WEAK FORMULATION OF SINGULAR DIFFERENTIAL EXPRESSIONS IN SPACES OF FUNCTIONS
WITH MINIMAL DERIVATIVES
M. A. El-GEBEILY Received 4 October 2004
A weak formulation for singular symmetric differential expressions is presented in spaces of functions which possess minimal differentiability requirements. These spaces are used to characterize the domains of the various operators associated with such expressions. In particular, domains of self-adjoint differential operators are characterized.
1. Introduction
Application of the general theory of self-adjoint operators to the spectral representation of operators associated with the formally self-adjoint differential expression
u= 1 w
n k=0
(−1)kpn−ku(k)(k) (1.1) was carried out to a completion by many researchers in this field. A complete account of this theory can be found in [1,11]. Account for the parallel theory of partial differen- tial and difference operators can be found in [2,5]. On the other hand, the differential expression (1.1) gives rise to the formal sesquilinear form
a(u,v)= n k=0
pn−ku(k)v(k) (1.2)
encountered in the course of studying weak formulations of differential equations. Unlike the differential expressions, the theory behind the sesquilinear forms (1.2) is not yet fully developed. The most general treatment we have so far is for the case when such forms are semibounded or sectorial [10]. The classical Lax-Milgram theorem which is widely used in treatments involving the bilinear forms (1.2) assumes that the underlying form is pos- itive and continuous. While such assumptions suffice to handle regular and some classes of singular differential expressions, they are not sufficient to handle the general singular expressions as they need not be semibounded. The importance of such a theory stems
Copyright©2005 Hindawi Publishing Corporation Abstract and Applied Analysis 2005:7 (2005) 691–705 DOI:10.1155/AAA.2005.691
from the many important applications it would have in areas such as the calculus of vari- ations and numerical solutions of differential equations. For some of these applications the reader is referred to the papers [3,4,7,9] and the references therein.
In [6] a variational formulation of the second order differential expression u= 1
w
−(pu)+qu (1.3)
was presented in regular as well as singular cases. Although no assumptions of semi- boundedness were made there, the treatment has two drawbacks. In a general setting, the presentation depended on the existence of a maximal space of definition inferred from Zorn’s lemma (see [6, page 43]). The difficulty with this space is the lack of a satisfactory concrete characterization to render it useful for further development. In a more special setting, the treatment relied on more concrete spaces but they require full differentiability assumptions and thus no use is made of the reduced order of differentiation granted by the variational setting ([6, page 48]). This makes the presentation particularly unattrac- tive if we want to devise Galerkin-like numerical methods to solve singular differential equations. These two drawbacks are eliminated in this work. We give here a weak for- mulation of the more general differential expression (1.1) in spaces which require differ- entiation properties dictated only by what is necessary for the sesquilinear form (1.2) to be meaningful. We also give full characterizations of various operators associated with the formal operatorin terms of these spaces. These characterizations include the most interesting operators associated with, namely, self-adjoint operators.
This paper is organized as follows. After this introduction we give a preliminary section in which the notation and the results frequently used in this work are given. The weak formulation of the problem is done inSection 3. In this section the working spaces are defined, the variational form of the problem is set and its equivalence to the original problem is established. In Section 4some further properties of the defined spaces are explored.
2. Preliminaries
The following notation will be used in this paper.D(a,b) denotes the space of test func- tions on the interval (a,b),−∞ ≤a < b≤ ∞, andL(a,b) its dual with respect to the fol- lowing topology. Denoting by·,·the pairing betweenD(a,b) andL(a,b), a functional f ∈L(a,b) if and only if for each compact interval [α,β] there is a constantCand an integerr≥0 such that
f,v≤C sup
0≤k≤r
v(k)∞ (2.1)
for every functionv∈D(a,b) with support in [α,β] (Candr generally dependent on [α,β]).L2w(a,b) denotes the Hilbert space of complex-valued square integrable functions on the interval (a,b) with respect to the almost everywhere positive weightw. The inner product and norm in this space are denoted by·,·wand · w, respectively.AC(k)(a,b) denotes the space of functions that are absolutely continuous on any compact subinterval of (a,b) together with their derivatives up to orderkinclusive.AC(a,b) is used in place of
AC(0)(a,b).L1loc(a,b) denotes the space of functions which are integrable on every finite sub-interval [α,β] of (a,b). Thekth classical derivative of a functionuwill be denoted as usual byu(k)whereas the notationu[k]will be used to denote thekth pseudo-derivative ofudefined by the formulae
u[k]=u(k) fork=1, 2,. . ., (n−1); u[n]=p0u(n); u[n+k]=pku(n−k)−
u[n+k−1] fork=1, 2,. . .,n, (2.2) (see also [11]).
Consider the formally self-adjoint differential expression u= 1
w n k=0
(−1)kpn−ku(k)(k) (2.3) defined on the interval (a,b), where w >0 almost everywhere on (a,b), the coefficient functionsp0,p1,. . .,pnare real valued and 1/ p0,p1,. . .,pn,w∈L1loc(a,b). Ifa,bare finite and the functions 1/ p0,p1,. . .,pn,ware integrable on (a,b) then this expression is said to be regular, otherwise it is singular.
The expressiondefines the following operators inL2w(a,b):
(1) The “maximal” operatorLwhose domainᏰis given by Ᏸ= u∈L2w(a,b) :u[k]∈AC(a,b),k=1, 2,. . ., (2n−1), 1
wu[2n]∈L2w(a,b)
, Lu=u.
(2.4)
Note thatu=(1/w)u[2n].
(2) The operatorL0whose domainᏰ0is given by Ᏸ0=
u∈Ᏸ:uhas compact support in (a,b),
L0u=u. (2.5)
(3) The “minimal” operatorL0whose domainᏰ0is given by Ᏸ0=
u∈Ᏸ: [u,v]ab=0∀v∈Ᏸ, (2.6) where [u,v]ab=[u,v](b)−[u,v](a) and [u,v](x) is the Lagrange expression
[u,v](x)= n k=1
u(k−1)(x)v[2n−k](x)−u[2n−k](x)v(k−1)(x). (2.7)
Note that (see [11]) [u,v](a) and [u,v](b) both exist for allu,v∈Ᏸ.
All three operators are densely defined and the following relationships hold among them
L0⊂L0=L0=L∗⊂L=L∗0, (2.8)
wheredenotes operator closure. In particular, the operatorsL0,L0are symmetric and the operators L0,Lare closed. For λ∈C, Im(λ)=0, putᏺλ=Ker(L−λI). Since the operator has real coefficients,u∈Dif and only ifu∈D andLu=λuif and only if Lu=λu. The common dimensiondof the spacesᏺλandᏺλis called thedeficiency index of the operatorL0. In fact, 0≤d≤2nand is independent ofλas long as Im(λ)=0. Now for a fixedλ∈C\R, the subspacesᏰ0,ᏺλandᏺλare linearly independent (see [8,11]) and
Ᏸ=Ᏸ0ᏺλᏺλ. (2.9)
For anyu∈Ᏸwrite
u=u0+uλ+uλ, (2.10)
whereu0∈Ᏸ0,uλ∈ᏺλanduλ∈ᏺλ. Then
Lu=L0u0+λuλ+λuλ. (2.11)
Formula (2.9) shows thatL0is self-adjoint if and only ifd=0.
Various characterizations of the domainsᏰ of self-adjoint extensionsLof the operator L0are given in [11] and elsewhere. We state here two characterizations which will be used in this work.
Theorem2.1. Any self-adjoint extensionLof the operatorL0is characterized by a unitary transformationU:ᏺλ→ᏺλsuch that
Ᏸ=Ᏸ0(U+I)ᏺλ,
Lu=L0u0+ (λU+λI)uλ. (2.12) In other words, there is a one to one correspondence between self-adjoint extensions ofL0and unitary transformations fromᏺλtoᏺλ.
Theorem2.2. SupposeᏰis the domain of definition of a self adjoint extensionLofL0. Then there exist functionsw1,w2,. . .,wd∈Ᏸsuch that
(1)w1,w2,. . .,wdare linearly independent moduloᏰ0, (2) [wi,wj]ba=0,i,j=1, 2,. . .,d,
(3)Ᏸ = {u∈Ᏸ: [u,wj]ba=0, j=1, 2,. . .,d}.
Conversely, for a set of functionsw1,w2,. . .,wd satisfying the conditions in Part 1 and 2 above, the setᏰ defined as in Part 3 is the domain of definition of a certain self adjoint extensionLofL0.
In what follows we summarize some results from [6] which will also be needed in this work. From now on, when we state that a complex number exists or is defined we also mean that it is finite. For functionsu,v∈AC(n−1)(a,b), we introduce the formal
sesquilinear form
a(u,v)= b
a
n k=0
pn−ku(k)v(k), (2.13)
if the integral exists. Let us also introduce the brackets {u,v}(x)= −
n k=1
u[2n−k](x)v(k−1)(x) (2.14) and note that
[u,v](x)= {u,v}(x)− {v,u}(x). (2.15) In a similar fashion to the Lagrange expressions we put{u,v}ba= {u,v}(b)− {u,v}(a).
Suppose the functionsu,v∈L2w(a,b) possess enough pseudo-derivatives to form the ex- pressionsa(u,v),u,vwand{u,v}ba, then
a(u,v)= u,vw− {u,v}ba. (2.16) Obviously, if all parts of the above equation exist, then
a(u,v)= u,vw (2.17)
if and only if{u,v}ba=0. For convenience, the following theorem is reproduced from [6].
Theorem2.3. For everyu∈Ᏸ0andv∈Ᏸ,a(u,v)exists and
L0u,vw=a(u,v)= u,Lvw. (2.18) Proof. LetV1=Ᏸ0equipped with the graph topology of the operatorL0. ThenV1is a Hilbert space. Lety∈Ᏸ0. Then
a(y,v)=L0y,vw≤L0ywvw
≤L0yV1vw. (2.19)
Hence,a(·,v) is continuous onᏰ0in the topology ofV1. SinceV1is the closure ofᏰ0in this topology, thena(·,v) is continuous onV1. On the other hand, sinceL0is the closure ofL0, there exists a sequence{un}inᏰ0such thatun→uandL0un→L0uinL2w(a,b).
Therefore,un→uinV1. Thusa(un,v)→a(u,v). That is,a(u,v) exists. Also a(u,v)=limaun,v=limL0un,vw=
L0u,vw= u,Lvw. (2.20)
It immediately follows from (2.18) that
{u,v}ba= {v,u}ba=0 (2.21) for all u∈Ᏸ0 andv∈Ᏸ. Hence the description (2.6) of the domain of the minimal operatorᏰ0may be sharpened to
Ᏸ0=
u∈Ᏸ:{u,v}ba= {v,u}ba=0∀v∈Ᏸ. (2.22) 3. Weak formulation
Note that the first and last expressions in (2.18) require 2n pseudo derivatives to be formed whereas the middle expression requires onlynderivatives. We are thus led to con- sidering the problem of obtaining a weak formulation for the expressionin spaces that require onlynderivatives. In this section we give such a formulation within the frame- work of the spaceL2w(a,b). As stated in the introduction, no assumptions are being made about the semiboundedness of the operators or the forms involved.
Define the following dense subspaces ofL2w(a,b):
ᐂ=
u∈L2w(a,b) : supp(u)⊂(a,b) compact,u∈AC(n−1)(a,b),u(n)∈L1(a,b), ᐆ=
u∈L2w(a,b) :u∈AC(n−1)(a,b), u[n]∈L∞loc(a,b), ᐆ0=
u∈ᐆ:{v,u}ba=0∀v∈Ᏸ.
(3.1) Some comments on the choice of the above spaces are now in order. The choice of the spaceᐂwas mainly motivated by the requirement thatᏰ0⊂ᐂ. This requirement, to- gether with the general assumptions we made about the coefficient functions, grant only the local integrability of the derivatives of the functions inᐂ. The spaceᐆ is so cho- sen to include the spaceᏰwhose functions have 2n−1 absolutely continuous pseudo- derivatives on the interval (a,b). Consequently, for a functionu∈Ᏸ,u[n]=p0u(n)∈ AC(a,b). From this one could infer a localLpproperty for anyp, 1≤p≤ ∞. The choice ofL∞loc(a,b) is forced by the natural duality with the properties of the spaceᐂin order to insure the existence of the integralsabu[n]v(n). Finally the spaceᐆ0is chosen to include Ᏸ0and, at the same time not to exceed the differentiability properties granted by func- tions in the spaceᐆ. It will be shown below that these spaces are dense inL2w(a,b) and give rise to a satisfactory theory for the weak formulation of the singular differentiable operators.
One is interested, in general, in solving variational equations of the form
a(u,v)= f,vw, (3.2)
where f ∈L2w(a,b) andvvaries in some convenient spaceᐃ. The equality (3.2) means that a continuity requirement with respect to the norm · whas to be imposed on the forma(u,·) overᐃ. As we will see, this continuity requirement plays a crucial role in recovering the domains of definition of the operators associated with. Since this is the
only continuity property we are going to need, the phrase “with respect to norm · w” will be dropped from this point on.
Lemma3.1. a(·,·)is defined onᐆ×ᐂ.
Proof. Letu∈ᐆ,v∈ᐂand suppose that supp(v)=[α,β]⊂(a,b).
b
au[n]v(n)= β
αu[n]v(n)≤u[n]L∞(α,β)v(n)L1(a,b) (3.3) and fork=0, 1,. . ., (n−1)
b
a pn−ku(k)v(k)= β
α pn−ku(k)v(k)≤u(k)v(k)L∞(α,β)
β
α
pn−k. (3.4)
Hence,a(u,v) exists.
Lemma3.2. Foru∈ᐆ0andv∈Ᏸ
a(u,v)= u,Lvw. (3.5)
Proof. Foru∈ᐆ0andv∈Ᏸ,u,Lvw exists and, from the definition ofᐆ0,{v,u}ba=0, hence (see the Preliminaries)a(u,v) is defined and the result follows from (2.16).
Theorem3.3. For f ∈L2w(a,b), the following are equivalent:
(I)u∈Ᏸ,Lu= f,
(II)u∈ᐆ,a(u,v)= f,vw∀v∈ᐂ.
In this case we may write
a(u,v)= Lu,vw ∀v∈ᐂ. (3.6)
Proof. Suppose (I) holds. By the definition of Ᏸ, u,u[n] ∈AC(a,b). Hence, u[n] is bounded on any compact subinterval of (a,b). Therefore,u[n]∈L∞loc(a,b). That is,u∈ᐆ.
Next letv∈ᐂand suppose that supp(v)=[α,β]⊂(a,b). Then, with the help of the def- initions (2.2) of pseudoderivatives,
f,vw= Lu,vw= β
αu[2n]v= β
α pnuv−
u[2n−1]v,
= β
αpnuv− β
α
u[2n−1]v
since β
αpnuvexists
= β
αpnuv+ β
αu[2n−1]v
= ···
= n k=0
β
α pn−ku(k)v(k)= β
α
n k=0
pn−ku(k)v(k)=a(u,v).
(3.7)
On the other hand, suppose (II) holds. Suppose v∈D(a,b). Since u∈ᐆ then (pn−ku(k))(k)∈L(a,b), 0≤k≤n (see (2.1)). Hence, nk=0(−1)k(pn−ku(k))(k)∈L(a,b).
On the other hand
f,vw=a(u,v)= b
a
n k=0
pn−ku(k)v(k)
= n k=0
β
αpn−ku(k)v(k)
= n k=0
(−1)(k)pn−ku(k)(k),v
= n
k=0
(−1)kpn−ku(k)(k),v
.
(3.8)
Sincew f ∈L1loc(a,b), we get
u[2n]=w f inL1loc(a,b). (3.9)
We proceed to show thatu∈Ᏸ.u∈L2w(a,b) by the definition ofᐆ. From (3.9) we get u[2n−1]=pnu−w f . (3.10) Since the right-hand side of the above equation is integrable over any compact subinter- val of (a,b) it follows thatu[2n−1]∈AC(a,b). In a similar fashion and with the help of the recursion (u[2n−k−1])=pn−ku(k)−u[2n−k],k=0, 2,. . ., (n−1) we get thatu[2n−k]∈ AC(a,b),k=1, 2,. . .,n. The definition ofᐆgivesu[n−k]∈AC(a,b),k=1, 2,. . .,n. From
this and (3.9) again we get thatu∈ᏰandLu=f.
Corollary3.4. Foru∈Ᏸ, the mappinga(u,·)is continuous onᐂ.
Proof. Foru∈Ᏸwe have byTheorem 3.3
a(u,v)= Lu,vw ∀v∈ᐂ. (3.11)
Hence,a(u,·) is continuous onᐂ.
Next we will show thatᏰis precisely the subspace ofᐆfor which the continuity prop- erty of the previous corollary holds. Before establishing this we need the following prop- erty.
Lemma3.5. Ᏸ0⊂ᐆ0∩ᐂ.
Proof. Letu∈Ᏸ0. Clearlyusatisfies the two properties defining the spaceᐆ. On the other hand, let
p0u(n)=g. (3.12)
Thengis absolutely continuous on the support ofu. Furthermore, u(n)= g
p0, (3.13)
therefore the local integrability of 1/ p0implies the integrability ofu(n). Thus,u∈ᐂ.
We remark here that the above lemma asserts also that the spacesᐆ,ᐆ0,ᐂare dense inL2w(a,b).
Theorem3.6. Ᏸ= {u∈ᐆ:a(u,·)is continuous onᐂ}.
Proof. Denote the right-hand side of the above equation byᏰ1. Foru∈Ᏸ1define the antilinear functionalGu(·) onᐂby
Gu(v)=a(u,v). (3.14)
Then Gu(·) is continuous on ᐂ. Since ᐂ is dense inL2w(a,b) we can extend Gu(·) to all of L2w(a,b). Hence, by the Riesz representation theorem, there is a unique element Tu∈L2w(a,b) such that
Gu(v)= Tu,vw ∀v∈ᐂ. (3.15)
Now notice thatᏰ⊂Ᏸ1and foru∈Ᏸwe have
Tu,vw=a(u,v)= Lu,vw ∀v∈ᐂ. (3.16) This means that the operatorTis densely defined and agrees withLonᏰ. That is,L⊂T.
It follows thatT∗⊂L∗=L0. ThereforeT∗is a symmetric closed operator. Forv∈Ᏸ0
with supp(v)=[α,β],u∈Ᏸ1we have
Tu,vw=a(u,v) sinceᏰ0⊂ᐂ
= b
a
n k=0
pn−ku(k)v(k)
= β
α
n k=0
pn−ku(k)v(k)
= n k=0
β
αpn−ku(k)v(k)
= n k=0
(−1)k β
αupn−kv(k)(k)
=
u,L0vw.
(3.17)
This means thatv∈Ᏸ(T∗) andT∗v=L0v. Thus we have the chain of operators L0⊂ T∗⊂L0. This yieldsT∗=L0and, hence,T⊂T∗∗=L∗0 =L.
In analogy with this result, we have the following theorem.
Theorem3.7. SupposeIm(λ)=0, then
(1)ᏺλ= {u∈ᐆ:a(u,v)=λu,vw∀v∈ᐂ} (2)Ᏸ0= {u∈ᐆ0:a(u,·)is continuous onᐂ}.
Proof. (1) This part is an immediate consequence of Theorems3.3and3.6, and the den- sity ofᐂinL2w(a,b).
(2) Let
Ᏹ0=
u∈ᐆ0:a(u,·) is continuous onᐂ. (3.18) Ifu∈Ᏸ0thenu∈ᐆ0and
a(u,v)= Lu,vw
=
L0u,vw ∀v∈ᐂ, (3.19)
that is, a(u,·) is continuous on ᐂ. Hence,u∈Ᏹ0. On the other hand, ifu∈ Ᏹ0, thena(u,·) is continuous onᐂand can be extended by continuity to all of L2w(a,b). In particulara(u,·) is continuous onᏰand, byLemma 3.2,
a(u,v)= u,Lvw ∀v∈Ᏸ. (3.20)
Hence, the mapping v→ u,Lvw is continuous onᏰ. Therefore,u∈D(L∗)= D(L0)=Ᏸ0.
As was stated in the preliminaries, the subspacesᏰ0,ᏺλ,ᏺλare linearly independent.
Since the spaceᐆ0is a superspace ofᏰ0, the question now arises as to whether the same is true about the spacesᐆ0,ᏺλ,ᏺλ. The affirmative answer is a special case of the following lemma.
Lemma3.8. A set of functionsw1,w2,. . .,wk∈Ᏸare linearly independent moduloᏰ0if and only if they are linearly independent moduloᐆ0.
Proof. The sufficiency part of this lemma is obvious since Ᏸ0 is a subspace of ᐆ0. To show the necessity part, assume the functionsw1,w2,. . .,wk∈Ᏸare linearly independent moduloᏰ0and there exist complex numbersα1,α2,. . .,αksuch that
ϕ k i=1
αiwi∈ᐆ0. (3.21)
Sinceϕ∈Ᏸwe can write
ϕ=ϕ0+ϕ1 (3.22)
withϕ0∈Ᏸ0andϕ1∈ᏺλ+ᏺλ. It follows thatϕ1∈ᐆ0, and, since we also haveϕ1∈Ᏸ, we have byTheorem 3.3
aϕ1,v=
Lϕ1,vw ∀v∈ᐂ. (3.23)
Hence, by Part 2 ofTheorem 3.7,ϕ1∈Ᏸ0. Thusϕ1=0 andϕ∈Ᏸ0. This necessarily gives
α1=α2= ··· =αk=0.
We next give a characterization of self-adjoint extensions ofL0 in terms of unitary operators between the spacesᏺλ andᏺλand the spaceᐆ0. The following theorem may be regarded as a counterpart ofTheorem 2.1.
Theorem3.9. SupposeLis a self-adjoint extension of the operatorL0with domain of defi- nitionᏰ and corresponding unitary operatorU. Define the spaceᐆ by
ᐆ=ᐆ0(U+I)ᏺλ. (3.24)
Then
Ᏸ =
u∈ᐆ:a(u,·)is continuous onᐂ. (3.25) Conversely, ifU:ᏺλ→ᏺλis a unitary operator andᐆ is defined by (3.24), then the set Ᏸ defined by (3.25) is the domain of definition of a certain self-adjoint extensionLofL0. Proof. Denote the right-hand side of (3.25) byᏰ1. It is straightforward to check that Ᏸ ⊂Ᏸ1. On the other hand, foru∈Ᏸ1, writeu=u0+ (U+I)uλ. Forv∈ᐂwe get
a(u,v)=au0,v+λU+λIuλ,vw. (3.26) The continuity ofa(u,·) and(λU+λI)uλ,·w onᐂimply the continuity ofa(u0,·) on ᐂ. Sinceu0∈ᐆ0, we get, by the second part ofTheorem 3.7, thatu0∈Ᏸ0. Hence,u∈Ᏸ. The converse statement follows from the characterization inTheorem 2.1and the first part of this theorem since the definition ofᏰ implies that
Ᏸ =Ᏸ0(U+I)ᏺλ. (3.27)
4. Further properties and characterizations
In this section, we give further properties and alternative characterizations of the weak spacesᐆ,ᐆ0 and the domains of self-adjoint extensions ofL0in terms of the so called
“boundary condition functions.”
It was shown in the previous section thata(·,·) is defined onᐆ×ᐂ. SinceᏰ0⊂ᐂ, thena(·,·) is defined onᐆ×Ᏸ0and, for a fixedu∈Ᏸ0, the mappingv→a(v,u) is con- tinuous onᐆ. The question is, how far can we push the spaceᏰ0and retain continuity onᐆ? The answer is in the corollary to the following lemma.
Lemma4.1. For everyu∈ᐆandv∈Ᏸ0,a(u,v)exists, a(u,v)=
u,L0vw (4.1)
and, consequently,{v,u}ba=0.
Proof. The proof is similar to thatTheorem 2.3withᏰreplaced byᐆ.
Corollary4.2. For everyu∈Ᏸ0, the mappingv→a(u,v)is continuous onᐆ.
We also have the following weakened definition of the spaceᐆ0.
Lemma4.3. ᐆ0consists precisely of all functionsu∈ᐆwhich for a fixed non-realλsatisfy
{ϕ,u}ba=0 (4.2)
for all functionsϕ∈ᏺλ+ᏺλ.
Proof. Equation (4.2) is necessary sinceᏺλ+ᏺλ⊂Ᏸ. On the other hand, suppose a func- tionu∈ᐆsatisfies (4.2) for allϕ∈ᏺλ+ᏺλ. Letv∈Ᏸand writev=v0+ϕforv0∈Ᏸ0
andϕ∈ᏺλ+ᏺλ. Then, usingLemma 4.1, we get{v,u}ba= {v0,u}ba+{ϕ,u}ba=0. Hence,
u∈ᐆ0.
Corollary4.4. ᐆ=ᐆ0ᏺλᏺλ.
Proof. We remark first that, byLemma 3.8,ᐆ0ᏺλᏺλis a direct sum.
Clearlyᐆ0ᏺλᏺλ⊂ᐆ. On the other hand, letu∈ᐆand assumeϕ1,ϕ2,. . .,ϕ2d
form a basis forᏺλᏺλ. We claim that the matrix ({ϕk,ϕi}ba) has full rank. To see this, assume the contrary. Then there exist scalarsθ1,θ2,. . .,θ2d, not all zeros, such that
2d i=1
θi ϕk,ϕib
a=0, k=1, 2,. . ., 2d. (4.3) Define the functionv=2d
i=1θiϕi. It follows from the above equation that{ϕk,v}ba=0, k=1, 2,. . ., 2d. Hence, by theLemma 4.3,v∈ᐆ0. Sinceϕ1,ϕ2,. . .,ϕ2d are linearly inde- pendent modulo ᐆ0, we must haveθ1=θ2= ··· =θ2d =0, which is a contradiction.
Now letα1,α2,. . .,α2dbe the solutions of the linear system ϕk,uba=
2d i=1
αi
ϕk,ϕi
b
a, k=1, 2,. . ., 2d, (4.4) and letϕ=2d
i=1αiϕi,u0=u−ϕ. It is easy to check that{ϕk,u0}ba=0,k=1, 2,. . ., 2d.
Therefore,u0∈ᐆ0, from which we get thatᐆ0ᏺλᏺλ⊃ᐆ.
Lemma4.5. Supposeϕ1,ϕ2,. . .,ϕ2d are2dfunctions inᏰwhich are linearly independent moduloᐆ0. Then
ᐆ0=
u∈ᐆ:ϕk,uba=0,k=1, 2,. . ., 2d,,
ᐆ=ᐆ0spanϕ1,ϕ2,. . .,ϕ2d. (4.5) Proof. Choose aλ∈Cwith Im(λ)=0 and letψ1,ψ2,. . .,ψ2dbe a basis forᏺλᏺλ. Then we can write
ϕk=θk+ 2d i=1
αkiψi, k=1, 2,. . ., 2d, (4.6)
whereθk∈Ᏸ0andαki’s are scalars. We claim that the 2d×2dmatrix [αki] has full rank.
To show this assume that there exist scalars γ1,γ2,. . .,γ2d such that2di=1αikγi=0, k= 1, 2,. . ., 2d. It follows that 2di=1γiϕi = 2d
i=1γiθi. That is, 2di=1γiϕi ∈ Ᏸ0. Since ϕ1,ϕ2,. . .,ϕ2d are linearly independent moduloᏰ0, thenγ1=γ2= ··· =γ2d=0. Hence, we can write
ψk=θk+ 2d i=1
βkiϕi, k=1, 2,. . ., 2d, (4.7) withθk∈Ᏸ0. The results now follow from (4.6), (4.7),Lemma 4.3and its corollary.
We turn now to characterizations of domains of self-adjoint extensions ofL0that par- allelTheorem 2.2. It was shown in [11] that the domain of definitionᏰ of self adjoint extensionsLofL0are characterized by functionsw1,w2,. . .,wd∈Ᏸsatisfying conditions 1, 2 ofTheorem 2.2such that
Ᏸ=Ᏸ0spanw1,w2,. . .,wd. (4.8) Define the space
ᐆ=
u∈ᐆ:wi,uba=0,i=1,. . .,d. (4.9) Lemma4.6. For everyu∈ᐆandv∈Ᏸ, a(u,v)exists,{v,u}ba=0and
a(u,v)=
u,Lvw. (4.10)
Proof. Letu∈ᐆandv∈Ᏸ and write v=v0+
d i=1
αiwi (4.11)
withv0∈Ᏸ0. UsingLemma 4.1we get {v,u}ba=
v0,uba+ d i=1
αiwi,uba=0. (4.12) Furthermore, sinceu,Lvwexists, it follows thata(u,v) exists. Equation (4.10) now fol-
lows from (2.16).
The following two theorems give a characterization of a class of self-adjoint extensions ofL0.
Theorem 4.7. Suppose Ᏸ is the domain of definition of a self adjoint extensionL ofL0
corresponding to the functionsw1,w2,. . .,wd. Define spaceᐆby (4.9) and the domain Ᏸ1=
u∈ᐆ:a(u,·)is continuous onᐂ, (4.13)
then
(1)Ᏸ1⊂Ᏸ,
(2)Ᏸ1=Ᏸ if and only if{wi,wj}ba=0,i,j=1, 2,. . .,d.
Proof. (1) The proof of this part follows the same lines of that of the second part of Theorem 3.7.
(2) IfᏰ1=Ᏸ thenw1,w2,. . .,wd∈Ᏸ1⊂ᐆ. Therefore, {wi,wj}ba=0,i,j=1, 2,. . .,d.
On the other hand, if{wi,wj}ba=0,i,j=1, 2,. . .,d, then, foru∈Ᏸ⊂ᐆwe may write
u=u0+ d i=1
αiwi (4.14)
withu0∈Ᏸ0. It is easy to check that{wi,u}ba=0, i=1,. . .,d implyingu∈ᐆ.
Hence,Ᏸ⊂Ᏸ1.
The foregoing theorem tells us that domains of the type (4.9) cannot be hoped to char- acterize all self-adjoint extensions ofL0. They rather characterize extensions for which the boundary condition functions satisfy{wi,wj}ba=0,i,j=1, 2,. . .,d. This class of ex- tensions will be called Class I. The following converse theorem applies to this class.
Theorem4.8. Suppose there exist functionsw1,w2,. . .,wd∈Ᏸsuch that (1)w1,w2,. . .,wdare linearly independent moduloᐆ0
(2){wi,wj}ba=0,i,j=1, 2,. . .,d.
Then the set Ᏸ =
u∈ᐆ:wj,uba=0, j=1, 2,. . .,danda(u,·)is continuous onᐂ (4.15) is the domain of definition of a certain Class I self-adjoint extensionLofL0.
Proof. Conditions 1, 2 above give thatw1,w2,. . .,wdare linearly independent moduloᏰ0
and [wi,wj]ba=0,i,j=1, 2,. . .,d. Then, byTheorem 2.2and (4.8), the set
Ᏸ1=Ᏸ0spanw1,w2,. . .,wd (4.16) is the domain of definition of a certain self-adjoint Class I extensionLofL0. Hence, by Theorem 4.7,
Ᏸ1=
u∈ᐆ:wj,uba=0, j=1, 2,. . .,danda(u,·) is continuous onᐂ. (4.17)
That is,Ᏸ1=Ᏸ.
For the more general conditions [wi,wj]ba=0,i,j=1, 2,. . .,dwe may defineᐆ by ᐆ=ᐆ0spanw1,w2,. . .,wd
, (4.18)
