By .~KE

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ARKIV F(JR MATEMATIK Band 8 nr 6

1.68 13 1 Read_ 23 October 1968

Complementary r e m a r k s about the limit point and limit circle theory

By .~KE

PLEIJEL

A B S T R A C T

A m e t h o d to o b t a i n b o u n d s for the n u m b e r of solutions w i t h regular b e h a v i o u r a t infinity of selfadjoint o r d i n a r y differential equations is given. I n this method, which is presented for a p o l a r problem, t h e sequence of nested ellipsoids in a previous p a p e r of mine is replaced b y the use of certain projections. A t t h e same time I t a k e the o p p o r t u n i t y to complete t h e unsatisfactory bibliography of m y earlier paper.

O. Introduction

The first complete generalization of H. Weyl's spectral theory of ordinary differential operators was given by K. Kodaira in a celebrated paper in the American Journal of Mathematics [8]. I n the same year 1950 I. l~I. Glazman's treatment appeared.

The problem has also been considered b y E. A. Coddington [I], b y W. N. E v e r i t t [5-6] and others. T. Kimura and M. Takahasi [7] have treated the eigenvalue problems of the equation L u =2u in full generality i.e. for any formally selfadjoint ordinary differential operator L.

Bounds for the n u m b e r of integrable-square solutions of L u =~u, I m 2 ~ 0 , were deduced b y W. N. E v e r i t t in three papers [2-4] b y different methods. I n m y previous article [9] the corresponding results were obtained for the polar equation Lu=]cpu, where p does not have a constant sign. I n this case/5 is assumed to have a positive definite Dirichlet integral and the solutions are requested to give finite values to this Dirichlet integral instead of being square integrable. The present paper gives a different approach to the same result. The method was presented at the Scandi- navian Congress of Mathematicians a t Oslo in 1968. As L. H6rmander remarked in a private communication at the congress, the W e y l - E v e r i t t result for Lu=2u is a consequence of the existence of closed symmetric extensions of L in E~. For general polar problems no extension theory of this kind has been established.

1. The method

Let L be a linear, formally selfadjoint, ordinary differential operator defined and with "sufficiently regular" coefficients on the half-infinite interval a ~<x < ~ . I t is assumed t h a t L has a Dirichlet integral ~(., .) so t h a t

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A. PLEIJEL, Complementary remarks about the limit point and limit circle theory

f~Lu ~ =

ⁱ^i[B(~,

v)] + ~(u, v).

⁽¹⁾

This Dirichlet integral shall be increasing (non-decreasing) with b a n d positive definite in G m for large values of b. The n u m b e r m is the half of the necessarily even order M = 2 m of L. Assume I M P > 0 . On the solution space l = { u [ L u = 2 p u } the formula (1) leads to

~ - ~

i ~(u,v)=~(u,v)-h.(u, v), (2)

where h(u, v) is a quadratic (Hermitean) form in u, v a n d certain of their derivatives a n d with coefficients depending on x . A s in Pleijel [9],it is p r o v e d t h a t h has t h e m a x i - m u m signature Ira, m] a n d t h a t it assumes this signature on the solution space.

L e t P be a linear subspace of l on which ha(u, u) is non-negative a n d which has the m a x i m u m dimension m. As a consequence of (2), the f o r m hb(u, u) is positive definite on P if a < b < ¢0. On the linear subspace of t

2¢(b) -- {~] h,(~, P) = 0},

the f o r m ha(u, u) is negative definite and the dimension of N(b) is m. I t follows t h a t l = P 4 N ( b ) as a direct sum. L e t ~ E l a n d write v/=V(b)+U(b ), where V(b)EP,

U(b) eN(b). Since h~(P, N(b))=0, we find for a n y V E P t h a t

hb(~-- V, ~ - - V) = h~(v 2 - V(b), yJ- V(b)) +hb(V(b) - V, V(b) - V). (3) Since V ( b ) - V E P , the last t e r m is non-negative a n d h~(v2-V(b), yJ-V(b)) is the m i n i m u m of ha(v 2 - V, ~ - V) when V varies in P. L e t a<b<b'. Formula (2) shows t h a t hb(u, u) increases with b. We conclude t h a t

ha,(yJ - V(b'), yJ - V(b')) >1 ha(y~ - V(b'), y~ - V(b'))

= ha(v 2 - V(b), v 2 - V(b)) +hb(V(b) - V(b'), V(b) - V(b')), (4) so t h a t h~(yJ-V(b), v 2 - V(b)) is also increasing. B u t h~(yJ-V(b), v 2 - V(b))=ha(U(b),

U(b)) < O. Thus

lira ~(y~ - V(b), v 2 - V(b)) = G(~) <. O. (5)

b...-~ ¢~

According to (4)

hb.(~p-- V(b'), v 2 - V(b'))--hb(~p-- V(b), V - V ( b ) )

>1 ha(V(b ) - V(b'), V ( b ) - V(b')) >1 hx(V(b) - V(b'), V ( b ) - V(b')) ~ O, (6) where X is a m o m e n t a r i l y fixed number, a < X < b <b'. F r o m (6) it follows t h a t

lim hx(V(b ) - V(b'), V ( b ) - V(b'))=O.

b , b "--.),r~

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.4.RKI~r FOR ~ T E ~ 4 . T I K . B d 8 nr 6 B u t hx defines a positive definite metric on the finite dimensional and therefore closed space P. Hence, with a certain V in P ,

lim V(b) = V e P

holds in the sense t h a t limb-~ hx(V(b)-V, V(b)-V)=O for every X > a . F r o m (6) we can then conclude t h a t

lim tb(V(b ) - V, V(b)- V)=O (7)

b--~ co

b y performing the transitions to the limit b'-~ ¢o, b-~ co in this order.

From (3), (5), (7) it then follows t h a t limb-,~ hb(~ -- V, ~p- V) -- G(~). According to (2) this shows t h a t ~(~ - V, y~ - V) tends to a finite limit when b-~ ~ . Thus to a n y solution ~ of L u - ~ p u we can find another solution V in P such t h a t the difference yJ - V gives a finite value to the Dirichlet integral taken over a ~< x < ~ . F r o m this it follows t h a t there are at least m linearly independent solutions of L u =~lau which have finite Dirichlet integrals over a ~< x < co.

R E F E R E N C E S

1. CODDnCGTO~, E. A., The spectral representation of ordinary selfadjoint differential operators.

Ann. of Math. 60, 192-211 (1954).

2. EV~]~ITT, W. N., In~egrable-square solutions of ordinary differential equations. Quar~. J . Math. Oxford, Set. (2), 10, 145-155 (1959).

3. - - I n t e g r a b l e - s q u a r e solutions of ordinary differential equations II. Quart. J . Math.

Oxford, Set. ( 2 ) / 3 , 217-220 (1962).

4. - - Integrable-square solutions of ordinary differential equations :[II. Quart. J . Math.

Oxford, Ser. (2) 14, 170-180 (1963).

5. - - Singular differential equations I. The even order case, Math. Ann. 156, 9-24 (1964).

6. - - Singular differential equations II. Some self-adjoint even order cases. Quart. J . Math.

Oxford, Set. (2) 18, 13-32 (1967).

7. K I ~ , T. a n d TA,rA~SI, M., Sur les op~rateurs diff~rentiels ordinaires lin~aires formelle- m e n t a u t o a d j o i n t s I. Funkeialaj Ekvacioj, Serio I n t e r n a c i a 7, 35-90 (1965).

8. KODAIRA, K., On ordinary differential equations of a n y even order a n d the corresponding eigenfunction expansions, Amer. J . Math. 72, 502-544 (1950).

9. PLEIJEL, .~., Some remarks a b o u t t h e limit point a n d limit circle theory. Ark. fSr Mat. 7:41, 543-550 (1968).

10. TAIrAVrASI, M., On a n o n selfadjoint ordinary differential operator. Funkcialaj Ekvaeioj, Serio I n t e r n a c i a 8:3, 109-141 (1966).

T r y c k t den 14 maj 1969

Uppsala ][969. Almqvist & Wiksells Boktryckeri AB

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