Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
858
I IIIII
lraE .A Coddington
Hendrik .S .V ed Snoo
ralugeR Boundary Value smelborP
Associated with Pairs of
Ordinary Differential snoisserpxE
galreV-regnirpS
Berlin Heidelberg New York 1891
Authors
Earl A. Coddington
Mathematics Department, University of California
Los Angeles, California 90024/USA
Hendrik S. V. de Shoo
Mathematisch Instituut, Rijksuniversiteit Groningen
Postbus 800, 9700 AV Groningen, The Netherlands
AMS Subject Classifications (1980): 34 B ,xx 47 A 70, 49 G xx
ISBN 3-540407064 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10706-1 Springer-Verlag NewYork Heidelberg Berlin
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Preface
Numerous papers have been devoted to the study of eigenvalue
problems associated with pairs L,M of ordinary differential
operators. They concern the solutions f of Lf = hMf subject
to boundary conditions. In an earlier paper [9] we showed how
these problems have a natural setting within the framework of sub-
spaces in the direct sum of Hilbert spaces. In these notes we work
out in detail the regular case~ where the coefficients of the opera-
tors L and M are nice on a closed bounded interval ~ and M
is assumed to be positive definite, in the sense that
(Mr, f)2 ~ c2(f,f)2 , f ~ ~0(~) ~ for some constant c > 0. It is
hoped that this detailed knowledge of the regular case will lead to
a greater understanding of the more involved singular case, where
L and M are defined on an arbitrary, possibly unbounded, open
interval.
The work of E. A. Coddington was supported in part by the
National Science Foundation, and the work of H.S.V. de Snoo was
supported by the Netherlands Organization for the Advancement of
Pure Research (ZWO).
Earl A. Coddington
Los Angeles~ California
Hendrik S. V. de Snoo
Groningen, The Netherlands
November1980
Contents
Page
i. Introduction . . . . . . . . . . . . . . . . . . . . . 1
2. Seifadjoint extensions of ~ . . . . . . . . . . . . 21
3. Forms generated by selfadjoint extensions of ~ . . . 36
4. Hilbert spaces generated by positive selfadjoint
extensions of ~ . . . . . . . . . . . . . . . . . . . 54
5. Minimal and maximal subspaces for the pair L, M . . o 64
6. Intermediate subspaces . . . . . . . . . . . . . . . . 70
7. Spectra and eigenvalues . . . . . . . . . . . . . . . 106
8. Resolvents . . . . . . . . . . . . . . . . . . . . . . 138
9- Eigenfunction expansions for selfadjoint subspaces . . 168
10. Semibounded intermediate subspaces . . . . . . . . . . 183
ii. Some special cases . . . . . . . . . . . . . . . . . . 2O5
References . . . . . . . . . . . . . . . .... ~ • • 22o
Index . . . . . . . . . . . . . . . . . . . . . 224
i. Introduction.
It is well known that two hermitian n × n matrices K, H~ where H is posi-
tive definite, H • 0, can be simultaneously diagonalized. The key to the proof is
to consider C n, where C is the complex number field, as a Hilbert space ~H with
,
the inner product given by (f,g) = g Hf, where f,g e C n, considered as a space of
column vectors. Then the operator A = H-1K is selfadjoint in ~H' and the spectral
theorem readily yields the result. Of course such A, when K is not hermitian,
can also be investigated in ~H" We consider a similar problem where K, H are
replaced by a pair of ordinary differential expressions L and M, where M > 0 in
some sense. Two difficulties arise: (1) there are many natural choices for a self-
adjoint H • 0 generated by M, and hence many choices for ~H' and (2), once a
choice for H has been made, there are many choices for the analogue of A. In our
work we consider all possible choices for H • 0 and the analogue of A.
In [9] we initiated our study by considering a pair of ordinary differential
expressions L and M of orders n and v = 2~, respectively, acting on vector-
valued functions f: ~ ~- cm~ where ~ = (a,b) is an open real interval. As to L
and M we have
L= ~P~D n ,k M= ~%D k:M ,+ D:d/~,
k=O k=O
where Pk,Qk are m × m complex matrix-valued functions such that Pj • 6 C J(~),
j = 0,...,n, Qk ~ ck(~)' k = 0,..., v = 2~, and Qv(x) is invertible for all x c ,~
whereas Pn(X) is invertible for all x ~ ~ if n > v. We shall use the notations
already introduced in [9], and assume some familiarity with the main results proved in
that paper. In particular, we recall that if A is a linear manifold in ~2 = ~®~,
where ~ is a Hilbert space, its domain ~A) and range ~(A) are given by
~A) = [f e ~ I If,g] e A, some g c ,]~
~(A) = [g £ ~ I [f,g] e A, some f~].
denotes the operator in L2(~) given by
% : [[f,~] f ~ I Co(~)L
then we assumed that ~ was positive in the sense that there exists a positive
constant c(J) for each compact subinterval J C ~ such that
)i.1( (~'f)2 -> 'j,2)f'f(2))J(°< Co(V)" ~ f
Here we consider the regular case. By this we mean that ~ = (a~b) is a finite
Interval~ M is regular on 5 (that is~ the Qk can be extended to the closure
= [a,b] so that Qk e ck(~), k = 07..., v~ and %(x) is invertible for all
x e ~)~ and L is such that the Pj can be extended to ~ so that Pj e cJ(~)~
j = 0~...~n~ and Pn(X) is invertible on ? if n > v. When M is regular on
5 we identify the domain ~ ) = C0(5 ) of % with C0(~)~ the set of all
f e C~(~) which vanish outside some proper closed subinterval [c~d]~
< c < d < b. In the regular case the local inequality (i.i) implies a global one~
in that there exists a constant c > 0 such that
)2.1( ,rM( )f ~ 2 o2(f,f) ,2 c~(~); ~ f
see Remark 3, just prior to "Dirichlet integrals" in Section 3 of [9]. Therefore
it is (1.2) which will be the basic assumption about ~ in the regular case. This
implies that (-1)~Qv(x) > 0 (positive definite) for all x ~ ~ •
In Section 2 we consider a regular M of order v = 2~ on m ~, and character-
ize all intermediate operators H and their adjoints H* in L2(1) satisfying
M . c H c M in terms of homogeneous boundary conditions involving the quasi-
mln max
.
derivatives f ~] of f with respect to M. In the selfadjoint case H = H we
have
(1.3) H = [{f,Mf] s Mma x I Af[l ] " Bf[2 ] = 01 vm ,]
where A,B are vm x vm matrices satisfying
rank (A : B) = vm, BA = AB ,
and where f[l],f[2] are vm × 1 matrices with elements
~a),f[1](a),..., f[w-l](a), f(b),f[l](b) , .... f[~-l](b) and
f[v-l](a), .... f[~](a), - f[v-l](b), .... - f[~](b), respectively. In (i.3) the
null space of B,v(B), plays a particularly important role. If P is the ortho-
gonal projection of C vm onto v(B), we show in Theorem 2.4 that the H = H* in
(1.3) can also be described as
1
H = [[f,Mf] e Mma x I Pf[l] = 0vm' A~[1] = (I-P)f[2]]'
where H : A AH: B-A(I - P), and B is the generalized inverse of B. If
= (~l,...,~q) is a basis for v(B), the conditions Pf[l] 1 = O vm are equivalent
to ~ * f[l] = Oq, 1 and these equivalent conditions are called the essential boundary
conditions for H~ whereas the remaining conditions A~[1] = (I - P)f[2] are called
the natural boundary conditions for H.
In Section 3 we consider a fixed selfadjoint extension H of ~ and the space
~H : [f ~ AcP-l(~) ] Dpf c L2(m), ~*f[1] = OIl
generated by the essential boundary conditions for H. We have
(1.4)
(Hf'g)2 = g[l]A~[l] + (f'g)D' f e ~H), g ~ RH'
where ' )D is the Dirichlet form
b@
Ja j+l .
(f'g)D = ~ ~ (DDg)Qjk (Dkf)'
j=0 k=j-1
and the Qjk are such that
0=j ~=j-i
The right side of (1.4) makes sense for f,g RH, c and determines a semibilinear
form [ , ]H there,
(!.5) [f'g]H = ~l]AHf[!]+(f'g)D' f'g e ~H"
This form is analysed under the further assumption that (-l)~Qv(x) > 0, x ~ ~. In
Theorem 3.1 it is shown that [ , ]H is symmetric, bounded below, and closed, and
hence 0 M si bounded below. Moreover, ~H = ~[H], the set of all f c L2(~) for
which there exists a sequence fn e ~H) satisfying
nfII - llf 2 -~o, [fn " fm'fn - fm]H = nf(H( " fm)'fn - fm)2 *~ .O
For a sufficiently large ~ > 0 we can introduce an inner product ( ' )H on ~H
via
(f'g)H = [f'g]H + ~f'g)2' f'g ~ ~H'
and, with this, ~H becomes a Hilbert space such that ~H) si dense in H. R In
Theorem 3.4 a converse result is indicated. Given any set of q linearly independent
essential boundary conditions G f[1] * q = 01 and corresponding space
= [f s ~'i(7) I D ~f ~ 2(~), ~*f[1] = o~],
and given a symmetric semibilinear form
]g'fI + ~f[1] = gill 'D)g'f( f,g e ,~
on ~ (i.e., a hermitian R and functions Qjk' with Qjk(X) = Qkj(X) and
Q~(x) > ~0 x e ~)~ there exists a unique selfadjoint extension H of ~ in L2(5)
such that ~[H] = ~ and [f~g]H = [f~g]' for f~g e ~. The space
H19 = v(Mmax) ~D[H] n plays a key role; its dimension is determined in Theorem 3.5-
In case the Friedrichs extension ~ of ~
1 = 0
: [[f,Mf] s Mma x I f[l] vm ']
has a trivial null space we have dim 19 H = vm - q, and moreover
~[H] : ~[~] + ~H' a direct sum,
S×,f[ H=o, f~[~], ×cm H
Here $[~] is the closure of CO(V ) in ~D[H], considered as a Hilbert space.
If S si any symmetric operator in a Hilbert space ~ with inner product
( , ~) its lower bound m(S) is defined by
)S(er = inf[(Sf, )f I f e ~S), (f,f) = i].
If ~ is regular on ~ and ~) = ~MF) > O, so that
,fM( 2)f -> 0, ~ f ,)7(oC
then (-l)UQv(x) • 0 for x [~ • and the results of Section 3 are applicable.
Moreover, the von Neumarm extension N M of ~ is given by
: {{f,Mf] e Mma x I w[2]f[l ] - W[l]f[2 ] = 0vm],
where w : (Wl,..., Wwm ) is a basis for V(Mm~x) , and ~[~] = ~[~] + V(Mmax).
!
If H is a selfadjoint extension of ~ satisfying m(H) ~ 0, then ~[H] = ~H e)
1 1 1
and [f'g]H = (H2f'HZg)2 for f~g e ~[H], where ~ H is the positive square root
of H. If m(%) > 0, so that (1.2) is valid for 2 = c ~) = m(~) > 0~ then
(1.6) [f,f]H~_m(M0)(f,f)2 , f c ~H' H : MF,
and ~ = ~[MF] is a Hilbert space with the inner product [ , ]M F = ( ' )D'
which we denote by ~M" The relation (1.6) implies that V(MF) = [0], and this is
equivalent to the invertibility of the matrix ~[i]' so that
= {[f,~] ~ M=x I ANf[i = ] f[2]],
where
* , * ,-i *
N N = = A A <m[l )] w[2]"
If Mini n C H = H C Mma x and > re(H) 0, then 2H is a Hilbert space with the inner
~
product ( ' )H = [ ' ]H' which we denote by . We have
~H = ~M @ NH'
an orthogonal sum, dim ~H = vm - q, and on ~H the norm I III H is equivalent
to the norm II +fI given by
T i211f + : ~ I-~ ~IlfJDII + ~IIf~Dll ,
j=0
where
