Sign-indefinite second order differential operators on finite metric graphs

0.48 MB·English

by  Amru Hussein

SIGN–INDEFINITE SECOND ORDER DIFFERENTIAL
OPERATORS ON FINITE METRIC GRAPHS.
AMRUHUSSEIN
Abstract. Thequestionofself–adjointrealizationsofsign–indefinitesecond
orderdifferentialoperatorsisdiscussedintermsofamodelproblem. Operators
3 of the type − d sgn(x) d are generalized to finite, not necessarily compact,
dx dx
1 metric graphs. All self–adjoint realizations are parametrized using methods
0 from extension theory. Thespectral and scattering theory of the self–adjoint
2 realizations arestudiedindetail.

n
a
J

6
1. Introduction
1

Differential expressions of the form
]
h
p +1, x Ω+,
(1) divA()grad with A(x)= ∈
h- − · (−1, x∈Ω−,
t
a for domainsΩ=Ω Ω Rd appearin differentcontexts. The Poissonproblem
+ −
m ∪ ⊂
related to this operator is of physical relevance. It appears in the mathematical
[ description of light propagation through regions with thresholds of materials with

negative and positive refraction indices, see [5] and the references therein. The
2
v quasi–static limit of the Maxwell equation yields the mentioned Poisson problem.
4 Another field of application is solid-state physics. In the effective mass approxi-
4 mation the effective mass tensor can be negative as well, see for example [1] and
1
the references therein. This yields indefinite differential operators, which involve
4
expressions of the form (1) and in this context the Schrödinger equation is consid-
.
1
ered.
1
An approachthat has turned out to be very fruitful for the study of differential
2
1 equations which involve the expression divA()grad with strictly positive coeffi-
− ·
: cient matrices A(), is to consider self–adjoint operators associated with these dif-
v ·
ferentialexpressionandtheirspectralresolution. Thequestionis,ifitispossibleto
i
X find self–adjoint realizations also for sign–indefinite coefficients A(). Unbounded
·
r operators with sign–changing coefficients in the highest order terms are not well
a
studied. To make a starting point a new model problem is introduced: Laplace
operators on finite metric graphs with changing sign. The aim of this work is to
present generalizations of the operator
d d +1, x 0,
(2) sgn(x) with sgn(x)= ≥
− dx dx ( 1, x<0
−
to finite metric graphs. This initial problem can be tackled using extension the-
ory. This approach allows straight forward generalizations from intervals to finite
2010 Mathematics Subject Classification. Primary 34B45, Secondary 47B25, 34L05, 35P20,
35P25,81U15.
Key words and phrases. Indefinite differential operators on metric graphs, self–adjointness,
spectral theory, scattering theory.
1
2 AMRUHUSSEIN
metric graphs. It yields a formal operator which admits the block operator matrix
representation
∆ 0
τ = − ,
0 +∆
(cid:20) (cid:21)
where ∆ denotes the one dimensional Laplace operator, and all self–adjoint real-
−
isations are completely characterized. On finite metric graphs this gives a model,
which on the one hand is still explicitly solvable, but on the other hand allows to
describe more complicated geometries than the one dimensional case does, which
deals only with intervals. A similar procedure has been performed already very
successfully for the sign–definite one dimensional Laplacian. This has become fa-
mousunderthename“quantumgraphs”. Forfurtherreferencesonthistopicseefor
example[3]andthereferencesgiventherein. Itturnsoutthatthereisaone–to–one
correspondencebetweentheself–adjointrealizationsoftheLaplacianongraphsand
the self–adjoint realizations for the model–operator discussed here.
The aim is to construct self–adjoint realizations of expressions of the form (1)
and (2) in terms of extension theory and to develop a better understanding of the
spectral and scattering properties of sign–indefinite differential operators. Some
aspectsoftheproblemareincludedinthegeneralsettingofSturm-Liouvilletheory,
but they have been considered to have no practical relevance, as remarked in [29,
Section 17.E].
The motivation to consider expressions of the form (1) in the present paper
is caused mainly by the fast development in the field of so called metamateri-
als – materials with negative optical refraction index. This has been initiated by
V. G. Veselago, see [28], who was the first to consider – hypothetically – the effect
of a negative refraction index. He had predicted that this feature would yield new
effects including for example backward waves, an inverse law of refraction and an
inverse Doppler effect. Today metamaterials can be constructed as artificial ma-
terials, whose optical properties are made useful for applications. An overview of
recent developments is given in [27], recall also the article [5] and the references
given therein.
Note that the model problem (2) is part of a range of problems related to the
ordinary differential expressions
d2 d d
, sgn(x) ,
−dx2 −dx dx
d2 d d
sgn(x) and sgn(x) sgn(x) .
− dx2 − dx dx
Each is exemplary for certain difficulties and methods used. Usually the operators
written in the first line are considered in Hilbert spaces, whereas the operators in
thesecondlineareconsideredinKreinspaces,seeforexample[6,32]. Theoperator
sgn(x) d2 hasapplicationsintheeffectivemassapproximationtoo,compare[34].
− dx2
Thisnoteis organizedasfollows: firstthe basicconceptsareintroducedandthe
notation is fixed. The subsequent section is devoted to the characterization of all
self–adjoint boundary conditions. In short the corresponding question for Krein–
space–self–adjointness is discussed as well. Section 5 puts the question of self-
adjoint extensions into the context of indefinite quadratic forms, and in Section 6
theoperatorsdiscussedareputintothegeneralcontextofextensiontheory. Section
7givesexplicitformulaeforeigenvalues,resonancesandresolvents. InSection8the
scatteringpropertiesofthesystemarediscussed. Thewaveoperatorsarecomputed
aswellas the scatteringmatrix. The scatteringmatrixcanbe computedincertain
cases in terms of a generalized star product.
SIGN–INDEFINITE OPERATORS ON METRIC GRAPHS 3
The content of this note has developed in parallel to the work on the article
[16] which is in preparation, and it is part of the author’s PhD thesis, see [15,
Chapter 4]. Indefinite operators on graphs provide an illustration and can serve
as a source of examples for indefinite operators of the form (1) on manifolds and
intervals involving more general coefficients A().
·
Acknowledgements. IwouldliketothankDavidKrejčiříkforcommunicatingthe
problem of sign–indefinite differential operators, its physical background, helpful
discussionsaswellasremarksandespeciallyforkindhospitalityduringmyvisitto
the Doppler Institute in Prague in February 2012. I am grateful as well to Vadim
KostrykinandtoStephanSchmitzfromwhomIbenefitedinmanywaysduringthe
still ongoing work on the article [16].
2. Basic concepts
Instead of considering the operator given in (2) in the Hilbert space L2(R,dx),
one considers in the equivalent space
L2([0, ),dx ) L2([0, ),dx ) (=L2(R,dx))
∞ 1 ⊕ ∞ 2 ∼
the formal differential operator
τu=(+u′′, u′′).
1 − 2
This allows to define a closed symmetric operator with equal deficiency indices
(2,2). Hence there exist self-adjoint extensions. The operator τ is generalized
from intervals to finite metric graphs which delivers indefinite symmetric second
order differential operators with finite deficiency indices. In the following the ba-
sic concepts are presented. The notation is mainly borrowed from the works of
V.KostrykinandR.SchraderonLaplaciansonmetricgraphs,seeforexample[19].
2.1. Finite metric graphs. Here and in the following a graph is a 4-tuple
=(V, , ,∂),
G I E
where V denotes the set of vertices, the set of internal edges and the set
I E
of external edges, where the set is summed up in the notion edges. The
E∪I
boundary map ∂ assigns to each internal edge i an ordered pair of vertices
∈ I
∂(i) = (v ,v ) V V, where v is called its initial vertex and v its terminal
1 2 1 2
∈ ×
vertex. Each external edge e is mapped by ∂ onto a single, its initial, vertex.
∈ E
A graph is called finite if V + + < and a finite graph is called compact
| | |I| |E| ∞
if = holds.
E ∅
The graph is endowed with the following metric structure. Each internal
G
edge i is associated with an interval [0,a ] with a > 0, such that its initial
i i
∈ I
vertex corresponds to 0 and its terminal vertex to a . Each external edge e is
i
∈ E
associated to the half line [0, ), such that ∂(e) corresponds to 0. The numbers
∞
a are called lengths of the internal edges i and they are summed up into the
i
∈ I
vector a= a R|I|. The 2-tuple ( ,a) consisting of a finite graph endowed
{ i}i∈I ∈ + G
with a metric structure defines a metric space called metric graph. The metric on
( ,a) is defined via minimal path lengths on it.
G
One distinguishes two types of edges
= ˙ and = ˙ ,
+ − + −
E E ∪E I I ∪I
where ( ) and ( ) are called the positive (negative) external edges and
+ − + −
E E I I
positive (negative) internal edges, respectively. The set ( ) denotes
+ + − −
E ∪I E ∪I
the positive (negative) edges. This partition defines two sub–graphs
, , where =(V , , ,∂ )
G− G+ ⊂G G± ± I± E± |(E±∪I±)
4 AMRUHUSSEIN
with V = ∂( ). Let be a = a , then ( ,a ) are metric graphs.
± E±∪I± ± { i}i∈I± G± ±
For brevity set
n=2 + and m=2 + .
+ + − −
|I | |E | |I | |E |
2.2. Functionspaces. Givenafinitemetricgraph( ,a)oneconsiderstheHilbert
G
space
( , ,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
Again one has a decomposition into two subspaces
( , ,a)= ( , ,a ) ( , ,a ).
H E I H E+ I+ + ⊕H E− I− −
Any function ψ: ( ,a) C can be written as
G →
ψ(x )=ψ (x), where ψ : I C
j j j j
→
with
[0,a ], if j ,
j
I = ∈I
j
([0, ), if j .
∞ ∈E
One defines
ai ∞
ψ := ψ(x )dx + ψ(x )dx ,
i i e e
ZG i∈IZ0 e∈EZ0
X X
where dx and dx refers to integration with respect to the Lebesgue measure on
i e
the intervals [0,a ] and [0, ), respectively.
i
∞
By with j one denotes the set ofall ψ suchthat its derivative
j j j
W ∈E∪I ∈H
ψ isabsolutelycontinuousandψ′ issquareintegrable. Onthe wholemetricgraph
j j
one considers the direct sum
( , )= .
j
W ≡W E I W
j∈E∪I
M
By withj denotethesetofallψ suchthatψ anditsderivative
j j j j
D ∈E∪I ∈H
ψ′ are absolutely continuous and ψ′′ is square integrable. Let 0 denote the set of
j j Dj
all elements ψ with
j j
∈D
ψ (0)=0, ψ′(0)=0, for j ,
j
∈E
ψ (0)=0, ψ′(0)=0, ψ (a )=0, ψ′(a )=0, for j .
j j j j
∈I
In the space one defines the subspaces
H
= , and 0 = 0.
D Dj D Dj
j∈E∪I j∈E∪I
M M
Note that these spaces clearly decouple the edges of the graph. The coupling is
goingtobeimplementedintermsofboundaryconditions. Withthescalarproducts
, defined by
W
h· ·i
ϕ,ψ = ϕ,ψ + ϕ′,ψ′
W H H
h i h i h i
and , given by
D
h· ·i
ϕ,ψ = ϕ,ψ + ϕ′,ψ′ + ϕ′′,ψ′′
D H H H
h i h i h i h i
the spaces and , respectively become themselves Hilbert spaces.
W D
SIGN–INDEFINITE OPERATORS ON METRIC GRAPHS 5
2.3. Differential operator. in this article the formal differential operator
d2ψ (x), j , x I ,
(3) (τψ) (x)= −dx j ∈E+∪I+ ∈ j
j (+ddx2ψj(x), j ∈E−∪I−, x∈Ij
is considered. That is τ acts as ∆ on ( ,a ) and as +∆ on ( ,a ), where
− G+ + G− −
d2
(4) (∆ψ) (x)= ψ (x), j , x I .
j dx j ∈E∪I ∈ j
To emphasise the importance of the domains and to make the choice of the do-
mains transparent, the notation is used to write a differential operator as 2-tuple,
consisting of a differential expression and a domain. The operator
Tmin =(τ, )
0
D
is closed, densely defined and symmetric. Its adjoint in the Hilbert space is the
operator Tmax = Tmin ∗, H
Tmax =(τ, ).
(cid:0) (cid:1) D
Since the operator Tmin has equal deficiency indices
(d ,d )=(d,d), where d=n+m,
+ −
there are self-adjoint realizations T =T∗ of τ with
Tmin T Tmax,
⊂ ⊂
and all these realizations can be parametrized using boundary conditions.
2.4. Space of boundary values. Onthelinesof[19]oneintroducestheauxiliary
Hilbert spaces
( , )= − +
K± ≡K E± I± KE±⊕KI±⊕KI±
with =C|E±| and (±) =C|I±|. Eventually one defines
KE± ∼ KI± ∼
= and 2 = .
+ −
K K ⊕K K K⊕K
For ψ one defines the vectors of boundary values
∈D
ψ (0) ψ (0)
{ j }j∈E+ { j }j∈E− ψ
ψ = ψ (0) , ψ = ψ (0) , ψ = + ,
+ {ψj(a )}j∈I+  − {ψj(a )}j∈I−  "ψ−#
{ j j }j∈I+ { j j }j∈I−
 ψ′(0)   ψ′(0) 
{ j }j∈E+ { j }j∈E− ψ′
ψ′ = ψ′(0) , ψ′ = ψ′(0) , ψ′ = + .
+  {ψj′(a})j∈I+  −  {ψj′(a})j∈I−  "ψ′−#
{− j j }j∈I+ {− j j }j∈I−
   
The vector [ψ]=ψ ψ′ is an element of 2 = .
⊕ K K⊕K
3. Self–adjoint realizations
To measure how far the maximal operator Tmax is away from being self-adjoint
one evaluates the form j defined by
(G+,G−)
j (ψ,ϕ):= Tmaxψ,ϕ ψ,Tmaxϕ , for ψ,ϕ ,
(G+,G−) h i−h i ∈D
compare for example [14, Section 3]. Note that this form can be represented by a
matrix in the finite dimensional Hilbert space 2, that is
K
j (ψ,ϕ)= [ψ],J [ϕ] ,
(G+,G−) h (G+,G−) iK2
6 AMRUHUSSEIN
where the matrix
0 0 1 0
n
0 0 −0 1
J(G+,G−) =1 0 0 0m,
n
 0 −1m 0 0 
is written with respect to the decomposition of 2 =  ; 1
+ − + − n
denotes the identity operator in the n-dimensionaKl spaceK ⊕aKnd ⊕1Kth⊕e Kidentity
+ m
K
in the m-dimensional space , respectively. By an abuse of notation one denotes
−
by j also the sesquilinKear form defined by J on 2. Notice that
(G+,G−) (G+,G−) K
J∗ = J and J2 = 1
(G+,G−) − (G+,G−) (G+,G−) − K2
holds and therefore J is a Hermitian symplectic matrix in 2. The relation
(G+,G−) K
of J to the standard Hermitian symplectic matrix
(G+,G−)
0 0 1 0
n
0 0 −0 1
JG =1 0 0 −0m,
n
 0 1m 0 0 
 
 
is provided by the similarity relation
1 0 0 0
n
0 1 0 0
(5) JG =Hn,mJ(G+,G−)Hn,m, where Hn,m = 0 0m 1 0 .
n
 0 0 0 1m
 − 
A subspace 2 is called maximal isotropic with respect to j if
M⊂K (G+,G−)
j (ξ ,ξ )= ξ ,J ξ =0 for all ξ ,ξ
(G+,G−) 1 2 1 G+,G− 2 1 2 ∈M
and if there are no proper subs(cid:10)paces ( (cid:11) ′ having this property. Any subspace
2 with dim d, can be paMrametMrized as follows. Let A and B be linear
mMap⊂sKin . By (A,MB)≥one denotes the linear map from 2 = to defined
K K K⊕K K
by (A, B)(η η )=Aη +Bη , where η ,η . One sets
1 2 1 2 1 2
⊕ ∈K
= (A,B), where (A,B):=Ker(A, B).
M M M
From(5)onededucesaone–to–onecorrespondencebetweensubspacesin 2 which
K
are maximal isotropic with respect to the standard Hermitian symplectic form,
which is well studied, andsubspaces in 2 that are maximalisotropic with respect
K
to the Hermitian symplectic form j occurring here.
(G+,G−)
Theorem 3.1. All self-adjoint extensions of Tmin are uniquely determined by sub-
spaces 2,whicharemaximalisotropicwithrespecttotheHermitiansymplec-
M⊂K
ticformj . Allsuchmaximalisotropicsubspacesaregivenby = (A,B),
(G+,G−) M M
where A and B are linear maps in , which satisfy the two conditions
K
(1) Rank(A, B)=n+m, that is the rank is maximal and
1 0
(2) BJ A∗ =AJ B∗, where J := n defines a map in
n,m n,m n,m 0 1
m
(cid:20) − (cid:21)
= .
+ −
K K ⊕K
Any self–adjoint extension of Tmin is given by T(A,B)=(τ,Dom(T(A,B))) with
Dom(T(A,B))= ψ Aψ+Bψ′ =0 ,
∈D|
where A,B satisfy both conditions (1)(cid:8)and (2). (cid:9)
Remark 3.2.
(1) The condition Aψ+Bψ′ =0 is equivalent to [ψ] (A,B).
∈M
SIGN–INDEFINITE OPERATORS ON METRIC GRAPHS 7
(2) The operator T(A,B) is the operator with positive and negative edges
−
interchanged, but the coupling is implemented by the same boundary condi-
tions at the vertices.
(3) For = the operator T(A,B) is the self-adjoint operator plus or minus
∓
G ∅
Laplace ∆(A,B) on , where ∆(A,B) denotes the realization of ∆ with
±
domain ∓Dom(∆(A,B)G) = ψ Aψ+Bψ′ = 0 , compare for example
{ ∈ D | }
[19].
(4) Observe that the second condition in Theorem 3.1 can be re–formulated
in terms of relations in finite dimensional Krein spaces. Consider the
space equipped with the indefinite inner product [, ]= ,J . Then
n,m K
K · · h· ·i
BJ A∗ =BA[∗], where [∗] denotes the Krein space adjoint.
n,m
·
Proof of Theorem 3.1. It is well known that all self-adjoint extensions of Tmin,
which is a closed symmetric operator with finite and equal deficency indices, is
determinedbysubspaces of 2,whicharemaximalisotropicwithrespecttothe
M K
Hermitiansymplecticformj ,seeforexample[14]andthereferencestherein.
(G+,G−)
To provethe parametrizationlet now be a maximalisotropic subspaceof 2
M K
withrespecttotheHermitiansymplecticformdefinedbyJ . Thenby(5)the
(G+,G−)
space H is a maximal isotropic subspace of 2 with respect to the standard
n,m
M K
Hermitian symplectic form defined by J . The space H canbe parametrized
G n,m
M
by matrices A ,B , see for example [19], which satisfy the two conditions
0 0
(1) Rank(A , B )=n+m and
0 0
(2) A B∗ =B A∗.
0 0 0 0
Hence the original space can be represented as
M
1 0
=H Ker(A , B )=Ker A , B n .
M n,m 0 0 0 0 0 1m
(cid:18) (cid:20) − (cid:21)(cid:19)
The matrices
1 0
A=A and B =B n
0 0 0 1
m
(cid:20) − (cid:21)
satisfy the two conditions formulated in Theorem 3.1. (cid:3)
Remark3.3. TheproofofTheorem3.1showsthatifonehasaself–adjoint Laplace
operator ∆(A ,B ) parametrized by matrices A and B , then T(A,B) is self–
0 0 0 0
−
adjoint with A=A and B =B J . In particular there is a unique parametriza-
0 0 n,m
tion in terms of unitary operators in . For Laplace operators it is known that for
K
self–adjoint boundary conditions defined by operators A and B in there exists
0 0
K
a unique unitary map U in such that equivalent boundary conditions are defined
K
by
1 1
A′ = (U 1) and B′ = (U +1),
0 −2 − 0 2i
see for example [14, Section 3]. Hence for T(A,B) self–adjoint one can give an
equivalent parametrization in terms of the same unitary matrix U with
1 1
A′ = (U 1) and B′ = (U +1)J .
n,m
−2 − 2i
The parametrization by matrices A and B in Theorem 3.1 is not unique, since
two operators T(A′,B′) and T(A,B) are equal if and only if the corresponding
maximalisotropic subspaces (A′,B′) and (A,B) agree. Using the resultfrom
M M
[23, Theorem 6], for self-adjoint Laplacians on finite metric graphs one obtains a
unique parametrization.
8 AMRUHUSSEIN
Corollary 3.4. Let (A,B) 2 be maximal isotropic with respect to the Her-
M ⊂ K
mitian symplectic form j . Then there exists a unique orthogonal projection
(G+,G−)
P and a unique Hermitian operator L with P⊥LP⊥ =L, where P⊥ =1 P, such
−
that with
1 0
A′ =L+P, B′ =P⊥ n
0 1
m
(cid:20) − (cid:21)
one has (A,B)= (A′,B′), where P is the orthogonal projector onto
M M
KerBJ .
n,m
⊂K
Example 3.5. Let be the star graph consisting of two external edges, = 1
+
G E { }
and = 2 , glued together at one single vertex ∂(1)=∂(2), and consider
−
E { }
1 1 0 0
A= − and B = .
0 0 1 1
(cid:20) (cid:21) (cid:20)− (cid:21)
Since these matrices satisfy the two conditions formulated in Theorem 3.1 the op-
erator T(A,B) defined on is self-adjoint. One can identify the metric graph
G G
with the real line, and under this identification the operator T(A,B) corresponds to
the operator d sgn(x) d defined in L2(R) on its natural domain.
−dx dx
From the self–adjointness of T(A,B) one deduces that the eigenvalue equation
can have square integrable solutions only for real spectral parameter. The self-
adjoint operator T(A,B) can be interpreted as a realization of a system of the
type
∆ 0
τ = −
0 +∆
(cid:20) (cid:21)
which is given with respect to the decomposition = ( ) ( ).
+ + − −
E∪I E ∪I ∪ E ∪I
The operator T(A,B) generates a unitary group U(t)=e−itT(A,B) which provides
solutions for the initial value problem
i∂ T(A,B) u(x,t)=0, t R,
∂t − ∈
u(,0)=u .
0
(cid:26) (cid:0) · (cid:1)
Comparedtothe time–dependentbehaviourofgroupsgeneratedby(positive)self–
adjoint Laplacians ∆, the group generated by T(A,B) describes a time–reversed
−
behaviouronthenegativeedgesandtheforward–directedbehaviouronthepositive
edges.
4. Self-adjointness in Krein spaces
One can consider the same graph described in Example 3.5, but one imposes
boundary conditions defined by
1 1 0 0
A= − and B = .
0 0 1 1
(cid:20) (cid:21) (cid:20) (cid:21)
These define the non–self–adjoint operator T(A,B) and the self–adjoint operator
∆(A,B),compareforexample[19]. Here ∆(A,B)denotestheLaplaceoperator
d−efined in (4) with domain Dom(∆(A,B))−= ψ Aψ + Bψ′ = 0 . The
{ ∈ D | }
operator T(A,B) corresponds, again by identifying this graph with the real line,
to the non–self–adjoint operator sgn(x) d2 with domain H2(R) L2(R), where
− dx2 ⊂
H2(R)denotestheSobolevspaceofordertwo. Ageneralizationof sgn(x) d2 from
− dx2
intervalstofinitemetricgraphscanbe doneonthe samelinesasfor d sgn(x) d .
−dx dx
Define the Krein space
K= L2(Ij), with indefinite inner product [, ]K = ,Jn,m H,
· · h· ·i
j∈E∪I
M
SIGN–INDEFINITE OPERATORS ON METRIC GRAPHS 9
where with a slight abuse of notation one sets
+ψ , j ,
j + +
(J ψ) = ∈E ∪I
n,m j
( ψj, j − −.
− ∈E ∪I
The multiplication with J is the fundamental symmetry of the Krein space K.
n,m
Since
J Tmax = ∆ and J Tmin = ∆0
n,m n,m
− −
holds, where ∆0 is the restriction of ∆ to 0, one obtains
D
Proposition 4.1. The extensions of Tmin in with J T(A,B) self-adjoint are
n,m
H
defined exactly by those A,B for which ∆(A,B) is self-adjoint in .
− H
As already mentioned the self-adjoint realizations ∆(A,B) of the maximal
−
Laplacian ∆ are characterized completely, see for example [19]. The eigenvalue
problem fo−r operators of the type sgn(x) d2 on compact finite metric graphs,
− dx2
which have been described here in Proposition 4.1, is considered in [7].
Remark 4.2. Analoguetothecasetreatedin Proposition 4.1 generalizations of the
operator
d d
sgn(x) sgn(x)
− dx dx
from intervals to metric graphs can be given. One can search for those extensions
∆(A,B) of the minimal Laplacian ∆0 with J ∆(A,B) self-adjoint in (or
n,m
− − − H
equivalently ∆(A,B) Krein-space-self-adjoint in K). Analogue to Proposition 4.1
−
one obtains that the operator J ∆(A,B) is self-adjoint in if and only if
n,m
− H
T(A,B) is self-adjoint in . Consequently these extensions can be parametrized
H
using Theorem 3.1.
5. Indefinite form methods
An approach that has turned out to be very fruitful for sign–definite operators
in Hilbert spaces is the one using quadratic forms. Representationtheorems give a
one to one correspondencebetween closed semi–bounded forms and semi–bounded
self-adjointoperators,seeforexample[17,ChapterVI].Thequadraticformdefined
by d sgn(x) d is symmetric, but not semi–bounded. However there are gener-
−dx dx
alizations of the first representation theorem to indefinite quadratic forms. One
formulation is given in [12], for further information on indefinite quadratic forms
see also the references given therein, in particular the works [10] and [24]. The
application of results about indefinite quadratic forms to certain indefinite second
order differential operators on bounded domains is going to be discussed in [16].
Ananalogueconstructioncanbegivenforcertaincompactmetricgraphs. Herethe
result is stated and compared to the previously obtained results, where methods
from extension theory have been used.
Assumethat( ,a)isacompact star graph thatis onehasfinitely manyinternal
G
edges andtheinitialverticesofalledgesareunifiedinonevertexandallterminal
I
vertices are vertices of degree one. This means ∂ (i ) = ∂ (i ) for i ,i and
− p − q p q
∈ I
∂ (i ) = ∂ (i ), for i = i . On the compact star graph ( ,a) one considers the
+ p + q p q
6 6 G
spaces H1( ) and H1( ) , where
G ⊂W 0 G ⊂W
H1( ):= ψ ψ (0)=ψ (0), i,j
j i
G { ∈W | ∈I}
and
H1( ):= ψ H1( ) ψ (a )=0, j .
0 G ∈ G | j j ∈I
Define now the gradient operator in by
(cid:8) H (cid:9)
D: H1( ) , ψ ψ′,
0 G →H 7→
10 AMRUHUSSEIN
whose adjoint operator is
D∗: H1( ) , ψ ψ′.
G →H 7→−
The operator D and the space RanD are closed. Since RanD is closed, it is
⊂ H
itself a Hilbert space with the Hilbert space structure inherited from . Hence
H
u, u RanD,
Q: RanD, Qu= ∈
H→ (0, u RanD
⊥
isapartialisometry. TheadjointQ∗ istheembeddingfromRanD to . Notethat
H
(RanD)⊥ is the space of constant functions and therefore the operator Q becomes
more precisely
1
Qu=u u,
− a
i∈I i ZG
since Q maps onto the orthogonal complement of the constant functions.
P
The operators D and D∗ were used implicitly in [23] to apply form methods to
Laplaceoperatorsonfinitemetricgraphs. Thefollowingtheoremisagraphversion
of the result that is going to be presented in [16].
Theorem 5.1. Let ( ,a) be a compact star graph and
G
(a) A L∞( ;R), that is A L∞(I ;R), for each j , be such that
j j
∈ G ∈ ∈I
(b) the operator QM Q∗ : RanD RanD is boundedly invertible, where M
A A
→
is the multiplication operator
M : , φ A()φ.
A
H→H 7→ ·
Then
(i) there exists a unique self-adjoint operator with Dom( ) H1( ) such
L L ⊂ 0 G
that
ϕ, ψ = ϕ′,A()ψ′
H H
h L i h · i
holds for all ϕ H1( ) and all ψ Dom( ), where
∈ 0 G ∈ L
Dom( )= ψ H1( ) M Dψ H1( ) .
L { ∈ 0 G | A ∈ G }
For any ψ Dom( ) one has ψ = D∗M Dψ, the domain Dom( ) is a
A
∈ L L L
core for the gradient operator D;
(ii) the operator is boundedly invertible and its inverse −1 is compact. In
L L
particular, the spectrum of is purely discrete.
L
Roughly speaking, Theorem 5.1 states that the operator is a self–adjoint re-
L
alization of the formal differential expression d A() d with Dirichlet boundary
−dx · dx
conditions. TheproofofTheorem5.1reliesontherepresentationtheoremforindef-
inite quadratic forms [12, Theorem 2.3 and Lemma 2.2]. It is completely analogue
to the one that is going to be given in [16] and omitted here. The condition (b) of
Theorem 5.1 can be verified by
Proposition5.2. Let ( ,a) be a compact star graph and A(),A()−1 L∞( ;R).
G · · ∈ G
Then QAQ∗ is boundedly invertible if and only if
1
=0.
A 6
ZG
The proof is based on the fact that the space (RanD)⊥ = KerD∗ is the one
dimensional space of constant functions in H1( ). It is analogue to the one that
G
is going to be presented in [16] and omitted here. Applying Proposition 5.2 to the
function
+1, j ,
J : C, x ∈I+
n,m j
G → 7→( 1, j −,
− ∈I