fl By

ARKIV FOR MATEMATIK Band 7 nr 41

1.68 06 1 Read 10 April 1968

S o m e r e m a r k s a b o u t the limit point a n d limit circle t h e o r y By

J~kKE P L E I J E L

ABSTRACT

Let L be a formally selfadjoint differential operator and p a real-valued function, both on a ~<x< co. The deficiency indices are the numbers of solutions of Lu = ~ p u for Im ~t >0 and for Im ),< 0 which have a certain regularity at x = c~. (A) If p(x) >~ 0 this regularity means that the integral of p(x) [u] ~ converges at infinity. (B) If p changes its sign for arbitrarily large values of x but L has a positive definite Dirichlet integral it is natural to relate the regularity to this integral. Weyl's classical study of the deficiency indices is reviewed for (A) with the help of elementary theory of quadratic forms. Individual bounds are found for the deficiency indices also when L is of odd order. It is then indicated how the method carries over to (B).

O. I n t r o d u c t i o n

If q is a c o n t i n u o u s r e a l - v a l u e d f u n c t i o n o n a ~<x< co a n d ~ a n o n - r e a l p a r a - m e t e r , t h e n

- u " + q u = ;~u

(0.1)

has a t least one s o l u t i o n which is s q u a r e - i n t e g r a b l e on a < ~ x < ~ . This r e s u l t was d e d u c e d b y H. W e y l i n his f u n d a m e n t a l t r e a t i s e [3] o n spectral properties of o r d i n a r y differential e q u a t i o n s . I f t h e e q u a t i o n is replaced b y

- u " + q u = ,~ p u ,

(0.2)

where p a n d q are r e a l - v a l u e d f u n c t i o n s , W e y l ' s m e t h o d c a n be a p p l i e d e v e n if p t a k e s b o t h positive a n d n e g a t i v e v a l u e s b u t has a definite sign for sufficiently large v a l u e s of x. I f t h e last a s s u m p t i o n is a b a n d o n e d , b u t q is n o n - n e g a t i v e , i t seems n a t u r a l t o relate t h e b e h a v i o u r a t i n f i n i t y of t h e solutions t o t h e Dirichlet i n t e g r a l

f[l

^{'l +ql 5}

As a base for such considerations we shall p r e s e n t W e y l ' s m e t h o d i n its de- p e n d e n c e o n q u a d r a t i c b o u n d a r y forms. F o r (0.1) W e y l p r o v e d t h a t g i v e n a n y s o l u t i o n ~p o n e c a n f i n d a n o t h e r one, v, s a t i s f y i n g a b o u n d a r y c o n d i t i o n a t x = a (for i n s t a n c e v ( a ) = 0) such t h a t y ~ - v becomes s q u a r e - i n t e g r a b l e over a < ~ x < ~ . T h i s r e s u l t c a n be generalized to a n y f o r m a l l y s e l f a d j o i n t e q u a t i o n of e v e n o r d e r

A. PLEIJEL, Limit point and limit circle theory

(Everitt [1]). B u t for equations of odd order a set of solutions satisfying a b o u n d a r y condition at x = a is generally too small for the purpose. However, according to the elementary theory of quadratic forms one can instead use a linear set of solutions determined b y a quadratic b o u n d a r y inequality at x = a.

I n this way Weyl's result can be generalized to formally selfadjoint equations of a r b i t r a r y order.

The numbers of square-integrable solutions of L u = ~u on 0 ~<x< co for Im~t > 0 a n d for I m ~t< 0 are the deficiency numbers of the selfadjoint operator/5. A com- plete hilbert space t r e a t m e n t of real selfadjoint differential operators of a r b i t r a r y (even) order was first given b y I. M. Glazman [2]. I n a footnote Glazman ob- served t h a t the order (even or odd) of a differential operator is a lower bound for the sum of its deficiency numbers b u t gave no individual estimates.

After the presentation of Weyl's method we indicate how it carries over to L u = ,~pu when the integral

p u g

g

is indefinite in an essential w a y b u t the differential operator has a positive de- finite Dirichlet integral.

1. The differential operator L e t the linear differential operator

M 1 d

L = ~ A j ( x ) D j, D =

j~o ~ - - ~ dx

be defined on an open interval I containing a<~x< c~. The coefficients At(x ) are assumed continuous a n d so regular t h a t the adjoint

M

L * = ~ DJAj(x)

1 = 0

can be formed. L e t L be formally selfadjoint i.e. let L a n d L* coincide. Then Green's formula reads

I ~ / ~ - u L v = i [k(u, v)]. (1.1) Here the " b o u n d a r y f o r m " k(u, v) is linear in u a n d hermitean. Thus ]r is a quadratic form with coefficients depending on x. B y computation

m - - 1

k(u, v) = ~ (DJvBju + D ~ u B j v ) - A2,n+ID'~vDmu (1.2)

iffiO

when M = 2 m or M = 2 m + 1, where m is an integer. I n the case M = 2m it is understood t h a t A2m+l=0 in (1.2). The B j u (?'= 0, 1, ..., ( m - 1 ) ) , are linear in u.

The expression for k(u, u) can be written

m - 1 m - 1

k(u, u)= j~01 I D]u + B]ul2_j~=o 89 [DJu _ Btu[2_ A2m+i ]Dmu] 2

(1.3) with A 2 m + l = 0 if M = 2 m .

On a finite dimensional linear set of functions t h e q u a d r a t i c f o r m k has a certain signature [~r, v] or [z,, v,] for e v e r y x. This means t h a t on t h e linear set k(u, u) can be w r i t t e n as a sum of x squares (of absolute values) minus v squares of altogether ~r + v linearly i n d e p e n d e n t linear forms. F r o m (1.3) we conclude t h a t

where

m + = m - = m

~z<~m +, v ~ m - , (1.4)

if M = 2 m / (1.5)

m + = m + l , m - = m if M = 2 m + l a n d A2m+l(x)<O, (1.6) m + = m , m - = m + l if M = 2 m + l a n d A2m+l(x)>O. (1.7) Observe t h a t in all cases m + + m - = M.

2. Solution space W e shall s t u d y solutions of t h e differential e q u a t i o n

L u = ~ pu

(2.1)

on a<~x< 0r Here p denotes a real-valued f u n c t i o n which for instance is con- tinuous. I t should n o t vanish identically. I n our p r e s e n t a t i o n of W e y l ' s m e t h o d (w167 2-5) we suppose

p(x)>~O for a<<.x< ~ . (2.2)

I n (2.1) ~ is a non-real parameter. To fix the ideas we assume in general t h a t

I m 2 > 0. (2.3)

The highest order coefficient of a formally selfadjoint differential operator is always real. We assume AM(X)mO for all x in a<~x< ~ . Because of this the e q u a t i o n (2.1) has M linearly independent solutions on a<~x< ~ which f o r m t h e solu- tion space

= { u I L u = ~ p u }

of dimension M . A solution cannot vanish on an interval w i t h o u t being 0 every- where.

3. Signature o f the boundary form

If u E l i.e. if u satisfies L u = ~ p u a n d if we p u t v = u in t h e Green's formula (1.1), we o b t a i n

k~(u, u) - k~(u, u) = ~ - j~ p lu] ~. (3.1)

A. PLEIJEL, Limit point and limit circle theory

L e t us assume t h a t t h e interval ~ ~< x ~< fl is so large t h a t p does n o t vanish iden- tically on this interval. Because of (2.3) the factor ( ~ - ~ ) / i is positive. Thus t h e r i g h t - h a n d side of (3.1) is positive definite.

T h e signatures of k~ a n d k n on l are [7t~, ~ ] , [ze~, v~]. T h e left-hand side of (3.1) can be written as a sum of ztz + v~ squares minus ze~ + v~ squares. W e shall first prove t h a t M<~zn+v~. Assume ztn+v~<M. I f we equate to 0 t h e ~r~+ v~ squares in k ~ ( u , u ) - k ~ ( u , u ) , we o b t a i n a linear subspace of l which because of ~ + v~< M has a positive dimension. B u t this is contradicted b y the fact t h a t (3.1) implies u = 0 for e v e r y u belonging to t h e subspace. Thus M ~<Te~+ v~.

B u t according to (1.4), our last inequality can be completed to M ~<:t~ + v ~ <

m + + m - = M. Hence, zt~= m + a n d v~= m - . This is true for a r b i t r a r y values of

~r a n d fl in the interval I , where L is defined. This shows that the signature [~, v]

o/ k on l is independent o/ x and that

~1~+~ ~ m - ~

where m + and m - are defined in (1.5)-(1.7).

4. Weyl's method

L e t a ~<x ~< b be a finite interval. The signature of ka on t is [m +, m - ] . According to t h e t h e o r y of quadratic forms there exist linear subspaces of 1 of dimension m +, b u t of no higher dimension, on which ka is positive definite. L e t l~ + be such a subspace

d i m l + = m +, k~(u,u)>~O on 1 +, equality only if u = 0 . (4.1) Similarly, since t h e signature of kb on 1 is [ m + , m - ] we can chose a linear sub.

space l ; of 1 such t h a t

dim l~ = m - , k~ (u, u) ~< 0 on l~, equality only if u = 0. (4.2) F o r a = a , fl=b t h e f o r m u l a (3.1) takes the f o r m

kb(u,u)_ka(u,u)=~_ ~ fb ^{vlul (4.3)}

I f u E l ; N l + the left-hand side of (4.3) is non-positive according to (4.1) a n d (4.2). Since t h e r i g h t - h a n d side of (4.3) is n o n - n e g a t i v e this gives u = 0. T h u s l ; N 1 + = {0}. B u t d i m l ; + dim 1 + = m - + m + = M which is t h e dimension of I. I t so follows t h a t t h e solution space equals t h e direct s u m

l = l ; 4 1 +. (4.4)

L e t yJ E I. On a c c o u n t of (4.4) we write yj = u § v, where u E l~ a n d v E 1 +. I n s e r t u = ~ - v in (4.3). Since kb(u,u)~<0 it follows t h a t

ka(~V-v,v-v)+ ~-~ f~plv-v[~<O. (4.5)

L e t ~1, 9)2, " ' ' , Cfm + be a base in la + a n d write v as a linear combination

m +

v = ~ tj~j.

j = l

F o r all v in l~ + we have/ca(v , v)/> 0 with equality only if v = 0. Thus, when ~[m~t > 0 , the p a r t of t h e left-hand side of (4.5) which is quadratic in T = t 1, t2, ..., tin§ is positive definite on la +.

Geometrically (4.5) tells us t h a t T belongs to t h e interior or the b o u n d a r y of a m +-dimensional u n i t a r y ellipsoid Eb. Our reasoning shows t h a t t h e ellipsoid defined b y (4.5) is n o t e m p t y since there exists a function v for which (4.5) is satisfied. I f b' > b t h e definition b y (4.5) of t h e ellipsoids E~ shows t h a t E b, ~ Eb.

Thus there is at least one point T which belongs to all t h e c o m p a c t sets Eb.

L e t v = tlq~l + t2~2 § ... + tm+~Prn + be t h e function in 1 + which corresponds to such a T. F o r this v the inequality (4.5) is satisfied for all values of b. Consequently

[' plw-vl <o.

* J a I t follows t h a t J p l y ~ - v [ 2 <

i.e. t h a t ~ p - v E s ~ ; p ) . We have p r o v e d t h a t every solution ~p o/ L u = 2 p u can be "compensated" by a solution v in l + so that

~* = ~ - v ~ s (a, ~ ; p).

L e t Yh,~2 . . . . ,yJm- be a completion of ~1,~2 . . . era§ to a base of I. T h e n t h e c o m p e n s a t e d functions

~ = ~0j- v j e 1:2 (a, ~ ; p )

( j = 1, 2, . . . , m - ) , also f o r m a base together with ~1, q2 . . . q,n+.

Thus, L u = X T u has at least m - linearly independent solutions in ~2(a, ~ ; p ) . Under the condition I m ~t > 0 we have m - = m i/ the order o/ L is M = 2m, or i/

M = 2 m + l and A 2 m + l ( x ) < 0 , while m - = m + l i/ M = 2 m + l and A 2 m + l ( x ) > 0 . I / I m ) ~ < 0 the inequalities /or A2m+l(x) should be reversed.

5. Compensating f u n c t i o n s and boundary conditions

The signature of k a o n l is [m+,m-], where m + + m - = d i m l . According to t h e t h e o r y of quadratic forms there exist linear subspaees of I of dimension m i n (m +, m - ) , b u t of no higher dimension, such t h a t ka(u , v ) = 0 for e v e r y u a n d v belonging t o the subspace. L e t la ~ be such a m a x i m a l nullspace with respect to k a. Thus k ~ ( u , u ) = O if uEl~ If u E l ~ N l ~ (compare (4.2)) t h e relation (4.3) proves t h a t u = 0 .

The dimension of l~ is m - . I f M is even, or if M = 2 m + l a n d A2m +l (X) > 0, t h e n m + ~< m - a n d dim I ~ = m + (provided I m ~t > 0). I n these cases

1 = l; 4 l~ ~ (5.1)

A. PLEIJ'EL, Limit point and limit circle theory

B u t if (5.1) holds true we can (as Weyl actually did) use l ~ instead of 1 + in t h e reasoning of section 4 (after formula (4.4)). T h u s if M = 2m, or if M = 2m + 1 a n d A2m+l(x) > 0 , we can compensate a n y solution ~v of L u = ~tpu b y means of a solu- tion v in l ~ so t h a t ~ o - v becomes square-integrable with respect to p.

If M = 2 m + l a n d A2m+l(X)<O the relation (5.1) is n o t valid a n d a set la ~ is generally n o t sufficient to compensate every solution ~0 to square-integrability over a ~< x < c~,

A b o u n d a r y condition for L on a ~<x ~< b can be defined as a m a x i m a l null- space of t h e quadratic form

k b(u, v) - k a(u, v).

If M is even, a n y t w o subspaces of l which are m a x i m a l nullspaces l ~ a n d l ~ with respect to ka a n d k~ determine a b o u n d a r y condition. This is n o t true when M is odd b u t it seems a n y h o w fit to call a m a x i m a l nullspace with respect to ]Ca a boundary condition at x = a b o t h when M is even a n d odd. U n d e r this agree- m e n t the result of the present section can be f o r m u l a t e d as follows.

I / M = 2m or i/ M = 2 m + 1 and A2m+l(x) > 0 , we can use solutions which satis/y a boundary condition at x = a to compensate any 8olution o/ L u = ~ p u to square.

integrability on a <~x< ~ with respect to p. I / M = 2 m + l and

A2m+l(X)< O

this is not generally possible. These statements are true when I m ~ > 0 . The changes when I m ~t< 0 are obvious.

R e m a r k I. One m i g h t w a n t wider possibilities to find spaces of restricted dimen- sion which are sufficient to compensate a n y solution to square-integrability. F o r this we can note t h a t the m a x i m a l dimension of subspaces of l on which ka is positive (not necessarily definite) is also m +. A space l~ +'~ of this t y p e is suffi- cient for the compensation.

R e m a r k I I . F o r M = 2 t h e ellipsoids Ea are one-dimensional i.e. of t h e form

It- c(b) l -<< ~(b).

T h e y are circles in t h e complex t-plane a n d coincide with the classical W e y l circles. If b' > b t h e circle I t - c ( b ' ) ]

<e(b')

is contained in t h e circle [ t - c ( b ) [ <e(b).

I f for b-+ oo the circles shrink to a single point we are in W e y l ' s limit point case. I f t h e y shrink to a limit circle we are in the limit circle case a n d all solu- tions are square-integrable. If M > 2 the classification is similar b u t more elaborate.

6. Weyrs method for operators with positive Dirichlet integrals

A real, formally selfadjoint linear differential operator can be written L = ~ DJaj (x) D j

t=0

with real-valued functions aj(x). As in w 1, let L be given a n d sufficiently regular on an open interval I containing a ~<x< c~. B y partial integrations

f:

~ L u = i [B(u, v)] +

f:

D(u, v), (6.1)

~t

where

D(u, v) = ~ aj(x) D J u ~ v (6.2)

1=0 m L l _ _

and

B(u, v)= ~ DJvBju.

i = 0

The expressions

Bju

are linear in u. The Dirichlet form (6.2) is evidently her- mitean. I f for instance

aj(x)>~

0 ( ] = 0, 1 . . . . , m), the integral of

D(u, u)

from to fi is non-negative and increases when the interval :r ~<x < f l increases. We assume t h a t this is so a n d also t h a t if the interval ~ ~<x ~<fl is sufficiently large, then

f~

D ( u , u ) = O

holds true for a "regular" u only if

u(x)=

0 in a subinterval of g ~<x <ft. If these conditions are fulfilled we say t h a t L has a positive definite Dirichlet integral.

Interchanging u and v in (6.1) and taking the complex conjugate we obtain

uL-v = - i [B(v,

u)] +

D(u, v).

(6.3)

Let

L u = , ~ p u

and take

v = u .

Then (6.1) and (6.3) give

(6.4)

p[ul

2=

- i [B(u,

u)] +

D(u, u).

(6.5) F r o m (6.4), (6.5) it follows t h a t

h~(u, u) - h~(u, u) = 4 - ~ -['~D(u, u),

m - - 1

where

h(u, v) = ~. (Bju ,~DJv + ~DJu Bjv).

t = 0

(6.6)

(6.7) For the signature [z, v] of the evidently quadratic form h, considered on a n y finite dimensional linear set, it follows from (6.7) (see w 1) t h a t

xc~m, v ~ m .

I f I m ~ > 0 , the equality (6.6) shows t h a t

h~(u, u) - h:~(u, u)

is positive definite on the solution space

z= {ul Lu= pu}.

A reasoning similar to the one performed in w 3 proves t h a t the signature of h on 1 is independent of x and always equals [m, m].

A. PLEI,IEL, Limit point and limit circle theory

R e p e a t i n g , b u t w i t h (6.6) i n s t e a d of (4.3), w h a t was done i n w167 4 a n d 5 we arrive a t t h e result that every solution ~f of L u = 2 p u on a <~ x < c~ can be com- pensated by a solution v belonging to a m a x i m a l nullspace /or ha, so that u = y ~ - v gives a /inite value to

j

' ~ D(u, u).

c$

T h a t v belongs to a m a x i m a l nullspace for ha c a n be considered as a k i n d of b o u n d a r y c o n d i t i o n for v a t x = a. B u t i n general t h e b o u n d a r y c o n d i t i o n s a t x = a d e t e r m i n e d b y ha do n o t coincide w i t h t h e b o u n d a r y c o n d i t i o n s of w 5 which are r e l a t e d to ka. D i r i c h l e t ' s b o u n d a r y c o n d i t i o n a t a(DJu(a)= O, j = O, 1 . . . ( m - 1)) is a b o u n d a r y c o n d i t i o n a t a, b o t h with respect to ka a n d ha. Observe t h a t a m a x i m a l nullspace i n 1 w i t h respect to ha has d i m e n s i o n m. I / ~ denotes a hil- bert spa~e with norm

it follows that L u = 2 p u has at least m solutions in ~ . This is true i / L is real /ormally sel/adjoint of order 2m and has a positive de/inite Dirichlet integral. The result is essentially independent of p.

7. C o n c l u s i o n s

Clearly questions r e l a t e d to W e y l ' s result h a v e t h e i r similarities i n a t h e o r y a b o u t differential operators w i t h positive Dirichlet integrals. Some of these ques- t i o n s are easily settled, others give rise to c e r t a i n difficulties. I t is t h e i n t e n t i o n t o discuss some of t h e m i n a f o r t h c o m i n g p a p e r from t h e D e p a r t m e n t .

Mathematics Department, Uppsala University, Uppsala, Sweden

R E F E R E N C E S

1. EVERITT, W. •., Singular differential equations. I. The even order case. Math. Ann. 156 (1964), 9-24.

2. G L A Z ~ , I. M., On the theory of singular differential operators. Uspehi Mat. /gauk (N.S.) 5, no. 6 (40) (1950), 102-135. Also in A. M. S. Translations, series 1, vol. 4: Differential Equa- tions, pp. 331-372.

3. WEYL, H., ~ber gewbhnliche Differentialgleichungen mit Singularit~ten trod die zugehbrigen Entwicklungen willkiirlicher Funktionen. Math. Ann. 68 (1910), 220-269. Also in Selecta Hermann Weyl. Birkh~user Verlag, Basel and Stuttgart, 1956.

Tryckt den 30 augusti 1968 Uppsala 1968. Almqvist & Wiksells Boktryckeri AB