JJ II

(1)

volume 4, issue 3, article 56, 2003.

Received 21 November, 2002;

accepted 25 March, 2003.

Communicated by:A.M. Fink

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

SEPARATION AND DISCONJUGACY

R.C. BROWN

Department of Mathematics, University of Alabama-Tuscaloosa, AL 35487-0350, USA.

EMail:dbrown@gp.as.ua.edu

c

2000Victoria University ISSN (electronic): 1443-5756 130-02

(2)

Separation and Disconjugacy R.C. Brown

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of35

J. Ineq. Pure and Appl. Math. 4(3) Art. 56, 2003

http://jipam.vu.edu.au

Abstract

We show that certain properties of positive solutions of disconjugate second order differential expressionsM[y] =−(py⁰)⁰+qyimply the separation of the minimal and maximal operators determined byMinL²(Ia)whereIa= [a,∞), a > −∞, i.e., the property that M[y] ∈ L²(I_a) ⇒ qy ∈ L²(I_a). This result will allow the development of several new sufficient conditions for separation and various inequalities associated with separation. Some of these allow for rapidly oscillatingq. It is shown in particular that expressions M withW KB solutions are separated, a property leading to a new proof and generalization of a 1971 separation criterion due to Everitt and Giertz. A final result shows that the disconjugacy ofM−λq²for someλ >0implies the separation ofM.

2000 Mathematics Subject Classification: Primary: 26D10, 34C10; Secondary 34L99, 47E05

Key words: Separation, Symmetric second order differential operator, Disconjugacy, Limit-point.

The author wishes to thank Don B. Hinton for his encouragement and support in the preparation of this paper

1. Introduction

Consider the symmetric second order differential expression

(1.1) M[y] :=−(py⁰)⁰+qy

wherep >0,p⁰ andqare continuous on the intervalIa = [a,∞),a >−∞. M is said to be disconjugate if every nontrivial real solution has at most one zero in I_a . A sufficient condition (from Sturm’s comparison theorem) for disconjugacy is that q ≥ 0, and in this case one can show existence of two positive solutions u₁ and u₂ of M[y] = 0on I_a, called the principal and nonprincipal solution respectively, such that u⁰₁ ≤ 0 andu⁰₂ > 0on I_a. More generally, M is disconjugate onIaif and only if there exists a positive solutionuon the interior ofI_a. For proofs of these facts and additional discussion see Hartman [15, Corollaries 6.1 and 6.4].

Recall also that M determines several differential operators in the Hilbert spaceL²(I_a). In particular the “preminimal” and “maximal” operatorsL⁰₀ and Lare given byM[y]foryin the domainsD⁰₀ ≡C₀^∞(I_a), the space of infinitely differentiable functions with compact support in the interior ofIaand

D={y∈L²(I_a)∩AC_loc(I_a) :py⁰ ∈AC_loc(I_a);M[y]∈L²(I_a)}, where AC_loc stands for the real locally absolutely continuous functions on I_a andL²(I_a)denotes the usual Hilbert space associated with equivalence classes of Lebesgue square integrable functionsf, ghaving norm and inner product

kfk= Z

Ia

|f|² ¹₂

, [f, g] :=

Z

Ia

fg.¯

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of35

The “minimal operator”L₀with domainD₀is then defined as the closure ofL⁰₀. With the above definitions one can show that

(i) C₀^∞(I_a)⊂ D⁰₀ ⊂ D₀ ⊂ D, (ii) L^{0 ∗}₀ =L^∗₀ =L,

(iii) L^∗ =L₀,

(iv) D⁰₀,D₀, andDare dense inL²(I_a).

The regularity assumptions made in this paper onpandqare stronger than necessary to properly define L₀, L. In general one needs only to assume the so-called “minimal conditions" thatp⁻¹andqare locally integrable on(a,∞).

In this caseC₀^∞(I_a)may not be contained inD₀⁰ but the properties (ii)–(iv) will still hold. The maximal and minimal operators L andL₀ can also be defined relative to an arbitrary interval (a, b) where −∞ ≤ a < b ≤ ∞. If p⁻¹, q are Lebesgue integrable on some interval (a, c)or(c, b)for a < c < bthena or bare said to be “regular"; otherwise they are “singular". (Infinite endpoints however are considered singular even ifp⁻¹, qare integrable on(a, b).) Thus in our settingais regular and∞is singular–we signal this by writingI_a= [a,∞) rather than(a, b).

M is limit-point or LP at ∞if there is at most one solution of M[y] = 0 which is inL²(Ia), and limit-circle orLC at the point if both solutions are so integrable. This can be shown equivalent to each of the following properties

(i) {y, z}(∞) := lim_x→∞(yp¯z⁰−py⁰z)(x) = 0¯ for ally, z ∈ D.

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of35

(ii) D = D₀ ⊕span(φ₁, φ₂),whereφ₁, φ₂ ∈ D and have compact support in Ia. ThusDis a two dimensional extension ofD0.

It is clear that if M is disconjugate it is LP at ∞ since the nonprincipal solutionu₂ ∈/ L²(I_a). A stronger condition at∞thanLP is strong limit-point orSLP which means

x→∞lim(pyz)(x) = 0¯

∀y, z ∈ D. For a thorough development of these operator theoretic ideas see Naimark, [17, §17]. Discussion of the SLP concept may be found in Everitt, [7].

We turn now to the central concern of this paper.

Definition 1.1. M is said to be separated onD0 or on D—equivalentlyL0 or Lis separated—if qy ∈ L²(I_a). (Obviously also by application of the triangle inequality(py⁰)⁰ ∈L²(I_a).)

The following is an exercise in the Closed Graph Theorem (see e.g. [16]).

Proposition 1.1. Separation onD₀orDis equivalent to the inequality (1.2) Ak(py⁰)⁰k+Ckqyk ≤KkM[y]k+Lkyk.

for nonnegative constantsA, C, K andL.

The next result shows some connections between LP or SLP at ∞ and separation. Its proof may be found in [2].

Proposition 1.2. IfM is separated onD₀then it is separated onDifM isLP at∞. On the other hand, ifM is separated onDthen it isSLP at∞.

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of35

Remark 1.1. Two immediate consequences of Proposition 1.2 are (i) if M is LC at∞then it is not separated, (ii) ifM isLP but notSLP at∞thenM is not separated onD₀.

Several criteria for separation ofM given by Everitt and Giertz in a series of pioneering papers [8] – [12], also see Everitt, Giertz, and Weidmann [13], and Atkinson [1]. More recent results (that include weighted cases) may be found in Brown and Hinton [2],[3]. We quote three typical results.

Theorem A (Brown and Hinton [2]). Ifp⁻¹is locally integrable onI_a,pq≥0, q(x)is locally absolutely continuous, and

(1.3)

p^1/2q⁰(x) q^3/2(x)

≤θ <2, onI_athenM is separated onD.

Remark 1.2. The original version of TheoremAwithp = 1andq > 0is due to Everitt and Giertz [11]. The case of nontrivialpbutθ < 1is given in [9].

Theorem B (Brown and Hinton [3]). Suppose that p⁻¹ is locally integrable onIa,pq ≥ 0, andq, pare twice differentiable onIa. ThenM is separated on D₀ if

(1.4) lim sup

x→∞

(pq⁰)⁰

q² ≤θ <2.

Remark 1.3. Note that in the case p = 1 both TheoremsAand Bwork for a wide class of increasingq such asq(x) = exp(x), q(x) = exp(xⁿ)forn > 0,

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of35

q(x) = exp(exp(· · ·exp(x))· · ·), etc. On the other hand, both theorems fail if q is rapidly oscillatory, e.g.,q(x) = exp(x)(1 + sin(exp(x)). Note also that a consequence of TheoremBis that ifp= 1andq⁰⁰ ≤0(i.e.,qis concave down) thenM is separated.

Theorem C (Brown and Hinton [2]). Suppose p⁻¹ ∈ L¹_loc(Ia), pq ≥ 0, q is differentiable. ThenM is separated onD₀ if either

(1.5) sup

x∈Ia

(x−a) Z ∞

x

q⁰

q² =K1 < 1 4 or

(1.6) sup

x∈I_a

(x−a) Z ∞

x

(q⁰)² =K₂ <∞.

Remark 1.4. In this theorem we see that separation holds for any psatisfying weak conditions provided thatqis of slow enough growth. For exampleq(x) = x^β, β < ¹₂, satisfies (1.5) and q(x) = Klog(x) satisfies (1.6). These facts should not be particularly surprising since ifq = 1thenM would be separated for any p; consequently one can conjecture that the same ought to be true ifq has slow enough growth.

Recently Chernyavskaya and Schuster,[4] have given necessary and sufficient conditions using averaging techniques due to Otelbaev for the inequalities

KkM[y]kp,R≥ ky⁰⁰kp,R+kqykp,R

(1.7)

≥ kryk_p,R, (1.8)

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of35

where the norms are L^p norms onR, q ≥ 1and is locally integrable, r > 0is locally pintegrable,M[y] = −y⁰⁰+qy ∈ L^p(R), and1 ≤ p ≤ ∞. Note that (1.7) or (1.8) can hold on theL^panalog ofDonly ifM has noL^porr-weighted L^p solutions. Although the conditions in [4] seem challenging to implement they can be applied to rapidly oscillating potentials such as

(1.9) q(x) = exp(|x|) + exp(|x|)(1 + sin(exp(|x|)) for which both TheoremsAandBfail.

In this paper we show that certain pointwise properties of a positive solution of a disconjugate expression M imply that M is separated onD. This means in particular that separation occurs if M has a fundamental set of solutions, sometimes calledW KBsolutions, with a particular asymptotic behavior at∞.

Since the existence ofW KBsolutions follows from certain integral conditions satisfied by p and q, we are led to a test for separation that includes a well- known 1971 result of Everitt and Giertz as a special case. We also show that our approach leads to several other sufficient conditions for separation which do not require verification of properties of positive solutions of M. Some of these will work for rapidly oscillating potentials similar to (1.9). We look also at conditions that ensure that the mapping associated with the inequality (1.10)

√ hy

≤KkM[y]k+Lkyk

is compact wherehis a weight, i.e., a positive locally integrable function, which in turn will lead to a more general inequality (see (2.17) below) than (1.10). We also investigate “perturbation” results: if M₁[y] = −(py⁰)⁰ +q₁y is separated,

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of35

when is the same true of M₂[y] = −(py⁰)⁰ + q₂y when in some sense q₂ is

“close” toq1?

Although our tests for separation hold only inL²(I_a)and are sufficient but not necessary, they are easy to apply. Moreover we consider nontrivialpand on occasion allowqto be negative or even unbounded below which is a more general setting than in [4]. Finally, as already mentioned, the inequalities (such as (2.17) below) associated with separation may be more complicated than (1.7)–

(1.8).

We use the following notational conventions in the paper. Positive constants will be denoted by capital letters with or without subscripts such as C,K, K₁, etc. The value of a constant may change from line to line without a change in the symbol denoting it. If f andg are functionsf ∼ g denotes the asymptotic equivalence of f and g, i.e., limx→∞f /g = 1. L²(w;I_a) is the standard w- weighted Hilbert space with norm and inner product

kfk_w = Z

Ia

w|f|² ¹₂

, [f, g]_w = Z

Ia

wf¯g,

where w is a weight. The class of Lebesgue integrable or locally Lebesgue integrable functions onI_awill be denoted byL(I_a)orL_loc(I_a).

Remark 1.5. The Hilbert space theory (see e.g. [17] of the operatorsL₀ and L is usually developed on complex domains. Thus D is the space of locally absolutely continuous complex valued functionsf onIa such thatf andM[f] belong to L²(I_a)with similar changes in the definitions ofD⁰₀ andD₀. All the standard closure and adjoint properties ofL₀ andLremain true in both cases.

Since the chief tool in our development is the concept of disconjugacy which is

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of35

defined only for real-valued solutions of M, we will derive conditions for the separation ofM only for realD0 andD. However all our results go over to the complex case. This is seen from observation that iff =f₁ +if₂ ∈ Dthen

kM[f]k² =kM[f₁]k² +kM[f₂]k², kqfk² =kqf₁k²+kqf₂k².

ThereforeM(f), qf ∈L²(Ia)↔M[f1], M[f2], qf1, qf2 ∈L²(Ia).

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of35

2. Main Results

Theorem 2.1. Letp >0,qbeC¹ functions. SupposeM[y] =−(py⁰)⁰+qyhas a positive solution on the interior ofIasuch that

(pu⁰)⁰u≡qu² ≤2p(u⁰)², (2.1)

(1−δ)(u⁰)² ≤u⁰⁰u, δ ∈[0,1/3), (2.2)

p⁰u⁰ ≥0.

(2.3)

Thenq≥0andM is separated onD.

Proof. We need only show thatM is separated onD₀. BecauseM is disconjugate and as will be seen below (see (2.9))q ≥0,M isLP at∞and separation onDwill follow by Proposition1.2; in this case by Proposition1.1ywill satisfy an inequality of the form

kqyk² ≤Ckyk²+DkM[y]k² for certain positive constantsC, D.

Letz(t) = −u⁰/u. Thenzsatisfies the Riccati-type equation

(2.4) (pz)⁰ =pz²−q.

Since

(2.5) (pz)⁰ = −u(pu⁰)⁰+p(u⁰)² u²

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of35

(2.1) – (2.3) is equivalent to the properties

−pz² ≤(pz)⁰, (2.6)

z⁰ ≤δz², (2.7)

p⁰z ≤0.

(2.8)

To see this, note that from the definition ofzand (2.5) (2.1)⇔ −2p(u⁰)²

u² ≤ −u(pu⁰)⁰ u²

⇔ −p(u⁰)²

u² ≤ −u(pu⁰)⁰+p(u⁰)² u²

⇔(pz)⁰ ≥ −pz². Also

(2.2)⇔ −(1−δ)(u⁰)²

u² ≥ −uu⁰⁰ u²

⇔δ(u⁰)²

u² ≥ −uu⁰⁰+ (u⁰)² u²

⇔δz² ≥z⁰.

Finally, the definition ofz and (2.3) clearly implies thatp⁰z ≤0.

Next define the operators

L(y) =y⁰+zy, L^∗(y) =−y⁰+zy

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of35

wherey∈C₀^∞(I_a).

We now derive sufficient conditions for the “separation” ofL^∗. We have kL^∗(y)k² = [L^∗(y), L^∗(y)]

= [LL^∗(y), y]

= [−y⁰⁰+ (z²+z⁰)y, y]

= Z

Ia

(y⁰)²+ (z²+z⁰)y². Sincep⁰z ≤0we see that

(pz)⁰ =p⁰z+pz⁰ ⇒pz⁰ ≥(pz)⁰ ≥ −pz²

⇒z⁰ ≥ −z². Becausez⁰+z² is nonnegative the inequality

kL^∗(y)k² ≥ ky⁰k² holds. By the triangle inequality it also follows that

kzyk² ≤4kL^∗(y)k².

The remaining step is use the separation ofL^∗ to show thatM restricted to C₀^∞(Ia)is also separated. We first observe that

L^∗(pL(y)) =−(py⁰+pzy)⁰ +z(py⁰+pzy)

=−(py⁰)⁰+ [−(pz)⁰+pz²]y

=−(py⁰)⁰+qy.

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of35

A consequence of (2.7) – (2.8) is that

−(pz)⁰+pz² =−pz⁰−p⁰z+pz²

≥ −pz⁰+pz²

≥pz²(1−δ)

≥0.

Therefore both

(2.9) q≥0 and (pz)⁰ ≤δpz².

Now also

kM[y]k² = [L^∗(pL)(y), L^∗(pL)(y)]

=kL^∗(pL(y))k²

≥ 1

4kz(pL(y))k²

= 1

4[L^∗((zp)²L(y)), y]

= 1 4

−((zp)²y⁰)⁰+ (z⁴p²−(z³p²)⁰y, y

= 1 4

Z

Ia

[(zp)²(y⁰)² + (z⁴p²−(z³p²)⁰)y²].

(2.10)

Hence sincep⁰z ≤0andz⁰ ≤δz²,

z⁴p²−(z³p²)⁰ =z⁴p²−3z²z⁰p²−2z²p(p⁰z)

≥(1−3δ)z⁴p². (2.11)

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of35

But(pz)⁰ ≥ −pz², so

pz² =q+ (pz)⁰ ≥q−pz².

Hence alsopz² ≥q/2. Combining this with (2.10) and (2.11) gives the inequality

(2.12) kM[y]k² ≥ 1

8k√

pqy⁰k²+1−3δ 4 kqyk², which immediately yields the separation inequality

(2.13) 16

1−3δkM[y]k² ≥ kqyk².

A closure argument (cf. [2, Lemma 1]) shows that the same inequalities are true on the minimal domainD0.

Remark 2.1.

(i) It is well-known that the existence of a positive solutionu, the existence of a continuously differentiable solutionzof the inequality z⁰+z²/p+q ≤ 0, or the identityM[y] = L^∗(pL(y)) for y having a continuous second derivative are each equivalent to the disconjugacy of M on I; see e.g.

[15, Corollary 6.1, Theorem 7.2] or Coppel [5, p.6].

(ii) We may require that both the conditions q ≥ 0 and (2.1) – (2.3) hold

“eventually”, i.e. on I_a⁰ for sufficiently large a⁰ > a. In this case the restriction ofM toI_a⁰ will be separated on its maximal domain. Sinceqis

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of35

bounded on(a, a⁰]it is immediate that separation holds also forI_a(cf. [2, Remark 1 and Proposition 2]) although the corresponding inequality may be of the form (1.1) rather than (2.13).

(iii) If we retrace the proof of Theorem2.1withp= 1(2.1) – (2.3) becomes (2.14) (1−δ)(u⁰)² ≤u⁰⁰u≤2(u⁰)²

⇔(1−δ)(u⁰)² ≤qu² ≤2(u⁰)², δ∈[0,1/3), with a corresponding change in (2.6) – (2.8).

(iv) If q is positive andu satisfies (2.1) or (2.2) thenu⁰ is strictly positive or negative, for ifu⁰(x₀) = 0eitheru(x₀) = 0or one of(pu⁰)⁰ oruvanishes atx₀. In either caseq >0⇒u(x₀) = 0,implying thatu≡0.

In the remainder of the paper “separated” means separated onDunless the restriction toD₀ is stated. Also, in proving separation inequalities onD₀ such as (1.2) we will generally start with y ∈ C₀^∞(I_a)and omit the routine closure argument which extends the inequality toD0.

We now show that information about the asymptotic behavior of positive solutions of M[y] = 0 can yield criteria for separation based on the stable conditions ofpandq.

Theorem 2.2. Suppose that p, q are positive and twice differentiable with p⁰ nonnegative or nonpositive. Set

t(x) :=

Z x

a

rq p, µ(x) := (pq)^−1/4,

(17)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of35

and assume thatlimx→∞t(x) =∞,µ(pµ⁰)⁰ ∈L¹(I_a), andlim sup_x→∞|p⁰|/√ pq= δ < ¹₃. ThenM is separated.

Proof. By Coppel [6, Theorem 13], M has fundamental solutionsusuch that forx→ ∞

u∼=µexp(±t(x)), u⁰ ∼=±(pµ)⁻¹exp(±t(x)).

It follows that(pu⁰)⁰ ∼qy₁and so u(pu⁰)⁰ ∼qu² ∼

rp

q exp(±2t(x))∼p(u⁰)².

Clearly (2.1) is satisfied on I_a⁰ for sufficiently large a⁰ > a. To derive (2.2) observe that the asymptotic equivalence ofp(u⁰)²and(pu⁰)⁰ implies that

(u⁰)² ∼u⁰⁰u+ p⁰ pu⁰u.

But

(p⁰/p)(u⁰u/(u⁰)²) = p⁰u

pu⁰ ∼p⁰ u pp

q/pu²

∼p⁰µ² ≤ |p⁰|

√pq ≤δ+ < 1 3

asx → ∞. Thus for > 0and on someI_a⁰ witha⁰ sufficiently large we have that

(u⁰)² ≤(u⁰⁰u+ (δ+)(1 +)(u⁰)²)⇒(1−(δ+)(1 +))(u⁰)² ≤u⁰⁰u

(18)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of35

which obviously implies (2.2) ifis small enough. Finally, ifp⁰ ≥0we choose y1 =µ(x) exp(t(x))and ifp⁰ ≤0we choosey1 =µexp(−t(x)). In either case (2.3) holds. By Remark2.1(ii), the fact thatM isLP at∞, and Theorem2.1, separation follows.

In 1970 [8] Everitt and Giertz showed:

Corollary 2.3. Ifp= 1,q ≥d >0, and Z

Ia

q^−1/4

(q^−1/4)⁰⁰ <∞, thenM is separated.

Proof. Evidently this condition is a special case of Theorem2.1withp= 1, cf.

[6, Theorem 14].

Remark 2.2. The hypothesis of Corollary2.3can be shown to be equivalent to (see [6, p. 122]

Z

Ia

q^−3/2q⁰⁰ <∞,

unlessq(x) ∼ cx⁻⁴ andq⁰(x) ∼ −4cx⁻⁵ forca positive constant. But in this caseM is trivially separated onI_aifa >0.

A similar result using the asymptotic properties of solutions but requiring less smoothness onqis given by:

(19)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of35

Theorem 2.4. Suppose thatp= 1,q ≥d >0is differentiable, and Z

Ia

|q⁰|

q^3r/2−1/2 <∞ for somer,1≤r ≤2. ThenM is separated.

Proof. By a result of Hartman and Winter [15, p. 320]M has solutionsusuch that

u∼q^−1/4exp(±t(x)), u⁰ ∼ ±q^1/2u,

wheret(x) = Rx a

√q. Sinceu⁰⁰ ∼ q^3/4exp(±Rx

a q^1/2), it is clear that (2.14) is satisfied on someI_a⁰,a⁰ > a.

In most cases however it is difficult to verify (2.1) – (2.3) or (2.14) directly, which motivates us to seek an equivalent formulation of Theorem2.1for which knowledge of properties of positive solutions ofM[y] = 0is not required.

Theorem 2.5. Letp > 0and z beC¹(I)functions. Then if (2.6) – (2.8) hold andq=pz²−(pz)⁰. M[y]is separated and the inequality (2.13) holds.

Proof. The fact that Theorem 2.1 implies Theorem2.5 is clear. On the other hand, if we setu=e⁻^R^z, thenuis a positive solution ofM[y] = 0.z =−u⁰/u, and the conditions (2.1) – (2.3) hold as they are equivalent to (2.6) – (2.8). Thus all the assumptions of Theorem2.1are satisfied.

(20)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of35

Theorem 2.6. Suppose thatM is separated,q ≥d >0, and thathis a weight.

Assume further that either

(2.15) lim

x→∞

q² h =∞ orlim_x→∞h=∞, and

(2.16) Kk√

py⁰k ≥ h^θ/2y

for some θ > 1 and all y ∈ C₀^∞. Let G_M(y) := {(y, M[y]), y ∈ D}, equipped with the graph norm. Then the mappingλ:G_M →L²(h;I)given by λ(G_M(y) =yis compact, andM satisfies an inequality of the form

(2.17) kM[y]k+K()kyk ≥

√ hy

onDfor >0.

Proof. If (2.15) holds andM is separated, then by (1.2) of Proposition1.1there is an inequality of the form

L

Ckyk+ K

CkM[y]k ≥ kqyk

= Z

Ia

q² h

hy²

¹₂

≥n Z ∞

xn

hy² ¹₂

. (2.18)

(21)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of35

for any positive integer and where the sequence {x_n} → ∞. Let λ_n : G_M → L²(I_aⁿ)be given by the characteristic function on I_aⁿ composed with λ, where I_aⁿ = [a, x_n]. Since the solutions of M[y] = 0 and q are continuous on I_aⁿ, a Green’s function argument shows that the maps λ_n : G_M → L²(h;I_aⁿ) are compact. By (2.18) the λn converge in operator norm to a compact limit λ.

Also since q ≥ d > 0, q is closed, considered as a multiplication operator

˜

q : L²(I_a) → L²(I_a), and since M is separatedD ⊂ D(˜q). In this situation Corollary V.3.8 of Goldberg [14, p. 123] applies and gives (2.17).

Under the second set of conditions we have from the Cauchy-Schwartz inequality, integration by parts, and sincelimx→∞h=∞that on someIa⁰,a⁰ > a, and fory∈C₀^∞(I_a⁰)that

k(py⁰)⁰k kh^θ/2yk ≥ k(py⁰)⁰k kyk

≥[(py⁰)⁰, y]

=k√ py⁰k²

≥K⁻²kh^θ/2yk². Hence

k(py⁰)⁰k ≥K⁻²kh^θ/2yk

≥K⁻²kh^(θ−1)/2√ hyk

≥K⁻²nk√

hyk_(x_n,∞). SinceM is separated we obtain from (1.2) the inequality

L

Ckyk+ K

CkM[y]k ≥ k(py⁰)⁰k ≥K⁻¹n

√ hy

(xn,∞)

(22)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of35

on theC₀^∞(I_a)functions and therefore also onD₀; the proof thatλrestricted to D0 is compact continues as in the first part. But sinceDis a finite dimensional extension ofD₀,λis also compact.

Remark 2.3. Following Everitt and Giertz [8] we say thatqis in the classP(γ) orq ∈P(γ)if whenevery ∈ Dthen|q|^γ ∈ L²(Ia). Thus the separation of M onDis equivalent toq∈P(1). It is also easy to verify by thinking ofq=q₁+q₂ whereq₁(x)≤1andq₂(x)>1thatq∈P(γ)⇒q ∈P(β)for anyβ ∈(0, γ].

Suppose nowq∈P(1)andlimx→∞q =∞. Then from the first part of Theorem 2.6not only willq ∈P(θ),θ <1, but the “compactness” inequality (2.17) will hold if h = q^θ. If M is separated, q → ∞, and (2.16) holds for h = q² and θ > 1 then q ∈ P(θ), and we have the interesting consequence that the mappingλ :G_M →L²(q;I_a)is compact. In general, ifq ∈P(γ)andq → ∞ thenλ:G_M →L²(q^β;I_a)is compact.

A disadvantage of Theorem2.5is that althoughqhas the form pz² −(pz)⁰, since M is disconjugate, it may be difficult to determine z and to verify (2.6) – (2.8). We attempt to remedy this problem in the next three corollaries and obtain additional usable tests.

Corollary 2.7. IfM1[y] =−(py⁰)⁰+q1ywhereq1(z1) = (pz₁²−(pz1)⁰)satisfies the hypotheses of Theorem2.5 andM_1,c[y] = −(py⁰)⁰+q_1,cywhereq_1,c(z₁) = (pc²z₁²−(pcz₁)⁰)y,wherec > 1thenM_1,c[y]is separated. More generally, ifg is a differentiable function such thatg, g⁰ ≥0and

(2.19) g⁰(x)x²

g(x)² ≤1

(23)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of35

then if q₂ = z²₂ −z₂⁰ wherez₂ = g(z₁)M₂ is separated. Conversely, ifM₂[y]

satisfies the hypotheses of Theorem2.5and

(2.20) g⁰(x)x²

g(x)² ≥1, thenM₁is separated.

Proof. Let z2 = cz1. Then since c > 1, z2 satisfies (2.6) – (2.8) and q2 = pz₂²−(pz₂)⁰. Alsop⁰z₂ ≤0. Separation follows by Theorem2.5.

For the second part, sincez₂satisfies (2.6) – (2.8) and by (2.19) we have that z⁰₂ =g⁰(z₁)z⁰₁ ≥ −g⁰(z₁)z₁² ≥ −g(z₁)² =−z₂²

≤δg⁰(z₁)z₁² ≤δg(z₁)² =δz₂².

Thusz₂satisfies (2.6) – (2.8) and we can again apply Theorem2.5. On the other hand, using (2.20)

z₂⁰ ≥ −z₂² ⇔z₁⁰ ≥ −g(z₁)²

g⁰(z₁) ≥ −z₁², z₂⁰ ≤δz₂² ⇔z₁⁰ ≤ δg(z₁)²

g⁰(z₁) ≤δz₁².

Example 2.1. Letp= 1,z₁(x) =√

x, andq₁(x) =x− ¹₂

x⁻¹². Ifa > ³₂²₃ , then (2.6) – (2.8) is satisfied for some δ < ¹₃. If g(x) = exp(x²), (2.19) is

(24)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page24of35

satisfied for say a > 2. Taking z₂(x) = g(z₁) = exp(x) we get thatq₂(x) = exp(2x)−exp(x)and there is an inequality of the form

KkM₂[y]k ≥ kq₂yk

onD₀defined onI_a. ThatM₂ is separated onD₀also follows from TheoremA, but the inequality seems new.

The next two lemmas are useful.

Lemma 2.8. Suppose thatM₁[y] =−(py⁰)⁰+q₁yis separated onD₀. If lim sup

x→∞

q₂

q₁ <1 +γ, lim inf

x→∞

q₂

q₁ >1−γ,

whereγ is sufficiently small, thenM₂[y]is also separated onD₀. Proof. Choosea⁰large enough so that onIa⁰

q₂ q₁ −1

< γ.

Since M₂[y] = M₁(y) + (q₂ −q₁)y by the triangle inequality and inequality (1.2) we have that

Lkyk+KkM₂[y]k+K

q₁ q₂

q₁ −1

y

≥KkM₁[y]k+Lkyk

(25)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page25of35

fory∈C₀^∞. Hence onI_a⁰

Lkyk+KkM₂[y]k+Kγkq₁yk ≥Ckq₁yk

≥C(1 +γ)⁻¹kq₂yk.

Thus

Lkyk+KkM₂[y]k ≥dkq₂yk,

whered= (1 +γ)⁻¹(C−Kγ),which is positive for small enoughγ.

Lemma 2.9. Suppose that M₁[y] = −(py⁰)⁰ +q₁y satisfies the separation in- equality (2.17) with h = q₁² for any > 0 onD₀. If also there are constants K₁, K₂ >0such thatK₁ ≤ |q₁/q₂| ≤K₂thenM₂[y] =−(py⁰)⁰+q₂ysatisfies the same separation inequality onD₀withh =q₂²for sufficiently small >0.

Proof. Since

M₂[y] =M₁[y] +q₂

1−q₁ q2

y fory∈C₀^∞(Ia), we arrive at the inequality

kM₂[y]k+

q₂

1− q₁ q₂

y

+K()kyk ≥kM₁[y]k+K()kyk

≥ kq₁yk ≥K₁kq₂yk for any >0. Hence also

kM₂[y]k+K()kyk ≥d₁kq₂yk, whered1 = (K1−(1 +K2) >0for small enough.

(26)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page26of35

Remark 2.4. Takingq₂ =−q₁ andK₁ =K₂ = 1in Lemma2.9, we see that if M1 satisfies (2.17) then so doesM2 which means that we can have separation for a potential q which is negative and unbounded below provided the expres- sion constructed with potential|q|satisfies (2.17).

Example 2.2. Supposep(x) = 1andq1(x) = exp(x). Then by TheoremAorB M₁ is separated. Lett_n(x) = exp(exp(· · ·exp(x))· · ·)be an-fold iteration of exp(x)and setq₂(x) = exp(x)(1 +sin(t_n(x)), > 0. Then TheoremsAand Bdo not apply because (1.3) and (1.4) are unbounded. However, by Lemma2.8 M₂ is separated if is sufficiently small. Clearlyt_n(x)can be replaced by any other rapidly increasing function.

Example 2.3. Letp₁(x) = exp(x)andq₁(x) =x^1/3onI_a. By TheoremCM₁is separated. It is easy to verify thatp₁ andq₁satisfy the Muckenhoupt condition

sup

x∈Ia

Z ∞

x

p⁻¹₁ Z x

a

q^θ₁ <∞, θ >1,

and therefore (cf. Opi´c and Kufner [18, Theorem 6.2]) the Hardy inequal- ity Kkexp(x/2)y⁰k ≥ kq^θ/2₁ yk holds on C₀^∞. Therefore from the second part of Theorem 2.6 we obtain an inequality of the form (2.17). If now q₂(x) =

−q₁(x)(2 + sin(exp(xⁿ)) we will have from Lemma 2.9 the same kind of in- equality but with

M₂[y] =−(exp(x)y⁰)⁰−x^1/3(2 + sin(exp(xⁿ))y.

Theorem 2.10. Ifp >0,z is aC¹ function,p⁰z ≤0and

−K1z² ≤z⁰ ≤K2z²

(27)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page27of35

for positive constantsK₁, K₂ then the operators M_1,c[y] =−(py⁰)⁰+q_1,cy, M_2,c[y] =−(py⁰)⁰+q_2,cy,

whereq_1,c=c²pz²−c(pz)⁰andq_2,c =c²pz²are separated for sufficiently large c≥1.

Proof. To prove thatM_1,cis separated we retrace the proof of Theorem2.1. Let Lc(y) =y⁰+czyandL^∗_c(y) =−y⁰ +czy,wherey∈C₀^∞(I). Then

kL^∗_c(y)k² = Z ∞

1

(y⁰)²+ (cz⁰+c²z²)y². Ifc≥K1,thencz⁰+c²z² ≥0and as before,

kczk² ≤4kL^∗_c[y]k². LikewiseL^∗_c(pL_c(y)) =M_1,c[y]and

q_1,c =−pcz⁰−p⁰cz+pc²z²

≥ −pcz⁰+pc²z²

≥pcz²(c−K₂)≥0

ifc > K₂. From the definition ofq_1,cwe also have that(pz)⁰ ≤K₂pz². And so kM_1,c[y]k² ≥ 1

4 Z

Ia

[(czp)²(y⁰)² + (c⁴z⁴p²−((cz)³p²)⁰y²]

≥ Z

Ia

[c⁴z⁴p² −3c³z²z⁰p²]y²

(28)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page28of35

≥

1− 3K₂ c

Z

Ia

c⁴p²z⁴y² forc >3K₂. Now also

k(pcz)⁰yk+kM_2,c[y]k ≥ kM_1,c[y]k ≥K₃

c²pz²y

whereK₃ =p

1−3K₂/c, so that kM_2,c[y]k ≥K₃

c²pz²y

− k(pcz)⁰yk

≥ r

1−3K₂ c −

rK₂ c

!

c²pz²y .

Since the constant is positive for large enoughcthe inequality (2.13) forM_2,c[y]

is established. Since

q_1,c

q_2,c = (1−(pz)⁰(c²pz²)

≤

1 + K2

c²

≥

1−K2

c²

,

Lemma 2.8 may be applied to conclude thatM_1,c is separated and satisfies an inequality like (2.13).

(29)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page29of35

Example 2.4. If p⁰ is of constant sign, letz = −sgn(p⁰)p

q/pthenp⁰z ≤ 0as required and q_2,c = c²q. A calculation shows that the hypothesis of Theorem 2.10becomes

−2K1 ≥ p⁰

√pq −p^1/2q⁰ q^3/2

sgn(p⁰)≤2K2. Equivalently we can require that

η= sup

x∈Ia

p⁰

√pq −p^1/2q⁰ q^3/2

<∞

to conclude that M_d[y] = −(py⁰)⁰ +dqy is separated for sufficiently large d.

For example, if p(x) = q(x) = exp(x²)both TheoremAandBfail for anyM_d yetη= 0and so we have an inequality of the form

Kk −(exp(x²)y⁰)⁰+dexp(x²)yk ≥ kdexp(x²)yk for large enoughd.

Corollary 2.11. Letp, z, h, andg be functions such thatp > 0andp, zareC¹, p⁰z ≤0,z⁰ ≤δz² forδ ∈[0,1/3),h ≥d >0,g is bounded, and

(2.21) lim

x→∞

h(pz)⁰ pz²

= 0,

then M₁[y] = −(py⁰)⁰ +q₁y, where q₁ = pz² −(pz)⁰ is separated onD and M₂[y] =−(py⁰)⁰+q₂, whereq₂ =pz²+hg(pz)⁰is separated on at least onD₀. If we assume additionally that

(2.22) lim

x→∞pz² =∞,