Singular Sturm-Liouville problems: The Friedrichs extension and comparison of eigenvalues

SINGULAR STURM-LIOUVILLE PROBLEMS: THE FRIEDRICHS EXTENSION AND COMPARISON OF EIGENVALUES H.-D. NIESSEN and A. ZETTL [Received 6 August 1990—Revised 21 March 1991]

ABSTRACT

A new characterization of singular self-adjoint boundary conditions for Sturm-Liouville problems is given. These are an exact parallel of the regular case. They are given explicitly in terms of principal and non-principal solutions. The special nature of the Friedrichs extension is clearly apparent and highlighted. Inequalities among the eigenvalues of different boundary conditions, separated and coupled, are obtained. Most of all we want to stress the method of proof. It is based on a very elementary transformation which transforms any singular non-oscillatory limit-circle endpoint into a regular one.

1. Introduction The purpose of this paper is to show that to every Sturm-Liouville problem which is bounded below there exists a very simple transformation, namely multiplication by a function, which transforms it into another Sturm-Liouville problem in such a way that every (finite) limit-circle endpoint is transformed into a regular one. Then many basic properties of singular problems can be inferred from corresponding regular ones. For example, we show that the characterizations of the Friedrichs extension of singular Sturm-Liouville problems in terms of principal and also in terms of non-principal solutions, due to Rellich [13] and Kalf [8] follow easily from the regular case. The inequalities among the eigenvalues for different boundary conditions given by Jorgens in [7] and the recent ones given by Weidmann in [15], both for regular problems, are extended to singular non-oscillatory eigenvalue problems in the limit-circle case. These seem to be new. Also, the asymptotic formula for the distribution of the eigenvalues which is well known for regular problems (see, for example, [1]) is generalized to these singular eigenvalue problems. All this follows readily from our transformation because with this transformation it is apparent which singular problem transforms into which regular one. The organization of the paper is as follows. After the introduction in § 1, principal and non-principal solutions are discussed in § 2, followed in § 3 by the study of the transformation Tv which serves as a main tool. In § 4 the relationship between semi-boundedness of the minimal operator, oscillation, and the Friedrichs extension is studied. The comparison of eigenvalues and the asymptotic formula for the distribution of the eigenvalues is given in §5, examples are discussed in § 6, and § 7 is devoted to the proof of Lemma 2.3. This proof is given here because it is long and technical. Part of this work was done while the second-named author was a visiting Professor at the University of Essen. His stay was supported by a DFG-grant. 1991 Mathematics Subject Classification: 34B24, 34C10, 47A20, 47E05. Proc. London Math. Soc. (3) 64 (1992) 545-578.

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2. Principal and non-principal solutions Let M be the differential expression defined by Afy : = - [ - ( / ? / ) ' + qy] onl = (a,b),

with - » *£ a < b ^ ».

(2.1)

Throughout this paper we assume that p,q,w:

/-»R,

p,tv>0a.e,

- , q, w e L 1OC (/). P

(2.2)

Consider the equation A/y = A_y, that is, -(py'Y

+ = ^wy

on I, with A e R.

(2.3)

The local integrability conditions of (2.2) ensure that the zeros of any non-trivial solution of (2.3) inside the interval / are isolated; thus they can accumulate only at an endpoint. If the zeros of one (and hence of every) non-trivial real-valued solution of (2.3) accumulate at the endpoint a (or b), then (2.3) is called oscillatory at a (respectively b), otherwise (2.3) is called non-oscillatory at a (respectively b). DEFINITION 2 . 1 . (See [6].) Let AeIR and let u, v be real solutions of (2.3). Then (a) u is called a principal solution at a if

(i) u(t) =£ 0 for t in {a, a) and some a in /, (ii) every solution y of (2.3) which is not a multiple of u satisfies u(t) = o(y(t)) asf->a.

(2.4)

(b) v is called a non-principal solution at a if (i) v{t) ¥=0 for t in (a, a) and some a in /, (ii) v is not a principal solution at a. Principal and non-principal solutions at b are defined similarly. To simplify the notation we state definitions and assertions only for the left endpoint a. Similar definitions and assertions always hold, and are freely used, for the right endpoint b. The notion of 'principal solution' has been introduced in [10]. The properties of principal and non-principal solutions have been investigated by Rellich [13]. Immediately clear from the definition is: 2.1. Let keU. If (2.3) has a principal solution u at a, then every non-zero real multiple of u is also a principal solution at a and no other solution is a principal solution at a. LEMMA

REMARK 2.1. By Lemma 2.1 the principal solution at an endpoint, if it exists, is unique up to a real constant multiplicative factor. Simple examples show that the same solution may be principal at one end and non-principal at the other end. Non-principal solutions are never unique, since if v is a non-principal solution and u is a principal solution (which in this case exists by Theorem 2.1) then v + cu is also a non-principal solution for any c in U. Clearly principal and non-principal solutions do not exist at an endpoint at which the equation (2.3) is oscillatory.

SINGULAR STURM-LIOUVILLE PROBLEMS

547

REMARK 2.2. If M is regular at a (see § 3 for the definition of regular) then for any solution y of (2.3), y and py' can be continuously extended to a and principal solutions u at a exist and satisfy the initial conditions: u(a) = 0, (/?«')(a. This shows that u is a principal solution at a.

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H.-D. NIESSEN AND A. ZETTL

THEOREM 2.2. Assume that (2.3) is non-oscillatory at a for some A in U. Let u, v be real non-trivial solutions of (2.3) and let a el be such that u{t), v(t) =£ 0 for a pv (iii) if u is a principal solution and v a non-principal solution at a, then there exists c inU, with c =£ 0, such that

('

u(t) = v(t)

C

—2>

where

a 0 so that

Applying Lemma 2.3 to [o, s] and to these values of A, B, A', B' and e, we obtain a real-valued function u such that U €

D{M, [°> s])>

u(o) = u(s) = (pu

(pu')(s) = (py

[pu'2^

\u(x•)\«* (x e [o, *]),

e.

This implies that

f {pu"

' + \q\u2}^E

+ E2

F

F

2

w.

J rw

Similarly, we choose p e (r, ^), >1 := 5 := fi' := 0, A' := (p.y)'(r) and e > 0 with e + e2j^\q\0(«), Now let ya(x)

for a < x ^ a,

!/(*) = y(x)

fora<x. tfya and yb are non-principal solutions at a and b, respectively, then (3) for zeD(N), (\/P)z' e L2(I) and lim^^^z^), \imx^bz(x) exist and are finite, (4) the domain of the Friedrichs extension NF is determined by D(NF) = {ze D(N)\ lim z(x) = lim z(x) = o). I

x->a

x-*b

J

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H.-D. NIESSEN AND A. ZETTL

Proof. By Lemma 3.5(1) there exist a, {5 e(a,b) with ocbz(x) = 0 can be proved. Therefore, in this case, limz(*) = limz(jc) = 0 for z e D(NF). x—*a

(3.4)

x—*b

Now let z belong to D(N). Then Lemma 2.3 implies that there exists za e D(N) which coincides with z in (a, a] and which vanishes in [/J, b). Then zb\—z — za belongs to D(N) as well, vanishes in (a, a], and coincides with z in [/?, b). If N is in the limit-circle case at a, then, by Lemma 3.5(3), N is pseudo-regular at a. Therefore, Remark 3.2 shows that linr^,, za(x) and lim^._»fl (Pza)(x) exist and are finite. Furthermore, {y/P)z'a = {\NP)Pz'a belongs to L2(/)> since l/VPeL 2 (/) and za vanishes in [/?, b). If Af is in the limit-point case at a, then za belongs to D(N0) and therefore to D(NF), too. Hence, by (2) and (3.4), (VP)z« belongs to L\l) and limx^flzfl(*) exists and is finite. Since za vanishes in [jS, b), linv_ 6 za(x) obviously exists. This shows that the assertion (3) holds for za. Similarly, it can be proved that zb, and therefore z = za + zb, fulfills (3). To prove (4), let

Dx:=\zeD(N)\ I

lim z{x) = lim z(x) = o|. x-*a

x—*b

)

If N is in the limit-circle case at a then from Lemma 3.5(3), Remark 3.2 and the definition of D1} it follows that [zx, z2]N(a) = 0 for all zlf z2e Dx. UN is in the limit-point case at a, then [zx, z2]N(a) = 0 for all zx, z2eD(N) [11, p. 78]. Similarly, [zx, z2]N(b) = 0 for zx, z2eDx. Therefore, the restriction S of N to Dx is symmetric. By (3.4), the Friedrichs extension NF of NQ is a restriction of 5. Hence, D(NF) = DX. 4. Oscillation, semiboundedness and the Friedrichs extension We start with a couple of abstract lemmata. LEMMA 4.1. Suppose S and T are symmetric densely defined operators in a Hilbert space such that SczT and dimD(S*)/£>(5)S'*|O(5,r)C: S* | D(BF)

=

B*\D(BF)

= BF.

(4-3)

Since SF and BF are self-adjoint, (4.3) implies that SF = BF. THEOREM 4.1. / / M is in the limit-circle case at a and if My = kay is non-oscillatory at a for some real ka, then My = ky is non-oscillatory at a for all real A.

Proof. By Corollary 3.3, there exists a positive function v e GP(I) such that the transformed operator N:=TVMTZX is pseudo-regular at a. Now Remark 3.2 shows that Nz = kz is non-oscillatory at a for every real A. Applying Lemma 3.4(2), we obtain that My = ky is non-oscillatory at a for every real A. In Corollary 2.1 it has been stated that My = ky is non-oscillatory at a and at b for all A < a if Mo is bounded below with lower bound a. The converse assertion also holds true: THEOREM 4.2. / / My = ky is non-oscillatory at a for some real ka and at b for some real kb, then MQ is bounded below and the domain of its Friedrichs extension MF is determined by

D(MF) = \yeD(M)\ I

lim ^ \ = lim ^ \ = o). x-*aVa(X)

x-*bV/,{X)

(4.4)

J

Here va and vb are arbitrary non-principal solutions of My = kay at a and of My = kby at b, respectively. Furthermore, the limits in (4.4) exist and arefinitefor every y e D(M) and L2(a, a) and (y/p)vb(^-)

e L2(p, b) for y e D(M), (4.5)

wb/

if va and vb do not vanish on (a, a] and [0, b), respectively. If ya and yb are any real solutions of My = kay and My = Afe.y, which have no zero in (a, a] and [j8, b), respectively, then l

) zL\a,a)

ya/

and (yJp)yb{L) Vfr/

€ L2(0, b) foryeD(MF).

(4.6)

SINGULAR STURM-LIOUVILLE PROBLEMS

559

Proof We may suppose that a < j3 and that ya and yb are positive on (a, a] and [)3, b), respectively. Then, by Lemma 3.6, there exists a positive function v e Gp(l) which coincides with ya on (a, a] and with yb on [j3, b). Let N:=TvMTZl be the transformed operator. Lemma 3.7(1) shows that NQ is bounded below. Thus, by Corollary 3.1, applied to Mx := MQ and Nt:=No, MQ is bounded below. Furthermore, Lemma 3.7(2) and Corollary 3.1 imply that for yeD(MF), (y/pMy/v)' = (y/P)(Tvy)' 6 L\I).

(4.7)

Since v coincides with ya on {a, a] and with yb on [0, b), this proves (4.6). If ya = va and yb — vb, then by Lemma 3.7(4) and Corollary 3.1, (4.4) holds true: D{MF) = TZlD{NF) = \ye D{M)\ lim Tvy(x) = lim Tvy(x) = ol. I

x—*a

x—*b

J

Furthermore, by Lemma 3.7(3), for all y e D(M), (4.7) is fulfilled and lim^

lim(rvy)(*) and iim^\ iim^\ = Mm (Tvy)(x)

x-*aVa{X)

x-*a

bV(X) x->bV b(X)

b

exist and are finite. THEOREM 4.3. Let My = Xy be non-oscillatory at a for some real X and let ua and va be principal and non-principal solutions of My = Xy at a which do not vanish in {a, oc\ Then, for every y 6 D(M), the following assertions are equivalent:

(4.8)

Km ^ 4 = 0, x-*a

Va(x) Wfl (^)

eL\a,(x),

(4.9)

eL\a,(x).

(4.10)

If M is also in the limit-circle case at a, then for every yeD(M) uniquely defined c, d e C such that (x) + dva(x)

forx^a,

there exist (4.11)

and the following assertions are equivalent to (4.8): [y,ua]M(a) = 0, y(x) = (c + o(l))ua(x) y = O(ua)

for x—»a and some c eC,

in (a, a].

(4.12) (4-13) (4.14)

If M is regular at a, then (4.8) is equivalent to y(a) = 0.

(4.15)

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H.-D. NIESSEN AND A. ZETTL

Proof. We may assume that M is regular at b. For otherwise we consider M on {a, b') for some b' e(a, b). This is possible by Lemma 2.3. Then My = Xy is non-oscillatory at b, too, and we can choose /? e (a, b) such that va has no zero in [/3, b). Then, by Lemma 2.3, to every y e D(M) there exists z e D(M) which coincides with y on (a, or] and which vanishes on [j8, 6). If y fulfills (4.8), then, by (4.4), z belongs to D{MF) and (4.6) applied to ya '= "a, yb •= va implies (4.9). With y := {pu'a)va - ua(pva)=t0, the identity (4.16) holds true. Since, by (4.5), (y/p)va(z/va)' eL2(a, a), (4.16) shows that (4.9) and (4.10) are equivalent for z and therefore for y, too. By Theorem 4.2, d:=\imx^a(z(x)/va(x)) exists and is finite. If (4.10) is fulfilled for y and hence for z, then Theorem 2.2(i) implies that d = 0, that is, (4.8) holds. Now let M be in the limit-circle case at a. Then ua and va belong to L2w(a, a). Therefore, by the variation-of-constants formula, there exist c, deC such that

ua(x)\ va{t)w{t)f{t)dt-va(x)\ Ja

ua(t)w(t)f(t) dt. Ja

Here, / := -y~\My - ky) e L2w{a, b) with Y\=(pu'a)va-ua{pv'a) (2.11), it follows that

eU.

From

i\va(t)\ fora Therefore,

Together with the fact that

this implies (4.11). Furthermore, c and d are uniquely defined since (4.11) and the equation lim*.^ {ua{x)lva(x)) = 0 imply that

To prove the equivalence of (4.8) and (4.12) we apply the transformation Tv, where v e GP(I) is a positive function which coincides with va on {a, a] (or with —va if va is negative on (a, a]). Such a function exists by Corollary 3.3. Then N := TvMTZl is pseudo-regular at a. Therefore, by Remark 3.2 and the facts that ua =£ 0 and lim(r«iO(x) = U m ^ T = 0, x-*a

x—*aVa\X)

SINGULAR STURM-LIOUVILLE PROBLEMS

561

we obtain

Then, since ua, va are real valued and ua belongs to D(M), we have, according to Lemma 3.2(8), [y, wo]w(«) = [Ty, Tua]N(a) = lim (Tvy)(x) lim (P(Tvua)')(x) - lim (P(Tvy)')(x) lim (Tvua)(x) x—*a

x—*a

x—*a

x—*a

= Mm (Tvy)(x) lim (P(Tvua)')(x). x—*a

x—*a

Hence, [y, ua]M(a) = 0 if and only if \imx^a(Tvy)(x) = 0, that is, if and only if

lim,_a (y(x)/va(x)) = 0. Since d = \imx^a(y(x)/va(x)) in (4.11), (4.8) is equivalent to (4.13). By (4.11) and since ua = o(va) for x—>a, (4.13) is equivalent to (4.14). Finally, if M is regular at a, then va(a) =£0 by Remark 2.2 and (4.8) reduces to y(a) = 0. COROLLARY 4.1. Suppose that the assumptions of Theorem 4.2 are fulfilled, let ua be a principal solution belonging to va, and let a el be chosen in such a way that ua and va do not vanish in (a, a]. Then the condition \imx_+a(y(x)/va(x)) = 0 in (4.4) can be replaced by the conditions (4.9) or (4.10). //, in addition, M is in the limit-circle case at a, then the condition lirn^o (y(x)/va(x)) — 0 in (4.4) can be replaced by any of the conditions (4.12) to (4.14). If M is in the limit-point case at a, then the condition \\mx_>a (y(x)/va(x)) = 0 in (4.4) is automatically fulfilled and may be dropped. A similar result holds for the right endpoint b.

Here the assertion concerning the limit-point case follows from the well-known fact [11, § 18.3] that at a limit-point endpoint no boundary condition is needed to determine self-adjoint extensions of Mo. An important feature of the characterization of the domain of the Friedrichs extension given by (4.8) or (4.9) or (4.10) is that it is independent of the limit-point/limit-circle classification. Thus it is not necessary first to determine this classification in order to impose the appropriate boundary conditions. The conditions (4.8) and (4.13)—in the limit-circle case and with stronger assumptions on the coefficients—are due to Rellich [13]. In the general case, (4.8) is due to Rosenberger [14], condition (4.9) has been obtained by Kalf [8], (4.10) seems to be new. Under more restrictive assumptions, Friedrichs [5] gives a condition which is slightly stronger than (4.9). Condition (4.15) shows that Theorem 4.2 reduces to the well-known fact that in the regular case the Friedrichs extension of the operator MQ (which then is always bounded below) is characterized by Dirichlet boundary conditions (see [12]). As Example 4 of [14] shows, in the limit-point case (4.8) is not equivalent to (4.13) and (4.14) (see Example I of § 6 with c = 6).

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H.-D. NIESSEN AND A. ZETTL

The condition given by Kalf in [8] is weaker than (4.9) in so far as he assumes , 0.

(4.19)

Define D(M ;Bl, B2) := {y e D(M)\ y satisfies (4.19) and is identically zero in a neighbourhood of a), M(BU B2)y := My for y e D(M ;BX, B2). Then M(BX, B2) is a symmetric operator in L2w(a, b) which is bounded below. Its Friedrichs extension has domain DF(M ; Bx, B2) = {y e D(M)\ y satisfies (4.19) and any one of the conditions(4.8), (4.9), (4.10), (4.12), (4.13) or (4.14)}. A similar result holds if the conditions at the endpoints are interchanged. REMARK 4.1. If the endpoint b is regular, then, as was seen in the proof of Lemma 4.3, (4.19) reduces to

Cxy(b) + C2{py'){b) = 0, with Cx, C2 real, C2 + C\>0. Theorem 4.4 with condition (4.13) for this case—and with stronger assumptions on the coefficients—is due to Rellich [13]. Our proof is different from his. Proof of Theorem 4.4. Using Remark 3.1 once more, we may suppose that (a, b)\s finite. Let v e GP(I) be a positive function which coincides with va near a and with vb near b. (We may suppose that va and vb are positive near a and b, respectively.) Let T:=TV, N:= TVMT~' and N(BU B2):= TVM(BU B2)T~\ Then Af is regular at a and at b. As has been shown in the proof of Lemma 4.3 (for a instead of b), there exist real C1} C2 which do not both vanish such that the domain of N(Blt B2) equals {z e D(N)\ Cxz{b) + C2(Pz')(b) = 0 and z is identically zero in a neighbourhood of a}. Then obviously N(Bl} B2) is symmetric and its closure N(BU B2) has domain {z e D(N)\ Cxz(b) + C2(Pz'){b) = 0 and z(a) = (Pz'){a) = 0}. By Theorem 2.2 of [12], N(Blt B2) is bounded below and the domain of its Friedrichs extension is determined by {z € D{N)\ Cxz{b) + C2(Pz'){b) = 0 and z(a) = 0}. Applying Lemma 3.2(7) and Corollary 3.1 with Mx := M(BX, B2), we obtain that M(BX, B2) is symmetric, bounded below and its Friedrichs extension has domain DF(M ; Bx, B2) = {ye D(M)\ y satisfies (4.19) and (4.8)}. Here by Theorem 4.3, we can replace (4.8) by any of the conditions (4.9), (4.10), (4.12), (4.13) and (4.14).

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H.-D. NIESSEN AND A. ZETTL

5. Eigenvalue comparisons in the non-oscillatory limit-circle case Throughout this section we assume that M is in the limit-circle case and that My = ky is non-oscillatory at each endpoint a, b for some (and hence for any) real A. Let Xa and Xb be real. Then by Theorem 2.1 there exist principal and non-principal solutions ua and va of My = kay at a and principal and non-principal solutions ub and vb of My = Xby at b. We may assume that va and vb are positive near a and b, respectively. Then, by Lemma 3.6, there exists a positive function v e GP(I) which coincides with va near a and with vb near b. By Remark 3.1, we may suppose that a and b are finite. Then Lemma 3.5 implies that N := TVMT~X is regular at a and at b. Therefore, by Remark 2.2, (Tua)(a) = (Tub)(b) = 0,

(P(Tua)')(a)*0,

(P(Tub)')(b)±0.

(5.1)

Hence, on multiplying ua and ub by suitable non-zero real numbers, we may assume that -l.

(5.2)

Furthermore, since Tva and Tvb are equal to 1 near a and b, respectively, (Tva)(a) = (Tvb)(b) = 1, (P(Tva)')(a) = (P(Tvb)')(b) = 0.

(5.3)

It is well known [11, § 18.2], that in this regular case Nx is a self-adjoint extension of No if and only if its domain D(NX) consists of all z eD(N) which satisfy a certain 'self-adjoint' boundary condition )=0

-

(5 4)

'

Here A, B are (2 x 2)-matrices of complex numbers satisfying the following two conditions: rank(/l : B) = 2,

(5.5)

AJA* = BJB*, with 7 = ^ ~ J ) ,

(5.6)

where A : B denotes the (2 x 4)-matrix obtained by placing B to the right of A and A* denotes the adjoint (conjugate transpose) of A. For Ax

A2\

n

/0

0

(5.4) reduces to the important special case of 'separated' boundary conditions: AlZ(a)+A2(Pz')(a)

= 0,

BlZ(b) + B2(Pz')(b) = Q.

(5.7) (5.8)

Then (5.5), (5.6) reduce to (AuA2)*0, AXA2 = AXA2)

(B1}B2)*0,

(5.9)

BXB2 = BXB2.

If Axj^0, we may multiply (5.7) by AX. This does not change (5.7), but AXA2 = AXA2 becomes equivalent to Ax and A2 being real for the new coefficients Ax and A2. A similar result is obtained by multiplication with A2 if

SINGULAR STURM-LIOUVILLE PROBLEMS

565

A{ = 0. Similarly, (5.8) may be multiplied by Bx or by B2. Then the selfadjointness of (5.7) and (5.8) is equivalent to (5.9) and Au A2 and Bx, B2 are real.

(5.10)

It is sometimes convenient to reformulate (5.7), (5.8), (5.9), (5.10) as follows: cos a. z(a)- sin a. (Pz')(«) = 0, for 0 ^ a