Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups

0.27 MB·English

by  Vadim Kostrykin

ProceedingsoftheConferenceon
QuantumGraphsandTheirApplications
Snowbird,Utah,
June18–June24,2005
6
0 LAPLACIANS ON METRIC GRAPHS: EIGENVALUES, RESOLVENTS AND
0
SEMIGROUPS
2

n
a VADIMKOSTRYKINANDROBERTSCHRADER
J

0
2 ABSTRACT. Themainobjectiveofthepresentworkistostudythenegativespectrumof
(differential)Laplaceoperatorsonmetricgraphsaswellastheirresolventsandassociated
1 heatsemigroups. Weproveanupperboundonthenumberofnegativeeigenvaluesanda
v lowerboundonthespectrumofLaplaceoperators.Alsoweprovideasufficientcondition
1 fortheassociatedheatsemigrouptobepositivitypreserving.
4
0
1
0
6 1. INTRODUCTION
0
Assuggestedbythetitle,themainobjectiveofthepresentworkistostudythenegative
/
h spectrum of (differential) Laplace operators on metric graphs as well as their resolvents
p
andassociatedheatsemigroups. Basic notionsrelatedtotheseoperatorsaresummarized
-
h inSection2below.Morecompleteaccountscanbefoundin[10],[12],[13].
t It is well known (see, e.g., [7], [19]) that there are deep interrelations between pro-
a
m pertiesofheatsemigroupsandspectralpropertiesoftheir generators. More importantly,
heatsemigroupsgeneratedbyLaplace-BeltramioperatorsonRiemannianmanifoldscarry
:
v alargeamountofinformationonthegeometryoftheunderlyingmanifolds.Asformetric
i
X graphssomeresultsinthisdirectioncanbefound,e.g.,in[22],[23]. Nevertheless,asys-
tematicanalysisofheatsemigroupsforLaplaceoperatorsonmetricgraphsisstillmissing.
r
a Inthisworkweperformafirststepinclosingthisgap.
Belowwewillproveanupperboundonthenumberofnegativeeigenvalues(Theorem
3.7)andalowerboundonthespectrumofLaplaceoperators(Theorem3.10).Inparticular,
theupperboundprovidesaverysimplesufficientconditionfortheLaplaceoperatortobe
nonnegative.ThelowerboundimprovesarecentresultbyKuchment[13].
Concerningresolventsandheatkernelswewillestablishsufficientconditionsforthem
tobepositivitypreserving.ToachievethiswewillprovideaclosedexpressionforGreen’s
functionintermsoftheboundaryconditionsdefiningthecorrespondingLaplaceoperator.
As a consequence for a class of boundary conditions associated with positive maximal
isotropic subspaces (Definition 4.4 below) we prove positivity of Green’s function (see
Theorems 4.6 and 5.1). The proof of Theorem 5.1 utilizes walks on a graph, a concept
introduced in our paper [12] as a main technical tool for solving the inverse scattering
2000MathematicsSubjectClassification. 34B45,34L15,47D03.
Keywordsandphrases. Differentialoperatorsonmetricgraphs,eigenvalueproblems,positivitypreserving
semigroups.
1
2 V.KOSTRYKINANDR.SCHRADER
problemonmetricgraphs. Bystandardargumentstheseresultsimplythattheassociated
heatsemigroupispositivitypreserving.
ForearlierworkonheatsemigroupsgeneratedbyLaplaceoperatorsonmetricgraphs
andtheirapplicationtospectralanalysiswereferto[2],[4],[8],[9],[17],[18],[22],[23].
Green’sfunctionshavebeenstudiedin[2],[3],[10],[20].
Thereisawell-knownapproachtoprovethesimplicityofthelowesteigenvaluebased
on the analysis of heat semigroups (see, e.g., [21]). Although we do not pursue this in
detail,merelyasanillustration,foraclassofboundaryconditionsweprovesimplicityof
the lowest eigenvalue of Laplace operatorson metric graphs without internal edges (see
Proposition6.4below).
InSection7weconsiderLaplaceoperatorsonmetricgraphswithnointernaledges,for
whichtheheatkernelcanbecomputedexplicitly.
Acknowledgements. The authors are indebted to S. Fulling, H. Hofer, M. Karowski,
W. Kirsch, M. Loss, and M. Taylor for useful comments. We thank the anonymousre-
ferees for careful reading of the manuscript and helpful suggestions. The participation
of the first author (V.K.) at the conference “Quantum Graphs and Their Applications”
held in the summer 2005 in Snowbird, Utah, USA has been supported by the Deutsche
ForschungsgemeinschaftandbytheNationalScienceFoundation.
2. BASIC STRUCTURES
In this section we revisit the theory of Laplace operators on a metric graph . The
G
materialpresentedhereisborrowedfromourprecedingpapers[10]and[12].
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
Throughoutthewholeworkwe willassumethatthegraph isconnected,thatis, for
G
anyv,v′ V thereisanorderedsequence v = v,v ,...,v = v′ suchthatanytwo
1 2 n
∈ { }
successiveverticesinthissequenceareadjacent. Inparticular,thisimpliesthatanyvertex
ofthegraph hasnonzerodegree,i.e.,foranyvertexthereisatleastoneedgewithwhich
G
itisincident.
Wewillendowthegraphwiththefollowingmetricstructure. Anyinternaledgei
∈ I
will be associated with an interval [0,a ] with a > 0 such that the initial vertex of i
i i
correspondsto x = 0 and the terminal one - to x = a . Any externaledge 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
calledametricgraph( ,a).
G
Givenafinitegraph = (V, , ,∂)withametricstructurea = a considerthe
i i∈I
G I E { }
Hilbertspace
(2.1) ( , ,a)= , = , = ,
E I E e I i
H≡H E I H ⊕H H H H H
e∈E i∈I
M M
LAPLACIANSONMETRICGRAPHS 3
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
Let∆0bethedifferentialoperator
d2
(2.2) ∆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
(2.3) ( , )= (−) (+)
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
(2.4) [ψ]:=ψ ψ′ d ,
⊕ ∈ K
withψandψ′ definedby
ψ (0) ψ′(0)
(2.5) 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
(2.6) J =
I 0
(cid:18)− (cid:19)
withIbeingtheidentityoperatoron . Considerthenon-degenerateHermitiansymplectic
K
form
(2.7) ω([φ],[ψ]):= [φ],J[ψ] ,
h i
where , denotesthescalarproductind =C2(|E|+2|I|).
h· ·i K∼
Alinearsubspace ofd iscalledisotropiciftheformωvanisheson identically.
M K M
Anisotropicsubspaceiscalledmaximalifitisnotapropersubspaceofalargerisotropic
subspace.Everymaximalisotropicsubspacehascomplexdimensionequalto +2 .
|E| |I|
4 V.KOSTRYKINANDR.SCHRADER
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
(2.8) (A,B):=Ker(A,B).
M
Theorem2.1. Asubspace d ismaximalisotropicifandonlyifthereexistlinear
M ⊂ K
mapsA, B : suchthat = (A,B)and
K→K M M
(i) themap(A,B): d hasmaximalrankequalto +2 ,
(2.9) K→K |E| |I|
(ii) AB† isself-adjoint,AB† =BA†.
Definition2.2. The boundaryconditions(A,B) and (A′,B′)satisfying (2.9) are equiv-
alent if the corresponding maximal isotropic subspaces coincide, that is, (A,B) =
M
(A′,B′).
M
Theboundaryconditions(A,B)and(A′,B′)satisfying(2.9)areequivalentifandonly
ifthereisaninvertiblemapC : suchthatA′ = CAandB′ = CB (seeProposi-
K → K
tion3.6in[12]).
ByLemma3.3in[12],asubspace (A,B) d ismaximalisotropicifandonlyif
M ⊂ K
(2.10) (A,B)⊥ = (B, A).
M M −
Wementionalsotheequalities
(A,B)⊥ = Ker(A,B) ⊥ =Ran(A,B)†,
(2.11) M
(A,B)=(cid:2)Ran( B,A)(cid:3)†.
M −
In the terminology of symplectic geometry (see, e.g., Section 2.3 in [16]) the equalities
(2.11)havethefollowinginterpretation:Thematrix(A,B)†isa(Lagrangian)frameforthe
maximalisotropicsubspace (A,B)⊥andthematrix( B,A)†isaframefor (A,B).
M − M
Thereisanalternativeparametrizationofmaximalisotropicsubspacesofd byunitary
K
transformationsin (see[11]andProposition3.6in[12]). 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.
Therefore,Mwe⊂willKusethenotationS(k; )forS(k;∈A,B|)Ew|ith|I|(A,BK)= .
M M M
There is a one-to-one correspondence between all self-adjoint extensions of ∆0 and
maximal isotropic subspaces of d (see [10], [12]). In explicit terms, any self-adjoint
K
extensionof∆0isthedifferentialoperatordefinedby(2.2)withdomain
(2.13) Dom(∆)= ψ [ψ] ,
{ ∈D| ∈M}
where is a maximal isotropic subspace of d . Conversely, any maximal isotropic
subspacMe of d defines through (2.13) 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
LAPLACIANSONMETRICGRAPHS 5
FromTheorem2.1itfollowsthatthedomainoftheLaplaceoperator∆( ,a)consists
M
offunctionsψ satisfyingtheboundaryconditions
∈D
(2.14) Aψ+Bψ′ =0,
with(A,B)subjectto(2.8)and(2.9). Hereψandψ′aredefinedby(2.5).
Withrespecttotheorthogonaldecomposition = (−) (+) anyelementχ
K KE ⊕KI ⊕KI
of canberepresentedasavector
K
χ
e e∈E
{ }
(2.15) χ= χ(−) .
{ i }i∈I
χ(+)
{ i }i∈I
Considertheorthogonaldecomposition 
(2.16) =
v
K L
v∈V
M
with the linearsubspaceofdimensiondeg(v)spannedbythoseelements(2.15)of
v
L K
whichsatisfy
χ =0 if e isnotincidentwiththevertex v,
e
∈E
(2.17) χ(−) =0 if v isnotaninitialvertexof i ,
i ∈I
χ(+) =0 if v isnotaterminalvertexof i .
i ∈I
Obviously,thesubspaces and areorthogonalifv =v .
SetdLv := Lv ⊕Lv ∼=LvC12deg(Lv)v.2Obviously,eachdL1v6inhe2ritsasymplecticstructure
fromd inacanonicalway,suchthattheorthogonaldecomposition
K
d =d
v
L K
v∈V
M
holds.
Definition2.3. Given thegraph = (V, , ,∂), boundaryconditions(A,B) satisfy-
G G I E
ing (2.9) are called local on if the maximalisotropic subspace (A,B) of hasan
G M K
orthogonalsymplecticdecomposition
(2.18) (A,B)= (v),
M M
v∈V
M
with (v)beingmaximalisotropicsubspacesofd .
v
M L
Otherwisetheboundaryconditionsarecallednon-local.
ByProposition4.2in[12],giventhegraph = (V, , ,∂),theboundaryconditions
G G I E
(A,B)satisfying(2.9)arelocalon ifandonlyifthereisaninvertiblemapC :
G K →K
and linear transformations A(v) and B(v) in such that the simultaneous orthogonal
v
L
decompositions
(2.19) CA= A(v) and CB = B(v)
v∈V v∈V
M M
arevalid.
Given a graph = (V, , ,∂) to any vertex v V we associate the graph =
v
G I E ∈ G
( v , , ,∂ )withthefollowingproperties
v v v
{ } I E
(i) =˘ ,
v
I
(ii) ∂ (e)=vforalle ,
v v
∈E
6 V.KOSTRYKINANDR.SCHRADER
(iii) =deg (v),thedegreeofthevertexvinthegraph ,
|Ev| G G
(iv) thereisaninjectivemapΨ : suchthatv ∂ Ψ (e)foralle .
v v v v
E →E∪I ∈ ◦ ∈E
Boundary conditions (A(v),B(v)) on each of the graphs induce local boundary
v
G
conditions(A,B)onthegraph with
G
A= A(v) and B = B(v).
v∈V v∈V
M M
Example2.4(Standardboundaryconditions). Givenagraph with V 2andminimum
G | |≥
degree not less than two, define the boundary conditions (A(v),B(v)) on for every
v
G
v V bythedeg(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
   
   
Obviously,A(v)B(v)† =0and(A(v),B(v))hasmaximalrank.Thus,foreveryv V the
∈
boundary conditions (A(v),B(v)) define self-adjoint Laplace operators ∆(A(v),B(v))
onL2( ). Thecorrespondingunitarymatrices(2.12)aregivenby
v
G
2
(2.20) [S(k;A(v),B(v))] = δ
e,e′ e,e′
deg(v) −
withδ Kroneckersymbol.
e,e′
Theboundaryconditions(A(v),B(v))inducestandardlocalboundaryconditions(A,B)
onthegraph .
G
ThefollowingresultisCorollary5in[13].
Lemma2.5. Boundaryconditions(A,B)satisfying(2.9)areequivalenttotheboundary
conditions(A,B)with
(2.21) A=P +L, B =P⊥ ,
b b Ker B Ker B
whereP isthe orthogonalprojectionin ontoKer B, P⊥ := I P its
Ker B b K b Ker B − Ker B
complementaryprojection,andL: theself-adjointoperatorgivenby
K→K
L=(B )−1AP⊥ .
|RanB† (KerB)
NotethatAB† =LandKerL KerB.
⊃ L+ik
Inparticular,ifKerB = 0 suchthatL = B−1A,thenS(k;A,B) = (cf.
bb { } −L ik
Proposition3.19in[12]). −
Corollary 2.6. Assume that det(A κB) = 0 for some κ > 0. Then S(iκ;A,B) is
self-adjoint.IfL 0,thenS(iκ;A,−B)isac6 ontractionforallκ >0.
≤
Proof. Theassumptiondet(A κB)=0combinedwiththefactthatthesubspaceKerB
reducestheoperatorLimplies−thatL 6 κisinvertible.ByLemma2.5wehave
−
S(iκ;A,B)=S(iκ;A,B)= P P⊥ (L κ)−1(L+κ)P⊥
(2.22) − Ker B − Ker B − Ker B
= 2P (L κ)−1(L+κ),
Ker B
b b − − −
LAPLACIANSONMETRICGRAPHS 7
whichisself-adjoint.From(2.22)itfollowsthat
S(iκ;A,B) =max 1, (L κ)−1(L+κ) .
k k { − }
IfL 0,thenbythespectraltheorem (L κ(cid:13))−1(L+κ) 1f(cid:13)orallκ >0. (cid:3)
≤ − (cid:13) ≤ (cid:13)
3. EIGENVALUE(cid:13)(cid:13)SOF LAPLACE OPER(cid:13)(cid:13)ATORS
Ifψ = ψ isaneigenfunctionoftheoperator ∆(A,B,a)corresponding
j j∈I∪E
totheeige{nval}uek2 ∈RH 0 ,Imk 0,thenitisnecessarily−oftheform
∈ \{ } ≥
(3.1) ψ (x;k)= sjeikxj forj ∈E,
j (αjeikxj +βje−ikxj forj .
∈I
Thevectorss = s ,α = α (−),andβ = β (+) satisfy
{ e}e∈E ∈ KE { i}i∈I ∈ KI { i}i∈I ∈ KI
thehomogeneousequation
s
(3.2) Z(k;A,B,a) α =0
 
β
with  
(3.3) Z(k;A,B,a)=AX(k;a)+ikBY(k;a),
where
I 0 0 I 0 0
(3.4) X(k;a)= 0 I I and Y(k;a)= 0 I I .
0 eika e−ika 0 eika e−−ika
−
Thediagonal  matricese±ikaaregivenby  
|I|×|I|
(3.5) [e±ika]jk =δjke±ikaj for j,k .
∈ I
Theconversestatementisalsotrueandwehavethefollowingresult.
Lemma 3.1. A ψ given by (3.1) is an eigenfunctionof ∆( ,a) correspondingto the
eigenvaluek2 R 0 ifandonlyif (3.2)possessesano−ntrivMialsolution.Themultiplicity
ofthiseigenva∈luei\s{eq}ualtodimKerZ(k;A,B,a).
Proof. If k2 < 0, the claim is obvious, since ψ Dom(∆( ,a)). If k2 > 0, any
∈ M
solutionof (3.2) satisfies s = 0 (see the proofof Theorem3.1 in [10]). Therefore, ψ
∈
Dom(∆( ,a)). (cid:3)
M
Asprovenin[10,Theorem3.1]thesetofzerosofdetZ(k;A,B,a)isdiscrete.
Denote
0 0 0
(3.6) T(k;a):= 0 0 eika
0 eika 0 
withrespecttotheorthogonaldecomposition(2.3). 
Theorem3.2. Assumethat = ˘ . Thenλ = k2 = 0suchthatdet(A+ikB) = 0isan
I 6 6 6
eigenvalueof ∆( (A,B),a)withmultiplicitymifandonlyif1isaneigenvalueof
− M
S(k;A,B)T(k;a)
withgeometricmultiplicitym.
If =˘ ,thenλ= κ2 <0isaneigenvalueof ∆( (A,B))withmultiplicitymif
andoInlyif0isaneigen−valueofA κB withgeome−tricMmultiplicitym.
−
8 V.KOSTRYKINANDR.SCHRADER
Remark3.3. In[10]itisshownthatdet(A+ikB) = 0anddet(A ikB) = 0holdfor
allk>0. 6 − 6
Proof. Observethatifdet(A+ikB)=0,then
6
AX(k;a)+ikBY(k;a)=(A+ikB)R (k;a)+(A ikB)R (k;a)
+ −
−
(3.7) =(A+ikB) I+(A+ikB)−1(A ikB)T(k;a) R (k;a)
+
−
=(A+ikB)(cid:2)I S(k;A,B)T(k;a) R+(k;a), (cid:3)
−
where (cid:2) (cid:3)
I 0 0
1
(3.8) R (k;a):= [X(k;a)+Y(k;a)]= 0 I 0 ,
+
2 0 0 e−ika
 
0 0 0
1
R (k;a):= [X(k;a) Y(k;a)]= 0 0 I .
−
2 − 0 eika 0
 
s
Hence,byLemma3.1 α isanontrivialsolutionto(3.2)ifandonlyif
 ∈K
β
 
s
(3.9) I S(k;A,B)T(k;a) α =0.
− e−ikaβ
(cid:2) (cid:3)
If =˘ ,thenequation(3.2)simplifiesto(A+ikB)s=0withs . (cid:3)
E
I ∈K
3.1. Positive Laplacians. Let d be a maximal isotropic subspace. Let (A,B)
M ⊂ K
be arbitraryboundaryconditionssatisfying (A,B) = . Let n (AB†) (n (AB†),
+ −
M M
n (AB†),respectively)bethenumberofpositive(negative,zero,respectively)eigenvalues
0
ofAB†. BySylvester’sInertiaLaw
n (CAB†C†)=n (AB†), n (CAB†C†)=n (AB†)
± ± 0 0
foranyinvertibleC : . SinceforanysuchC
K→K
(CA,CB)= (A,B),
M M
wemaydefine
n ( ):=n (AB†), n ( ):=n (AB†)
± ± 0 0
M M
forarbitraryboundaryconditions(A,B)satisfying (A,B)= .
M M
Wementionseveralpropertiesofnumbersn ( )andn ( ). First,by(2.10),
± 0
M M
(3.10) n ( ⊥)=n ( ), n ( ⊥)=n ( ).
± ∓ 0 0
M M M M
Second,foranyunitarytransformationU in onehas
K
(3.11) n (dU )=n ( ), n (dU )=n ( ),
± ± 0 0
M M M M
wheredU denotesthe“double”ofU,
U 0
(3.12) dU = .
0 U
(cid:18) (cid:19)
Finally,wehavethefollowingobviousinequality
(3.13) n ( (A,B)) max dimKerA,dimKerB .
0
M ≥ { }
LAPLACIANSONMETRICGRAPHS 9
The numbers n ( ) and n ( ) admit the following equivalent characterization in
± 0
termsofthematricesMS(k; ). M
M
Proposition 3.4. Let d be maximal isotropic. For any k > 0 the number of
eigenvaluesofS(k; M))ly⊂ingKintheopenupper(lower,respectively)complexhalf-plane
isequalton ( )(Mn ( ),respectively).ThenumberofrealeigenvaluesofS(k; )is
+ −
M M M
equalton ( ).
0
M
Proof. Consider
1 1
AS =−2(S−I), BS = 2ik(S+I)
withS=S(k; ). NotethatS(k; )=S(k;AS,BS). Observingthat
M M
1 1
(3.14) ASBS† = 2kIm S≡ 4ik(S−S†),
weobtainthat
n (ImS(k; ))=n ( ), n (ImS(k; ))=n ( )
± ± 0 0
M M M M
foranyk>0. (cid:3)
Let d be a maximalisotropic subspace. For arbitraryϕ,ψ Dom∆( ,a)
M ⊂ K ∈ M
integratingbypartsweobtain
(3.15) ϕ, ∆( ,a)ψ = ϕ′,ψ′ + [ϕ],Q[ψ] ,
h − M iH h iHj h idK
j∈E∪I
X
0 I
whereQ= withrespecttotheorthogonaldecompositiond = .
0 0 K K⊕K
(cid:18) (cid:19)
Using(2.11)itisanelementaryexercisetocheckthattheorthogonalprojectionind
K
ontothesubspace isgivenby
M
B†
PM = −A† (AA†+BB†)−1(−B,A)
(3.16) (cid:18) (cid:19)
B†(AA†+BB†)−1B B†(AA†+BB†)−1A
− ,
≡ A†(AA†+BB†)−1B A†(AA†+BB†)−1A
(cid:18)− (cid:19)
wheretheblockmatrixnotationisusedwithrespecttotheorthogonaldecompositiond =
K
. Since[ϕ]and[ψ]in(3.15)belongto ,weobtain
K⊕K M
[ϕ],Q[ψ] = [ϕ],Q [ψ] ,
h idK h M idK
whereQ :=P QP . Using(3.16)wegetthat
M M M
B†
(3.17) QM =− −A† (AA†+BB†)−1AB†(AA†+BB†)−1(−B,A).
(cid:18) (cid:19)
Thus,wehaveproventhefollowingresult.
Proposition3.5. Foranymaximalisotropicsubspace d
M⊂ K
(3.18) ϕ, ∆( ,a)ψ = ϕ′,ψ′ + [ϕ],Q [ψ] ,
h − M iH h iHj h M idK
j∈E∪I
X
holdsforallϕ,ψ Dom∆( ,a).
∈ M
10 V.KOSTRYKINANDR.SCHRADER
Remark3.6. ObservethatQ dependsonthemaximalisotropicsubspace onlyand
M
M
notonthe particularchoiceofthematricesAandB parametrizing . Thus, usingthe
M
parametrizationofthemaximalisotropicsubspace bymatrices(A,B)definedin(2.21)
M
weobtain
P⊥ b b
(3.19) QM =− P−KerKBer+BL L(I+L2)−2 −PK⊥erB, PKerB +L ,
(cid:18) (cid:19)
wherewehaveusedtheequality (cid:0) (cid:1)
(3.20) AB† =L.
SinceKerL KerB,theequalityAψ+Bψ′ =0impliesthatL2ψ+Lψ′ =0.Therefore,
⊃ bb
foranyψ Dom(∆( ,a))weobtain
∈ M
P⊥ , P +Lb [ψ]b P⊥ ψ+(P +L)ψ′
− KerB KerB ≡− KerB KerB
(cid:0) (cid:1) =−(I+L2)PK⊥erBψ+PKerBψ′.
Hence,
L(I+L2)−2 P⊥ , P +L [ψ]= (I+L2)−1Lψ.
− KerB KerB −
Therefore,using(3.19)weobtainthat
(cid:0) (cid:1)
[ϕ],Q [ψ] = ϕ,Q ψ ,
h M idK h M iK
holdsforallϕ,ψ Dom(∆( ,a)),where
∈ M e
L 0
QM = −0 0
(cid:18) (cid:19)
with the block-matrix notationwitherespect to the orthogonaldecompositiond =
K K⊕
. Together with (3.18) this leads to the representation for the sesquilinear form of the
K
operator ∆( ,a)obtainedpreviouslybyKuchmentin[13,Theorem9].
− M
From (3.17) it follows that n (Q ) = n ( ). Thus, by Sylvester’s Inertia Law,
± M ∓
M
Proposition3.5immediatelyimpliesthat ∆( ,a) 0inthesenseofquadraticforms,
− M ≥
ifthemaximalisotropicsubspace d satisfiestheconditionn ( )=0.
+
M⊂ K M
Usingvariationalarguments,weproveamoregeneralresult:
Theorem3.7. Thenumberofnegativeeigenvaluesof ∆( ,a)countingtheirmultiplic-
itiesisnotbiggerthann ( ). If = ˘ , then ∆(− )Mhaspreciselyn ( )negative
+ +
M I − M M
eigenvalues.
Fortheproofweneedthefollowingsimplelemma.
Lemma 3.8. Assume that (A,B) satisfies (2.9). The function f(κ) := det(A κB)
−
haspreciselyn ( (A,B))positivezeroesandn ( (A,B))negativezeroes(counting
+ −
multiplicity).IfκM=0isazerooff,thenitsmultipliciMtyequalsRankB n ( (A,B))
+
− M −
n ( (A,B)) n ( ).
− 0
M ≤ M
Proof. ByLemma2.5theboundaryconditions(A,B)areequivalenttotheboundarycon-
ditions(A,B)definedin(2.21). Nowassumethataκ >0isazerooff withmultiplicity
m 1. By(3.20),wehave
≥
(3.21) b b det(A κB)=cdet(A κB)=cdet(L κ)
− − −
with a nonzero constantc C. Thus, κ is an eigenvalueof L with multiplicity m. By
(3.20)itisalsoaneigenvalu∈eofAB†. b b
Thecaseofnegativeeigenvaluescanbeconsideredinexactlythesameway.