Formally self-adjoint quasi-differential operators and boundary value problems

0.14 MB·English

by  Andrii Goriunov

FORMALLY SELF-ADJOINT QUASI-DIFFERENTIAL OPERATORS AND BOUNDARY
VALUE PROBLEMS
ANDRIIGORIUNOV,VLADIMIRMIKHAILETS,KONSTANTINPANKRASHKIN
ABSTRACT. We develop the machinery of boundary triplets for one-dimensional operators generated by
formallyself-adjointquasi-differentialexpressionof arbitraryorderon a finite interval. The techniqueare
then used to describe all maximal dissipative, accumulative and self-adjoint extensions of the associated
minimaloperatoranditsgeneralizedresolventsintermsof theboundaryconditions. Somespecificclasses
areconsideredingreaterdetail.
1. INTRODUCTION
3
1 Many problems of the modern mathematical physics and the quantum mechanics lead to the study of
0
differential operators with strongly singular coefficients such as Radon measures or even more singular
2
distributions, see the monographs [2, 3] and the very recent papers [10, 13, 11, 12] and the references
n
therein. In such situations one is faced with the problem of a correct definition of such operators as the
a
J classical methods of the theory of differential operators cannot be applied anymore. It was observed in

4 the recent years that a large class of one-dimensional operators can be handled in a rather efficient way

with the help of the so-called Shin–Zettl quasi–derivatives[4, 24]. The class of such operators includes,
]
A
forexample,theSturm-Liouvilleoperators actingin L ([a,b],C)bytherule
2
F
. (1) l(y)=−(py′)′+qy,
h
t where thecoefficients p and q satisfytheconditions
a
m
1 Q Q2
[ , , ∈L ([a,b],C),
p p p 1

3
v Qistheantiderivativeofthedistributionq,and[a,b]isafiniteintevral. Thecondition1/p∈L ([a,b],C)
1
0
impliesthatthepotentialfunctionq maybea finitemeasureon [a,b],see[18].
1
8 Forthetwo-termformaldifferentialexpression
1
. (2) l(y)=imy(m)+qy, m≥3,
5
0 where q = Q′ and Q ∈ L ([a,b],C), the regularisation with quasi-derivatives was constructed in [19].
2 1
1 Similarlyonecan studythecase
v: q=Q(k), k≤ m ,
i 2
X
where Q∈L ([a,b],C)if m is even and k=m/2, and Q∈hL (i[a,b],C)otherwise, and all the derivatives
2 1
r
a ofQ are understoodin thesenseofdistributions.
In the present paper we consider one-dimensional operators generated by the most general formally
self-adjoint quasi-differential expression of an arbitrary order on the Hilbert space L ([a,b],C), and the
2
mainresultconsistsinanexplicitconstructionofaboundarytripletfortheassociatedsymmetricminimal
quasi-differentialoperator. Themachineryofboundarytriplets[15]isausefultoolinthedescriptionand
the analysis of various boundary value problems arising in mathematical physics, see e.g. [6, 8, 9], and
we expect that the constructions of the present paper will be useful, in particular, in the study of higher
orderdifferentialoperatorson metricgraphs [5].
The quasi-differential operators were introduced first by Shin [25] and then essentially developed by
Zettl[26],seealsothemonograph[14]andreferences therein. Thepaper[26]providesthedescriptionof
2010MathematicsSubjectClassification. 34B05,34L40,34B38,47N20.
Keywordsandphrases. quasi-differentialoperator;distributionalcoefficients;self-adjointextension;maximaldissipative
extension;generalizedresolvent.
Thefirstandsecondauthorswerepartiallysupportedbythegrantno.01-01-12ofNationalAcademyofSciencesofUkraine
(underthejointUkrainian-RussianprojectofNASofUkraineandRussianFoundationofBasicResearch).
1
2 ANDRIIGORIUNOV,VLADIMIRMIKHAILETS,KONSTANTINPANKRASHKIN
all self-adjoint extensions of the minimal symmetric quasi-differential operator of even order with real-
valuedcoefficients. Itisbasedontheso-calledGlasman-Krein-Naimarktheoryandisratherimplicit. The
approach of the present work gives an explicit description of the self-adjoint extensions as well as of all
maximaldissipative/accumulativeextensionsintermsofeasily checkableboundaryconditions.
Thepaperisorganised as follows. In Section 1 we recall basicdefinitionsand knownfacts concerning
theShin–Zettlquasi-differentialoperators. Section2 presentstheregularizationoftheformaldifferential
expressions(1)and(2)usingthequasi-derivatives,andsomespecificexamplesareconsidered. InSection
3 the boundary triplets for the minimal symmetric operators are constructed. All maximal dissipative,
maximal accumulative and self-adjoint extensions of these operators are explicitly described in terms
of boundary conditions. Section 4 deals with the formally self-adjoint quasi-differential operators with
real-valued coefficients. We provethat every maximaldissipative/accumulativeextensionof theminimal
operator in this case is self-adjoint and describe all such extensions. In Section 5 we give an explicit
description of all maximal dissipative/accumulativeand self-adjoint extensions with separated boundary
conditionsforaspecial case. In Section6 wedescribeallgeneralized resolventsoftheminimaloperator.
Some results of this paper for some particular classes of quasi-differential expressions were announced
withoutproofin [16, 17].
2. QUASI-DIFFERENTIAL EXPRESSIONS
In this section we recall the definition and the basic facts concerning the Shin–Zettl quasi-derivatives
and thequasi-differentialoperators onafiniteinterval,see[14,26]foramoredetaileddiscussion.
Let m ∈ N and a finite interval [a,b] be given. Denote by Z ([a,b]) the set of the m×m complex
m
matrix-valuedfunctions A whoseentires (a )satisfy
k,s
1)a ≡0,s>k+1;
k,s
(3) 2)a ∈L ([a,b],C), a 6=0 a. e. on [a,b],
k,s 1 k,k+1
k=1,2,...,m,s=1,2,...,k+1;
such matrices will be referred to as Shin–Zettl matrices of order m on [a,b]. Any Shin–Zettl matrix
A defines recursively the associated quasi-derivatives of orders k ≤ m of a function y ∈ Dom(A) in the
followingway:
D[0]y:=y,
k
D[k]y:=a−1 (t) (D[k−1]y)′−(cid:229) a (t)D[s−1]y , k=1,2,...,m−1,
k,k+1 k,s
s=1 !
m
D[m]y:=(D[m−1]y)′− (cid:229) a (t)D[s−1]y,
m,s
s=1
and theassociateddomainDom(A)isdefined by
Dom(A):= y D[k]y∈AC([a,b],C),k=0,m−1 .
n (cid:12) o
The above yields D[m]y∈L ([a,b],C). T(cid:12)he quasi-differential expression l(y) of order m associated with
1 (cid:12)
A isdefined by
(4) l(y):=imD[m]y.
Letc∈[a,b]anda ∈C,k=0,m−1. Wesay thatafunctionysolvestheCauchyproblem
k
(5) l(y)−l y= f ∈L ([a,b],C), (D[k]y)(c)=a , k=0,m−1,
2 k
if yis the first coordinate ofthevectorfunction w solvingthe Cauchy problem forthe associated thefirst
ordermatrixequation
(6) w′(t)=Al (t)w(t)+j (t), w(c)=(a 0,a 1,...a m−1)
FORMALLYSELF-ADJOINTQUASI-DIFFERENTIALOPERATORSANDBOUNDARYVALUEPROBLEMS 3
where wedenote
0 0 ... 0
0 0 ... 0
(7) Al (t):=A(t)− ... ... ... ... ∈L1([a,b],Cm×m),
 0 0 ... 0 
 
 i−ml 0 ... 0 
 
 
and j (t):= 0,0,...,0,i−mf(t) T ∈L ([a,b],Cm). Thefollowingstatementis provedin [26].
1
Lemma 1. U(cid:0)nder theassumptio(cid:1)ns(3), theproblem(5)hasa uniquesolutiondefined on[a,b].
Thequasi-differentialexpression l(y)givesriseto theassociated maximalquasi-differentialoperator
L :y7→l(y),
max
Dom(L )= y∈Dom(A) D[m]y∈L ([a,b],C) .
max 2
n (cid:12) o
in the Hilbert space L2([a,b],C), and the associated mi(cid:12)nimal quasi-differential operator is defined as the
(cid:12)
restrictionof L ontotheset
max
Dom(L ):= y∈Dom(L ) D[k]y(a)=D[k]y(b)=0,k=0,m−1 .
min max
n (cid:12) o
If the functionsak,s are sufficiently smooth, then(cid:12)all the brackets in the definition of the quasi-derivatives
(cid:12)
can be expanded, and we arrive at the usual ordinary differential expressions, and the associated quasi-
differentialoperators becomedifferentialones.
Let us recall the definition of the formally adjoint quasi-differential expression l+(y).The formally
adjoint(also calledtheLagrangeadjoint)matrix A+ forA∈Z ([a,b])isdefined by
m
A+ :=−L −1ATL ,
m m
where AT istheconjugatetransposedmatrixto A and
0 0 ... 0 −1
0 0 ... 1 0
 . . . . . 
L m := .. .. .. .. .. .
 0 (−1)m−1 ... 0 0 
 
(−1)m 0 ... 0 0 
 
 
Onecan easilyseethatL −1 =(−1)m−1L .
m m
Wewecan definetheShin–Zettlquasi-derivativesassociatedwithA+ which willbedenoted by
D{0}y,D{1}y,...,D{m}y,
and theyact on thedomain
Dom(A+):= y D{k}y∈AC([a,b],C),k=0,m−1 .
n (cid:12) o
Theformallyadjointquasi-differentialexp(cid:12)ressionisnpw defined as
(cid:12)
l+(y):=imD{m}y
andwedenotetheassociatedmaximalandminimaloperatorsbyL+ andL+ respectivelyThefollowing
max min
theorem isprovedin[26].
Theorem 1. The operators L , L+ , L , L+ are closed and densely defined in L ([a,b],C), and
min min max max 2
satisfy
L∗ =L+ , L∗ =L+ .
min max max min
If l(y)=l+(y),then theoperatorL =L+ is symmetricwith thedeficiency indices(m,m),and
min min
L∗ =L , L∗ =L .
min max max min
Wealso willrequirethefollowingtwo lemmaswhoseproofcan befounde.g. in[14]:
4 ANDRIIGORIUNOV,VLADIMIRMIKHAILETS,KONSTANTINPANKRASHKIN
Lemma 2. Foranyy,z∈Dom(L ) thereholds
max
b
m
D[m]y·z−y·D[m]z dt = (cid:229) (−1)k−1D[m−k]y·D[k−1]z t=b
t=a
Za (cid:16) (cid:17) k=1 (cid:12)
(cid:12)
Lemma3. Forany(a ,a ,...,a ),(b ,b ,...,b )∈Cm thereexistsafunc(cid:12)tiony∈Dom(L )such
0 1 m−1 0 1 m−1 max
that
D[k]y(a)=a , D[k]y(b)=b , k=0,1,...,m−1.
k k
3. REGULARIZATIONS BY QUASI-DERIVATIVES
Let us consider some classes of formal differential expressions with singular coefficients admitting a
regularisationwiththehelpoftheShin–Zettlquasi-derivatives.
Considerfirst theformalSturm–Liouvilleexpression
l(y)=−(p(t)y′)′(t)+q(t)y(t), t ∈[a,b].
Theclassicaldefinitionofthequasi-derivatives
D[0]y:=y, D[1]y= py′, D[2]y=(D[1]y)′−qD[0]y
allows one to interpret the above expression l as a regularquasi-differential one if the function p is finite
almosteverywhereand, inaddition,
1
(8) ,q∈L ([a,b],C).
1
p
Somephysicallyinterestingcoefficientsq(i.e. havingnon-integrablesingularitiesorbeingameasure)are
not covered by the preceding conditions, and this can be corrected using another set of quasi-derivatives
as proposedin [?,24]. Set
D[0]y=y, D[1]y= py′−Qy,
(9) Q Q2
D[2]y=(D[1]y)′+ D[1]y+ y,
p p
where function Q is chosen so that Q′ =q and the derivative is understood in the sense of distributions.
Then theexpression
l[y]=−D[2]y
isa Shin–Zettlquasi-differentialoneifthefollowingconditionsaresatisfied:
1 Q Q2
(10) , , ∈L ([a,b],C).
1
p p p
In this case the expression l generates the associated quasi-differential operators L and L . One can
min max
easily see that if p and q satisfy conditions (8), then these operators coincide with the classic Sturm-
Liouvilleoperators, but the conditions (10) are considerably weaker than (8), and the class of admissible
coefficients is much larger if one uses the quasi-differential machinery. This can be illustrated with an
example.
a b
Example 1. Consider the differential expression (1) with p(t) =t and q(t)= ct , and assume c 6= 0.
Theconditions(8)are reduced to theset of theinequalities a <1 and b >−1, whiletheconditions(10)
holdfor
a −3
a <1 and b >max a −2, .
2
So we see that the use of the quasi-derivatives allowsnone to considoer the Sturm–Liouville expressions
withanypowersingularityofthepotential q ifitiscompensatedby an appropriatefunction p. (cid:3)
Remark 1. The formulas (9) for the quasi-derivatives contain a certain arbitrariness due to the non-
uniqueness of the function Q which is only determined up to a constant. However, one can show that if
Q:=Q+c, forsome constant c∈C, then L (Q)=L (Q) and L (Q)=L (Q), i.e. the maximal
max max min min
and minimaloperators donotdepend onthechoiceof c. (cid:3)
e e e
FORMALLYSELF-ADJOINTQUASI-DIFFERENTIALOPERATORSANDBOUNDARYVALUEPROBLEMS 5
Onecan easilysee thattheexpression
l+(y)=−(py′)′+qy
defines the quasi-differential expression which is formally adjoint to one generated by (1). It brings up
the associatedmaximaland minimaloperators L+ and L+ . Theorem 1 showsthat if p and q in (1) are
max min
real-valued, thentheoperator L =L+ issymmetric.
min min
It iswell-knownthatfortheparticularcase p≡1and q∈L ([a,b],C)onehas
2
Dom(L )=W2([a,b],C)⊂C1([a,b],C).
max 2
Thefollowingexampleshowsthatinsomecases allfunctionsinDom(L )\{0}arenon-smooth.
max
Example 2. Considerthedifferentialexpression(1)with
p(t)≡1, q(t)= (cid:229) a m d (t−m ),
m ∈Q∩(a,b)
where Qistheset ofreal rationalnumbersand
a m 6=0forall m ∈Q∩(a,b), and (cid:229) |a m |<¥ .
m ∈Q∩(a,b)
Then onecan take
Q(t)= (cid:229) a m H(t−m ),
m ∈Q∩(a,b)
with H(t) being Heaviside function, and Q is a function of a bounded variation having discontinuities at
every rational point of (a,b). Therefore, for every subinterval [a ,b ]⊂(a,b) and any y∈Dom(L ) ∩
max
C1([a ,b ],C)wehave
y′(m +)−y′(m −)=a m y(m ), m ∈Q∩[a ,b ].
Then a m y(m )=0 for all m ∈Q∩[a ,b ], which gives y(m )=0, and the density of {m }∩[a ,b ] in [a ,b ]
impliesy(t)=0 forallt ∈[a ,b ]. (cid:3)
Nowconsidertheexpression
l(y)=imy(m)(t)+q(t)y(t), m≥3,
assumingthat
m
q=Q(k), 1≤k≤ ,
2
(11) L ([a,b],C),mh=i2n,k=n;
2
Q∈
(L ([a,b],C)otherwise,
1
where the derivativesof Q are understoodin the senseof distributions. Introducethe quasi-derivativesas
follows:
D[r]y=y(r), 0≤r≤m−k−1;
k
D[m−k+s]y=(D[m−k+s−1]y)′+i−m(−1)s QD[s]y, 0≤s≤k−1;
(12) s
(cid:18) (cid:19)
(D[m−1]y)′+i−m(−1)k k QD[k]y, 1≤k<m/2,
D[m]y= k
((D[m−1]y)′+QD[m2]y+(cid:0)(−(cid:1)1)m2+1Q2y, m=2n=2k;
where k are the binomial coefficients. It is easy to verify that for sufficiently smooth functions Q the
j
equalityl(y)=imD[m]yholds. Alsoonecaneasilyseethat,underassumptions(11), allthecoefficientsof
(cid:0) (cid:1)
6 ANDRIIGORIUNOV,VLADIMIRMIKHAILETS,KONSTANTINPANKRASHKIN
the quasi-derivatives (12) are integrable functions. The Shin–Zettl matrix corresponding to (12) has the
form
0 1 0 ... 0 ... 0 0
0 0 1 ... 0 ... 0 0
 . . . . . . . . 
. . . . . . . .
. . . . . . . .
 −i−m k Q 0 0 ... 0 ... 0 0 
(13) A(t):= 0 0 i−m k Q 0 ... 0 ... 0 0 ,
 .(cid:0) (cid:1) .1 . . . . . . 
 . . . . . . . . 
. . . . . . . .
 (cid:0) (cid:1) 
 0 0 0 ... 0 ... 0 1 
 
 m 
 (−1)2Q2d 0 0 ... i−m(−1)k+1 k Q ... 0 0 
 2k,m k 
whered istheKroneckersymbol. Similarlytothepreviouscasetheinitialformaldifferentialexpression
ij (cid:0) (cid:1)
(2)can bedefined inthequasi-differentialform
l[y]:=imD[m]y,
and itgenerates thecorrespondingquasi-differentialoperators L andL .
min max
Remark 2. Again, the formulas forthe quasi-derivativesdepend on the choice of the antiderivativeQ of
orderkofthedistributionqwhichisnotonlydefineduptoaapolynomialoforder≤k−1. However,one
can showthatthemaximalandminimaloperators do notdepend on thechoiceofthispolynomial. (cid:3)
For k=1 the aboveregularization was proposed in [24], and for even m they were announced in [21].
Thegeneral case is presentedhere forthefirst time. Notethatifthedistributionq isreal-valued, thenthe
operator L issymmetric.
min
4. EXTENSIONS OF SYMMETRIC QUASI-DIFFERENTIAL OPERATORS
Throughout the rest of the paper we assume the Shin–Zettl matrix is formally self-adjoint, i.e. A =
A+. The associated quasi-differential expression l(y) is then formally self-adjoint, l(y)=l+(y), and the
minimal quasi-differential operator L is symmetric with equal deficiency indices by Theorem 1. So
min
one may pose a problem of describing (by means of boundary triplets) various classes of extensions of
L inL ([a,b],C).
min 2
Forthereader’s conveniencewegiveaveryshortsummaryofthetheoryofboundarytripletsbased on
theresultsofRofe-Beketov [23]and Kochubei[20], seealsothemonograph[15]and references therein.
LetT beacloseddenselydefinedsymmetricoperatorinaHilbertspaceH withequal(finiteorinfinite)
deficiency indices.
Definition1([15]). Thetriplet(H,G ,G ),whereH isanauxiliaryHilbertspaceandG ,G arethelinear
1 2 1 2
mapsfromDom(T∗)toH,iscalledaboundarytripletforT,ifthefollowingtwoconditionsaresatisfied:
(1) forany f,g∈Dom(L∗) thereholds
(T∗f,g) −(f,T∗g) =(G f,G g) −(G f,G g) ,
H H 1 2 H 2 1 H
(2) forany g ,g ∈H thereis avector f ∈Dom(T∗)such thatG f =g andG f =g .
1 2 1 1 2 2
The above definition implies that f ∈ Dom(T) if and only if G f = G f = 0. A boundary triplet
1 2
(H,G ,G ) with dimH = n exists for any symmetric operator T with equal non-zero deficiency indices
1 2
(n,n) (n≤¥ ), butitis notunique.
Boundary triplets may be used to describe all maximal dissipative, maximal accumulative and self-
adjoint extensions of the symmetric operator in the following way. Recall that a densely defined linear
operator T ona complexHilbertspace H is called dissipative(resp. accumulative)if
` (T f, f) ≥0 (resp.≤0), forall f ∈Dom(T)
H
anditiscalledmaximaldissipative(resp. maximalaccumulative)if,inaddition,T hasnonon-trivialdis-
sipative/accumulativeextensions in H . Every symmetric operator is both dissipative and accumulative,
andeveryself-adjointoperatorisamaximaldissipativeandmaximalaccumulativeone. Thus,ifonehasa
symmetricoperator T, then one can state the problem of describing its maximaldissipativeand maximal
accumulative extensions. According to Phillips’ Theorem [22] (see also [15, p. 154]) every maximal
FORMALLYSELF-ADJOINTQUASI-DIFFERENTIALOPERATORSANDBOUNDARYVALUEPROBLEMS 7
dissipative or accumulative extension of a symmetric operator is a restriction of its adjoint operator. Let
(H,G ,G ) beaboundarytripletfor T. Thefollowingtheoremis provedin [15].
1 2
Theorem 2. If K is a contraction on H, then the restriction of T∗ to the set of the vectors f ∈Dom(T∗)
satisfyingthecondition
(14) (K−I)G f +i(K+I)G f =0
1 2
or
(15) (K−I)G f −i(K+I)G f =0
1 2
isamaximaldissipative,respectively,maximalaccumulativeextensionofT. Conversely,anymaximaldis-
sipative(maximalaccumulative)extensionofListherestrictionofT∗ tothesetofvectors f ∈Dom(T∗),
satisfying(14)or (15),respectively,andthecontractionK isuniquelydefinedbytheextension. Themax-
imal symmetric extensions of T are described by the conditions (14) and (15), where K is an isometric
operator. These conditionsdefineaself-adjointextensionif K is unitary.
Remark 3. Let K and K betheunitaryoperators on H and lettheboundaryconditions
1 2
(K −I)G y+i(K +I)G y=0
1 1 1 2
and
(K −I)G y−i(K +I)G y=0
2 1 2 2
define self-adjoint extensions. These are two different bijective parameterizations, which reflects the
fact that each self-adjoint operators is maximal dissipative and a maximal accumulative one at the same
time. The extensions, given by these boundary conditions coincide if K = K−1. Indeed, the boundary
1 2
conditionscan bewritteninanotherform:
K (G y+iG y)=G y−iG y, G y−iG y∈Dom(K)=H,
1 1 2 1 2 1 2
K (G y−iG y)=G y+iG y, G y+iG y∈Dom(K)=H,
2 1 2 1 2 1 2
and theequivalenceoftheboundaryconditionsreads as K K =K K =I. (cid:3)
1 2 2 1
Let us get back to the quasi-differential operators. The following result is crucial for the rest of the
paper as it allows to apply the boundary triplet machinery to the symmetric minimal quasi-differential
operator L .
min
Lemma 4. DefinelinearmapsG , G fromDom(L )toCm as follows: form=2nand n≥2 weset
[1] [2] max
−D[2n−1]y(a), D[0]y(a),
..., ...,
   
(−1)nD[n]y(a), D[n−1]y(a),
(16) G y:=i2n , G y:=
[1]  D[2n−1]y(b),  [2]  D[0]y(b), 
   
 ...,   ..., 
   
 (−1)n−1D[n]y(b)   D[n−1]y(b) 
   
   
and form=2n+1andn∈N weset
−D[2n]y(a), D[0]y(a),
..., ...,
   
(−1)nD[n+1]y(a), D[n−1]y(a),
(17) G y:=i2n+1 D[2n]y(b), , G y:= D[0]y(b), ,
[1] [2]
   
 ....,   ..., 
   
 (−1)n−1D[n+1]y(b),   D[n−1]y(b), 
   
 a D[n]y(b)+b D[n]y(a)   g D[n]y(b)+d D[n]y(a) 
   
   
where
(−1)n (−1)n+1
a =1, b =1, g = +i, d = +i.
2 2
Then (Cm,G ,G )is aboundarytripletfor L .
[1] [2] min
8 ANDRIIGORIUNOV,VLADIMIRMIKHAILETS,KONSTANTINPANKRASHKIN
Remark 4. The valuesof the coefficients a , b , g , d fortheodd case may be replaced by an arbitrary set
ofnumberssatisfyingtheconditions
a g +ag =(−1)n, b d +bd =(−1)n+1, a d +bg =0,
(18)
b g +ad =0, ad −bg 6=0.
Proof. We need to check that the triplet (Cm,G ,G ) satisfies the conditions 1) and 2) in Definition 1
[1] [2]
forT =L andH =L ([a,b],C). Dueto Theorem1, L∗ =L .
min 2 min max
Letus startwiththecaseofevenorder. DuetoLemma2, for m=2n:
2n
(L y,z)−(y,L z)=i2n (cid:229) (−1)k−1D[2n−k]y·D[k−1]z t=b.
max max t=a
k=1
(cid:12)
(cid:12)
Denote (cid:12)
G =:(G ,G ), G =:(G ,G ),
[1] 1a 1b [2] 2a 2b
where
G y=i2n −D[2n−1]y(a),...,(−1)nD[n]y(a) ,
1a
(cid:16) (cid:17)
G y=i2n D[2n−1]y(b),...,(−1)n−1D[n]y(b) ,
1b
(cid:16) (cid:17)
G y= D[0]y(a),...,D[n−1]y(a) ,
2a
(cid:16) (cid:17)
G y= D[0]y(b),...,D[n−1]y(b) .
2b
(cid:16) (cid:17)
Onecalculates
n
(G y,G z)=i2n (cid:229) (−1)kD[2n−k]y(a)·D[k−1]z(a),
1a 2a
k=1
2n
(G y,G z)=i2n (cid:229) (−1)k−1D[2n−k]y(a)·D[k−1]z(a),
2a 1a
k=n+1
n
(G y,G z)=i2n (cid:229) (−1)k−1D[2n−k]y(b)·D[k−1]z(b),
1b 2b
k=1
2n
(G y,G z)=i2n (cid:229) (−1)kD[2n−k]y(b)·D[k−1]z(b),
2b 1b
k=n+1
whichresults in
n
G y,G z =i2n (cid:229) (−1)k−1D[2n−k]y·D[k−1]z t=b ,
[1] [2] t=a
k=1
(cid:12)
(cid:0) (cid:1) 2n (cid:12)
G y,G z =i2n (cid:229) (−1)kD[2n−k]y·D[k−1]z(cid:12)t=b ,
[2] [1] t=a
k=n+1
(cid:12)
(cid:0) (cid:1)
(cid:12)
andthismeansthatthecondition1)oftheDefinition1isfulfilled,andthe(cid:12)surjectivitycondition2)istrue
duetoLemma3.
Thecaseofoddorderis treated similarly. DuetoLemma2,for m=2n+1we have
2n+1
(L y,z)−(y,L z)=i2n+1 (cid:229) (−1)k−1D[2n−k]y·D[k−1]z t=b.
max max t=a
k=1
(cid:12)
(cid:12)
Denote (cid:12)
G =:(G ,G ,G ), G =:(G ,G ,G ),
[1] 1a 1b 1ab [2] 2a 2b 2ab
FORMALLYSELF-ADJOINTQUASI-DIFFERENTIALOPERATORSANDBOUNDARYVALUEPROBLEMS 9
where
G y=i2n+1 −D[2n]y(a),...,(−1)nD[n+1]y(a) ,
1a
(cid:16) (cid:17)
G y=i2n+1 D[2n]y(b),...,(−1)n+1D[n+1]y(b) ,
1b
(cid:16) (cid:17)
G y=i2n+1 a D[n]y(b)+b D[n]y(a) ,
1ab
(cid:16) (cid:17)
G y= D[0]y(a),...,D[n−1]y(a) ,
2a
(cid:16) (cid:17)
G y= D[0]y(b),...,D[n−1]y(b) ,
2b
G y=(cid:16)g D[n]y(b)+d D[n]y(a). (cid:17)
2ab
Onecalculates
n
(G y,G z)=i2n+1 (cid:229) (−1)k−1D[2n−k]y(a)·D[k−1]z(a),
1a 2a
k=1
2n+1
(G y,G z)=i2n+1 (cid:229) (−1)kD[2n−k]y(a)·D[k−1]z(a),
2a 1a
k=n+2
n
(G y,G z)=i2n+1 (cid:229) (−1)k−1D[2n−k]y(b)·D[k−1]z(b),
1b 2b
k=1
2n+1
(G y,G z)=i2n+1 (cid:229) (−1)kD[2n−k]y(b)·D[k−1]z(b),
2b 1b
k=n+2
(G y,G z)−(G y,G z)=
1ab 2ab 2ab 1ab
=i2n+1(−1)n D[n]y(b)·D[n]z(b)−D[n]y(a)·D[n]z(a) ,
(cid:16) (cid:17)
which shows that the condition 1) of Definition 1 is satisfied. Now take arbitrary vectors
f = (f )2n , f = (f )2n ∈C2n+1. Thelastconditionin(18)meansthat thesystem
1 1,k k=0 2 2,k k=0
ab +ba = f
n n 1,n
gb +da = f
n n 2,n
(cid:26)
has auniquesolution(a ,b ). Denoting
n n
a := f , b := f fork<n,
k 1,k k 2,k
a :=(−1)2n+1−kf , b :=(−1)2n−kf forn+1<k<2n
k 1,k k 2,k
we obtain two vectors (a ,a ,...,a ),(b ,b ,...,b ) ∈ Cm ≡ C2n+1. By Lemma 3, there exists a
0 1 m−1 0 1 m−1
functiony∈Dom(L ) suchthat
max
D[k]y(a)=a , D[k]y(b)=b , k=0,1,...,m−1,
k k
andduetoabovespecialchoiceofa andb onehasG y= f andG y= f ,sothesurjectivitycondition
[1] 1 [2] 2
ofDefinition1 holds. (cid:3)
Forthesakeofconvenience,weintroducethefollowingnotation. DenotebyL therestrictionof L
K max
onto the set of the functions y(t) ∈ Dom(L ) satisfying the homogeneous boundary condition in the
max
canonical form
(19) (K−I)G y+i(K+I)G y=0.
[1] [2]
Similarly, denote by LK the restriction of L onto the set of the functions y(t)∈Dom(L ) satisfying
max max
theboundarycondition
(20) (K−I)G y−i(K+I)G y=0.
[1] [2]
Here K is an arbitrary bounded operator on the Hilbert space Cm, and the maps G G are defined by
[1] [2]
the formulas (16) or (17) depending on m. Theorem 2 and Lemma 4 lead to the following description of
extensionsofL .
min
10 ANDRIIGORIUNOV,VLADIMIRMIKHAILETS,KONSTANTINPANKRASHKIN
Theorem 3. Every L with K being a contracting operator in Cm, is a maximal dissipative extension of
K
L . Similarly every LK with K being a contracting operator in Cm, is a maximal accumulative exten-
min
sion of the operator L . Conversely, for any maximal dissipative (respectively, maximal accumulative)
min
extension L of the operator L there exists a contracting operator K such that L = L (respectively,
min K
L=LK). The extensions L and LK are self-adjoint if and only if K is a unitary operator on Cm. These
K
correspondeencesbetween operators{K}andtheextensions{L}areallbijective. e
e
Remark5. Theself-adjointextensionsofasymmetricminimalquasi-differentialoperatorweredescribed
e
by means of the Glasman-Krein-Naimark theory in the work [26] and several subsequent papers. How-
ever, the description by means of boundary triplets has important advantages, namely, it gives a bijective
parametrization of extensionsby unitary operators, and onecan describe the maximaldissipativeand the
maximalaccumulativeextensionsinasimilarway.
5. REAL EXTENSIONS
Recall that alinearoperator L actingin L ([a,b],C)iscalled real if:
2
(1) Foreveryfunction f from Dom(L) thecomplexconjugatefunction f alsolies inDom(L).
(2) The operator L maps complex conjugate functions into complex conjugate functions, that is
L(f)=L(f).
If theminimalquasi-differential operatoris real, onearrives at thenatural question on howto describe
itsreal extensions. Thefollowingtheoremholds.
Theorem 4. Let m be even, and let the entries of the Shin–Zettl matrix A=A+ be real-valued, then the
maximalandminimalquasi-differentialoperatorsL andL generatedbyAarereal. Allrealmaximal
max min
dissipative and maximal accumulative extensions of the real symmetric quasi-differential operator L
min
of the even order are self-adjoint. The self-adjoint extensions L or LK are real if and only if the unitary
K
matrixK issymmetric.
Proof. As thecoefficientsofthequasi-derivativesarereal-valued functions,onehas
D[i]y=D[i]y, i=1,2n,
whichimpliesl(y)=l(y). Thusforany y∈Dom(L ) wehave
max
D[i]y∈AC([a,b],C), i=1,2n−1, l(y)∈L ([a,b],C), L (y)=L (y).
2 max max
ThisshowsthattheoperatorL isreal. Similarly,for y∈Dom(L )we have
max min
D[i]y(a)=D[i]y(a)=0, D[i]y(b)=D[i]y(b)=0, i=1,2n−1,
whichprovesthat L isa real as well.
min
Duetothereal-valuednessofthecoefficients ofthequasi-derivatives,theequalities(16)imply
G y=G y, G y=G y.
[1] [1] [2] [2]
As themaximal operatoris real, any ofits restrictionssatisfies thecondition2) oftheabovedefinitionof
areal operator, so wearereduced tocheck thecondition1).
Let L be an arbitrary real maximal dissipativeextension given by the boundary conditions (19), then
K
forany y∈Dom(L )thecomplexconjugateysatisfies (19)too,thatis
K
(K−I)G y+i(K+I)G y=0.
[1] [2]
By takingthecomplexconjugatesweobtain
K−I G y−i K+I G y=0,
[1] [2]
(cid:16) (cid:17) (cid:16) (cid:17)
and L ⊂ LK due to Theorem 3. Thus, the dissipative extension L is also accumulative, which means
K K
that it is symmetric. But L is a maximal dissipativeextensionof L . As the deficiency indices ofL
K min min
arefinite,theoperator L =LK mustbeself-adjoint. Furthermore,duetoRemark 3theequalityL =LK
K K
is equivalent to K−1 =K. As K is unitary, we have K−1 =KT, which gives K =KT. In a similar way
one can show that a maximal accumulative extension LK is real if and only if it is self-adjoint and K =
KT. (cid:3)