
   
   
Clearly,A(v)B(v)† =0and(A(v),B(v))hasmaximalrank. Thecorrespondingunitary
matrices(2.12)aregivenby
2
(2.24) [S(A(v),B(v))] = δ
e,e′ e,e′
deg(v) −
with δ Kronecker symbol. If deg(v) = 1, we set A(v) = 0, B(v) = 1 (Neumann
e,e′
boundaryconditions)suchthat(2.24)remainsvalid.
Thelocalboundaryconditions(A,B)onthegraph definedby(2.21)will becalled
G
standard boundaryconditions. We use the notation for the correspondingmaximal
st
M
isotropicsubspace.
Remark2.7. Consideragraphwithnointernallines =( v ,˘ , ,∂)and 2. By
G { } E |E|≥
Proposition2.1in [12], thesetofallisotropicsubspacessatisfyinganyofthe equivalent
conditions of Proposition 2.4, contains precisely four spaces, which correspond to the
boundaryconditionsinvariantwithrespecttopermutationsofedges: (I,0)(Dirichlet),
(0,I)(Neumann),standard ,andco-standard ⊥. M
M Mst Mst
Furthermore, byaresultin[30], inthesetofallisotropicsubspacessatisfyinganyof
the equivalentconditionsof Proposition 2.4, is the only one with the property that
st
M
everyfunctioninthedomainof∆( )iscontinuousatthevertexv.
M
Example 2.8 (Magnetic perturbationsof standard boundaryconditions). If the maximal
isotropicsubspace (A,B)satisfiesanyoftheequivalentconditionsofProposition2.4,
M
thenforanyunitaryU wehave
AU(BU)† =AB† =0.
Thus,themaximalisotropicsubspace U := (AU,BU)alsosatisfiestheconditions
M M
ofProposition2.4. Inparticular,since
S( U)=U†S( )U,
M M
wehavetherelation
(2.25) tr S( U)=tr S( ).
K K
M M
A special choice of unitary matrices U corresponds to magnetic perturbations of the
Laplaceoperator∆( ,a). Byaresultin[27]anymagneticperturbationoftheLaplace
M
operator ∆( ,a) is unitarily equivalent to ∆( U,a) with some U = U , where
v
M M
v∈V
M
everyU isunitaryanddiagonalwithrespecttothecanonicalbasisin ,
v v
L
U =diag eiϕj(v) .
v j∈S(v)
{ }
In particular, any magnetic perturbation(cid:16)of standard bou(cid:17)ndary conditions (see Example
2.6)satisfiestheconditionsofProposition2.4.
HEATKERNELSONMETRICGRAPHS 9
3. HEATKERNELAND WALKS ON THEGRAPH
3.1. TheResolvent. ThestructureoftheunderlyingHilbertspace (2.2)givesnaturally
H
risetothefollowingdefinitionofintegraloperators.
Definition3.1. TheoperatorKontheHilbertspace iscalledintegraloperatorifforall
j,j′ therearemeasurablefunctionsK (,H) : I I Cwiththefollowing
j,j′ j j′
∈E ∪I · · × →
properties
(i) K (x , )ϕ () L1(I )foralmostallx I ,
j,j′ j j′ j′ j j
· · ∈ ∈
(ii) ψ =Kϕwith
(3.1) ψ (x )= K (x ,y )ϕ (y )dy .
j j j,j′ j j′ j′ j′ j′
j′∈E∪IZIj′
X
The( + ) ( + )matrix-valuedfunction(x,y) K(x,y)with
|I| |E| × |I| |E| 7→
[K(x,y)] =K (x ,y )
j,j′ j,j′ j j′
iscalledtheintegralkerneloftheoperatorK.
Belowwewillusethefollowingshorthandnotationfor(3.1):
G
ψ(x)= K(x,y)ϕ(y)dy.
Z
Wedenote
I 0 0
(3.2) R(k;a):= 0 I 0 ,
0 0 e−ika
and  
0 0 0
(3.3) T(k;a):= 0 0 eika
0 eika 0 
withrespecttotheorthogonaldecomposition(2.4). Thediagonal matricese±ika
|I|×|I|
aregivenby
(3.4) [e±ika] =δ e±ikaj for j,k .
jk jk
∈ I
Lemma3.2. Foranymaximalisotropicsubspace d theresolvent
M⊂ K
( ∆( ;a) k2)−1 for k2 C spec( ∆( ;a)) with det(A+ikB)=0
− M − ∈ \ − M 6
is the integral operator with the ( + ) ( + ) matrix-valued integral kernel
r (x,y;k,a),Imk>0,admitting|Ith|ere|Epr|es×enta|Iti|on |E|
M
r (x,y;k,a)=r(0)(x,y,k)
M
(3.5) i
+ Φ(x,k)R(k;a)−1[I S(k; )T(k;a)]−1S(k; )R(k;a)−1Φ(y,k)T,
2k − M M
whereR(k;a)isdefinedin(3.2),thematrixΦ(x,k)isgivenby
φ(x,k) 0 0
Φ(x,k):=
0 φ (x,k) φ (x,k)
+ −
(cid:18) (cid:19)
withdiagonalmatricesφ(x,k)=diag eikxj j∈E,φ±(x,k)=diag e±ikxj j∈I,and
{ } { }
eik|xj−yj|
[r(0)(x,y,k)] =iδ , x ,y I .
j,j′ j,j′ 2k j j ∈ j
10 V.KOSTRYKIN,J.POTTHOFF,ANDR.SCHRADER
If =˘ ,thisrepresentationsimplifiesto
I
i
r (x,y,k)=r(0)(x,y,k)+ φ(x,k)S(k; )φ(y,k).
M 2k M
Theintegralkernelr (x,y;k,a)iscalledGreen’sfunctionorGreen’smatrix.
M
TheproofofLemma3.2isgivenin[28].
3.2. Walks on Graphs and Cycles. We recall the following definitions from [28]. A
nontrivialwalkw onthe graph fromj to j′ is anorderedsequence
G ∈ E ∪I ∈ E ∪I
formedoutofedgesandvertices
(3.6) j,v ,j ,v ,...,j ,v ,j′
0 1 1 n n
{ }
suchthat
(i) j ,...,j ;
1 n
∈I
(ii) theverticesv V andv V satisfyv ∂(j),v ∂(j ), v ∂(j′), and
0 n 0 0 1 n
∈ ∈ ∈ ∈ ∈
v ∂(j );
n n
∈
(iii) for any k 1,...,n 1 the vertex v V satisfies v ∂(j ) and v
k k k k
∈ { − } ∈ ∈ ∈
∂(j );
k+1
(iv) v =v forsomek 0,...,n 1 ifandonlyifj isatadpole.
k k+1 k
∈{ − }
Ifj,j′ thisdefinitionisequivalenttothatgivenin[29].
The∈nuEmbernisthecombinatoriallength w andthenumber
comb
| |
n
w = a >0
| | jk
k=1
X
isthemetriclengthofthewalkw.
Atrivialwalkonthegraph fromj toj′ isatriple j,v,j′ such
G ∈ E ∪I ∈ E ∪I { }
that v ∂(j) and v ∂(j′). Otherwise the walk is called nontrivial. In particular, if
∈ ∈
∂(j) = v ,v , then j,v ,j and j,v ,j are trivial walks, whereas j,v ,j,v ,j
0 1 0 1 0 1
{ } { } { } { }
and j,v ,j,v ,j arenontrivialwalksofcombinatoriallength1andofmetriclengtha .
1 0 j
{ }
Boththecombinatorialandmetriclengthofatrivialwalkarezero.
We will say thatthe walk (3.6) leavesthe edgej throughthe vertexv andentersthe
0
edgej′ throughthevertexv . Atrivialwalk j,v,j′ leavesj andentersj′ throughthe
n
{ }
samevertexv.
Foranygivenwalkw fromj toj′ we denotebyv (w)thevertex
−
throughwhichthewalkleavesthe∈edEge∪jIandby∈v E(w∪)Ithevertexthroughwhichthewalk
+
enterstheedgej′. Fortrivialwalksonehasv (w)=v (w).
− +
Assumethattheedgesj,j′ arenottadpoles.Forawalkwfromj toj′weset
∈E ∪I
x if v (w)=∂−(j),
dist(x ,v (w)):= j −
j − (aj xj if v−(w)=∂+(j),
−
and
x if v (w)=∂−(j),
dist(x ,v (w)):= j′ +
j′ + (aj xj if v+(w)=∂+(j).
−
Awalkw = j,v ,j ,v ,...,j ,v ,j′ traversesaninternaledgei ifj =ifor
0 1 1 n n k
some1 k n{. Itvisitsthevertexv ifv} = v forsome0 k n.∈ThIescoren(w)
k
≤ ≤ ≤ ≤