Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 340796 | https://doi.org/10.1155/2011/340796

Bing Xie, Jian Gang Qi, "Sufficient and Necessary Conditions for the Classification of Sturm-Liouville Differential Equations with Complex Coefficients", Abstract and Applied Analysis, vol. 2011, Article ID 340796, 15 pages, 2011. https://doi.org/10.1155/2011/340796

Sufficient and Necessary Conditions for the Classification of Sturm-Liouville Differential Equations with Complex Coefficients

Academic Editor: Nicholas D. Alikakos
Received23 Feb 2011
Accepted27 Apr 2011
Published07 Jul 2011

Abstract

This paper gives sufficient and necessary conditions for the classification of Sturm-Liouville differential equations with complex coefficients given by Brown et al. These conditions involve weighted Sobolev subspaces and the asymptotic behavior of elements in the maximal domain. The results of the present paper generalize the corresponding results for formally symmetric Sturm-Liouville differential equations to non-self-adjoint cases.

1. Introduction

Consider the Sturm-Liouville differential expression ๐œ๐‘ฆโˆถ=๐‘คโˆ’1๎‚ƒโˆ’๎€ท๐‘๐‘ฆ๎…ž๎€ธ๎…ž๎‚„[+๐‘ž๐‘ฆ=๐œ†๐‘ฆon๐‘Ž,๐‘),(1.1) where ๐‘,๐‘ž are both complex valued, ๐‘ค(๐‘ฅ) is a positive weight function, โˆ’โˆž<๐‘Ž<๐‘โ‰ค+โˆž, and ๐œ† is the so-called spectral parameter. We call ๐œ a formally symmetric differential expression if ๐‘,๐‘ž are both real valued; otherwise ๐œ is called formally nonsymmetric. In all cases, we call ๐œ a formally differential expression or operator.

Let ๐ฟ2๐‘ค denote the Hilbert space ๐ฟ2๐‘ค๎‚ป[๎€œโˆถ=๐‘ฆismeasurableโˆถ๐‘Ž,๐‘)โŸถโ„‚โˆถ๐‘๐‘Ž||||๐‘ค(๐‘ฅ)๐‘ฆ(๐‘ฅ)2๎‚ผd๐‘ฅ<โˆž(1.2) with inner product โˆซโŸจ๐‘ฆ,๐‘งโŸฉโˆถ=๐‘๐‘Ž๐‘ง(๐‘ฅ)๐‘ค(๐‘ฅ)๐‘ฆ(๐‘ฅ)d๐‘ฅ and the norm โ€–๐‘ฆโ€–2=โŸจ๐‘ฆ,๐‘ฆโŸฉ for ๐‘ฆ,๐‘งโˆˆ๐ฟ2๐‘ค. We call a solution ๐‘ฆ of (1.1) an ๐ฟ2๐‘ค-solution or square integrable solution if ๐‘ฆโˆˆ๐ฟ2๐‘ค. Set ๎€ฝ๐’Ÿ(๐œ)=๐‘ฆโˆˆ๐ฟ2๐‘คโˆถ๐‘ฆ,๐‘๐‘ฆ๎…žโˆˆACloc,๐œ๐‘ฆโˆˆ๐ฟ2๐‘ค๎€พ,(1.3) where ACloc=ACloc([๐‘Ž,๐‘),โ„‚) is the set of complex valued functions that are absolutely continuous on each compact subinterval of [๐‘Ž,๐‘). We call ๐’Ÿ(๐œ) the natural (or maximal) domain associated with the formally differential operator ๐œ.

The aim of the present paper is to study the asymptotic behavior of elements of ๐’Ÿ(๐œ). This is closely related to the classification of (1.1) according to the number of square integrable solutions of (1.1) in suitable weighted integrable spaces. The study of this problem has a long history started with the pioneering work of Weyl in 1910 [1]. When ๐‘(๐‘ฅ) and ๐‘ž(๐‘ฅ) are all real valued, Weyl classified (1.1) into the limit point and limit circle cases in the geometric point of view by introducing the ๐‘š(๐œ†)-functions, where we say that ๐œ or (1.1) is in the limit point case at ๐‘ if there exists exactly one ๐ฟ2๐‘ค-solution (up to constant multiple) for ๐œ†โˆˆโ„‚ with Im๐œ†โ‰ 0 and is in the limit circle case if all solutions belong to ๐ฟ2๐‘ค for ๐œ†โˆˆโ„‚ with Im๐œ†โ‰ 0. This work has been greatly developed and generalized to formally symmetric higher-order differential equations and Hamiltonian differential systems. For this line, the reader is referred to [2โ€“10] and references therein.

The same problem was also studied by Sims in 1957 for the case where ๐‘ž(๐‘ฅ) is complex valued [11]. He considered the case where ๐‘(๐‘ฅ)=๐‘ค(๐‘ฅ)โ‰ก1 and Im๐‘ž(๐‘ฅ) is semibounded and classified (1.1) into three cases. Recently, this work has been extensively generalized by Brown et al. [12] under mild assumptions on weighted function ๐‘ค(๐‘ฅ) and the complex valued coefficients ๐‘(๐‘ฅ), ๐‘ž(๐‘ฅ). They proved that there exist three distinct possible cases for (1.1).

For formally symmetric ๐œ, it is well known (see [13, 14]) that (1.1) is in the limit point case at ๐‘ if and only if ๎‚ƒ๐‘ฆ๐‘(๐‘ฅ)2(๐‘ฅ)๐‘ฆ๎…ž1(๐‘ฅ)โˆ’๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2๎‚„(๐‘ฅ)โŸถ0as๐‘ฅโŸถ๐‘(1.4) for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ). This kind of characterization (1.4) plays an important role in spectral theory of differential operators since (1.4) gives a natural boundary condition of functions in ๐’Ÿ(๐œ) at the end point ๐‘. In this case every self-adjoint extension associated with the differential expression needs not a boundary condition at ๐‘. The analogues of the result (1.4) are also valid for both formally symmetric higher-order differential equations and Hamiltonian differential systems (see, e.g. [4, 5, 7, 8, 15, 16]). By using the asymptotic behavior of elements in ๐’Ÿ(๐œ), the further classification of the limit point case into the strong limit point case and the weak limit point case for high-order scalar differential equations was given by Everitt et al. in [17โ€“19] and further studied in [14, 20]. It was generalized to Hamiltonian differential systems by Qi and Chen [21] and well studied in [22]. For real valued functions ๐‘(๐‘ฅ) and ๐‘ž(๐‘ฅ), we say that (1.1) is in the strong limit point case at the end point ๐‘ if, for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ), ๐‘(๐‘ฅ)๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2(๐‘ฅ)โŸถ0as๐‘ฅโŸถ๐‘.(1.5)

In the present paper, we attempt to set up the analogues of the results (1.4) and (1.5) for (1.1) with complex valued coefficients ๐‘ and ๐‘ž. In the classification of Brown et al. in [12], Cases II and III depend on the admissible rotation angles (see Theorem 2.1). The exact dependence is set up in Theorem 2.5. We find that the asymptotic behavior of elements in ๐’Ÿ(๐œ) also depends on the admissible rotation angles. So we first study the properties of the admissible angle set ๐ธ (defined in (2.10)) and prove that ๐ธ either contains a single point or is an interval. See Lemma 3.1. Then we introduce a pencil of Hamiltonian differential expressions with a new spectral parameter corresponding to (1.1) and set up the relationship between classifications of Hamiltonian differential expressions and (1.1). See Lemma 4.3. Applying the results mentioned in (1.4) and (1.5), we obtain sufficient and necessary conditions for Cases I and II involving weighted Sobolev spaces and the asymptotic behavior of elements in ๐’Ÿ(๐œ). See Theorems 4.1 and 4.11. The main results of the present paper cover the result (1.4) (see Remark 4.2) and indicate that (1.4) means (1.5) when ๐ธ has more than one point; see Corollary 4.9.

Following this section, Section 2 gives some preliminary knowledge for (1.1) with complex valued coefficients, and Section 3 presents properties of the admissible rotation angle set ๐ธ. The main results are given in Section 4.

2. Preliminary Knowledge

Throughout this paper, we always assume that(i)๐‘(๐‘ฅ)โ‰ 0, ๐‘ค(๐‘ฅ)>0 a.e. on [๐‘Ž,๐‘) and 1/๐‘,๐‘ž,๐‘ค are all locally integrable on [๐‘Ž,๐‘),(ii)๐‘ and ๐‘ž are complex valued, and ฮฉ=๎‚ปco๐‘ž(๐‘ฅ)[๎‚ผ๐‘ค(๐‘ฅ)+๐‘Ÿ๐‘(๐‘ฅ)โˆถ๐‘Ÿ>0,๐‘ฅโˆˆ๐‘Ž,๐‘)โ‰ โ„‚,(2.1) where co denotes the closed convex hull (i.e., the smallest closed convex set containing the exhibited set). Then, for each point on the boundary ๐œ•ฮฉ, there exists a line through this point such that every point of ฮฉ either lies in the same side of this line or is on it. That is, there exists a supporting line through this point. Let ๐พ be a point on ๐œ•ฮฉ. Denote by ๐ฟ an arbitrary supporting line touching ฮฉ at ๐พ, which may be the tangent to ฮฉ at ๐พ if it exists. We then perform a transformation of the complex plane ๐‘งโ†ฆ๐‘งโˆ’๐พ and a rotation through an appropriate angle ๐œƒ so that the image of ๐ฟ coincides with the imaginary axis now and the set ฮฉ is contained in the new right nonnegative half-plane.

For this purpose we introduce the set ๐‘† defined by ๎€ฝ๐‘†=(๐œƒ,๐พ)โˆถ๐พโˆ‰ฮฉโˆ˜๎€ฝ๐‘’,Re๐‘–๐œƒ๎€พ๎€พ(๐œ‡โˆ’๐พ)โ‰ฅ0โˆ€๐œ‡โˆˆฮฉ,(2.2) where ฮฉโˆ˜ is the interior of ฮฉ, and define the corresponding half-plane ฮ›๐œƒ,๐พ=๎€ฝ๎€ฝ๐‘’๐œ‡โˆˆโ„‚โˆถRe๐‘–๐œƒ๎€พ๎€พ(๐œ‡โˆ’๐พ)<0.(2.3) Then, ฮ›๐œƒ,๐พโŠ‚โ„‚โงตฮฉ. From the definition of ๐‘†, for all ๐‘ฅโˆˆ[๐‘Ž,๐‘) and 0<๐‘Ÿ<โˆž, ๎‚ป๐‘’Re๐‘–๐œƒ๎‚ธ๐‘ž(๐‘ฅ)๐‘ค(๐‘ฅ)+๐‘Ÿ๐‘(๐‘ฅ)โˆ’๐พ๎‚น๎‚ผโ‰ฅ0.(2.4)

The definition of ๐‘† is different from the corresponding one given by Brown et al. [12], but they are equivalent in describing square integrable solutions.

Besides, for (๐œƒ,๐พ)โˆˆ๐‘†๎€ฝ๐‘’Re๐‘–๐œƒ๎€พ||||๐‘’(๐œ‡โˆ’๐พ)โ‰ฅ0โŸบcos(๐œƒ+๐›พ)โ‰ฅ0where๐œ‡โˆ’๐พ=๐œ‡โˆ’๐พ๐‘–๐›พ.(2.5)

Using a nesting circle method based on that of both Weyl [1] and Sims, Brown et al. [12] divided (1.1) into three cases with respect to the corresponding half-planes ฮ›๐œƒ,๐พ as follows. The uniqueness referred to in the theorem and the following sections is only up to constant multiple.

Theorem 2.1 (cf. [12, Theoremโ€‰โ€‰2.1]). Given a (๐œƒ,๐พ)โˆˆ๐‘†, the following three distinct cases are possible. Case I. For all ๐œ†โˆˆฮ›๐œƒ,๐พ, equation (1.1) has unique solution ๐‘ฆ satisfying ๎€œ๐‘๐‘Ž๎‚ƒ๎€ฝ๐‘’Re๐‘–๐œƒ๐‘๎€พ||๐‘ฆ๎…ž||2๎€ฝ๐‘’+Re๐‘–๐œƒ๎€พ||๐‘ฆ||(๐‘žโˆ’๐พ๐‘ค)2๎‚„+๎€œ๐‘๐‘Ž๐‘ค||๐‘ฆ||2<โˆž(2.6) and this is the only solution satisfying ๐‘ฆโˆˆ๐ฟ2๐‘ค.Case II. For all ๐œ†โˆˆฮ›๐œƒ,๐พ, all solutions of (1.1) belong to ๐ฟ2๐‘ค, and there exists unique solution of (1.1) satisfying (2.6).Case III. For all ๐œ†โˆˆฮ›๐œƒ,๐พ, all solutions of (1.1) satisfy (2.6).

Since every ฮ›๐œƒ,๐พ is a half-plane, it holds that ฮ›๐œƒ1,๐พ1๎™ฮ›๐œƒ2,๐พ2โ‰ โˆ…(2.7) for (๐œƒ๐‘—,๐พ๐‘—)โˆˆ๐‘†, ๐‘—=1,2, with ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹). Note that (2.4) implies that, for 0<๐‘Ÿ<โˆž and ๐‘ฅโˆˆ[๐‘Ž,๐‘), ๎‚ป๐‘’Re๐‘–๐œƒ๎‚ต๐‘ž(๐‘ฅ)๐พ๐‘Ÿ๐‘ค(๐‘ฅ)+๐‘(๐‘ฅ)โˆ’๐‘Ÿ๎‚ถ๎‚ผโ‰ฅ0.(2.8) Letting ๐‘Ÿโ†’0 and ๐‘Ÿโ†’โˆž in (2.4) and (2.8), respectively, we have the following.

Lemma 2.2. For every (๐œƒ,๐พ)โˆˆ๐‘† and ๐œ†โˆˆฮ›๐œƒ,๐พ, there exists ๐›ฟ๐œ†(๐œƒ)>0 such that ๎€ฝ๐‘’Re๐‘–๐œƒ๎€พ๎€ฝ๐‘’(๐‘žโˆ’๐พ๐‘ค)โ‰ฅ0,Re๐‘–๐œƒ๎€พ(๐‘žโˆ’๐œ†๐‘ค)โ‰ฅ๐›ฟ๐œ†๎€ฝ๐‘’(๐œƒ)๐‘ค,Re๐‘–๐œƒ๐‘๎€พโ‰ฅ0(2.9) on [๐‘Ž,๐‘).

Using variation of parameters method, we can verify that, if all solutions of (1.1) belong to ๐ฟ2๐‘ค for some ๐œ†0โˆˆโ„‚, then it is true for all ๐œ†โˆˆโ„‚. This also means the following.

Lemma 2.3. If there exists a (๐œƒ0,๐พ0)โˆˆ๐‘† such that (1.1) is in Case I with respect to (with respect to for short) ฮ›๐œƒ0,๐พ0, then (1.1) is in Case I with respect to ฮ›๐œƒ,๐พ for every (๐œƒ,๐พ)โˆˆ๐‘†.

This indicates that Case I is independent of the choice of (๐œƒ,๐พ)โˆˆ๐‘†. But Cases II and III depend on the choice of (๐œƒ,๐พ)โˆˆ๐‘† in general, that is, there may exist (๐œƒ1,๐พ1),(๐œƒ2,๐พ2)โˆˆ๐‘† such that (1.1) is in Case II with respect to ฮ›๐œƒ1,๐พ1 and is in Case III with respect to ฮ›๐œƒ2,๐พ2. In order to make clear the dependence, we introduce the admissible angle set ๐ธ defined by ๐ธ={๐œƒโˆถโˆƒ๐พโˆ‰ฮฉโˆ˜,(๐œƒ,๐พ)โˆˆ๐‘†}.(2.10)

Remark 2.4. For given ๐œƒโˆˆ๐ธ, there exist many ๐พ such that (๐œƒ,๐พ)โˆˆ๐‘†. In fact, if ๐œƒ0โˆˆ๐ธ with (๐œƒ0,๐พ0)โˆˆ๐‘† for some ๐พ0โˆ‰ฮฉโˆ˜, then for all ๐พโˆˆ๐ฟ0, (๐œƒ0,๐พ)โˆˆ๐‘†๐ฟ0=๎€ฝ๎€ฝ๐‘’๐œ†โˆˆโ„‚โˆถRe๐‘–๐œƒ0๎€ท๐œ†โˆ’๐พ0๎€พ๎€ธ๎€พ=0.(2.11)

The exact dependence of Cases II and III on (๐œƒ,๐พ) can be given with the similar proof in [23, Theoremโ€‰โ€‰2.1].

Theorem 2.5 (cf. [23, Theoremโ€‰โ€‰2.1]). If there exists a (๐œƒ0,๐พ0)โˆˆ๐‘† such that (1.1) is in Case II with respect to ฮ›๐œƒ0,๐พ0, then (1.1) is in Case II with respect to ฮ›๐œƒ,๐พ for all (๐œƒ,๐พ)โˆˆ๐‘† except for at most one ๐œƒ1โˆˆ๐ธ(mod๐œ‹) such that (1.1) is in Case III with respect to ฮ›๐œƒ1,๐พ1.

Remark 2.6. Theorem 2.5 means that, if there exist ๐œƒ๐‘—โˆˆ๐ธ, ๐‘—=1,2, such that ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹) and (1.1) is in Case III with respect to ฮ›๐œƒ๐‘—,๐พ๐‘— for ๐‘—=1,2, then (1.1) is in Case III with respect to ฮ›๐œƒ,๐พ for all (๐œƒ,๐พ)โˆˆ๐‘†.

3. Properties of the Angel Set ๐ธ

This section gives some properties of the set ๐ธ, which will be used in the proof of our main results in Section 4. In what follows, we say that ๐ธ has more than one point if there exist ๐œƒ1,๐œƒ2โˆˆ๐ธ with ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹).

Lemma 3.1. Let ๐ธ be defined as in (2.10). (i)The set ๐ธ is connected in the sense of mod2๐œ‹. (ii)If ๐ธ has more than one point, then, for every ๐œ†โˆˆโ„‚โงตฮฉ, there exist ๐œƒ1,๐œƒ2โˆˆ๐ธ with ๐œƒ1<๐œƒ2 such that ๐œ†โˆˆฮ›๐œƒ,๐พ for ๐œƒโˆˆ(๐œƒ1,๐œƒ2)โŠ‚๐ธ.

Proof. (i) Suppose that ๐ธ has more than one point. Let ๐œƒ1,๐œƒ2โˆˆ๐ธ with ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹); then 0<๐œƒ2โˆ’๐œƒ1<๐œ‹ (mod2๐œ‹) or ๐œ‹<๐œƒ2โˆ’๐œƒ1<2๐œ‹ (mod2๐œ‹).
If 0<๐œƒ2โˆ’๐œƒ1<๐œ‹ (mod2๐œ‹) and (๐œƒ๐‘—,๐พ๐‘—)โˆˆ๐‘†, ๐‘—=1,2, then we claim that [๐œƒ1,๐œƒ2]โŠ‚๐ธ (mod2๐œ‹). Let ๐ฟ๐‘— be the line similarly defined as ๐ฟ0 with ๐พ0 and ๐œƒ0 replaced by ๐พ๐‘— and ๐œƒ๐‘—, ๐‘—=1,2. That is, ๐ฟ๐‘—=๎€ฝ๎€ฝ๐‘’๐œ†โˆˆโ„‚โˆถRe๐‘–๐œƒ๐‘—๎€ท๐œ†โˆ’๐พ๐‘—๎€พ๎€ธ๎€พ=0,๐พ๐‘—โˆ‰ฮฉโˆ˜,๐‘—=1,2.(3.1) Let ๐พ be the intersection point of ๐ฟ1 and ๐ฟ2. Set ||||๐‘’๐œ‡โˆ’๐พ=๐œ‡โˆ’๐พ๐‘–๐›พ(๐œ‡,๐พ),๐œ‡โˆˆฮฉ.(3.2) It follows from (2.5) that ๎€ท๐›พcos(๐œ‡,๐พ)+๐œƒ๐‘—๎€ธโ‰ฅ0,๐œ‡โˆˆฮฉ,๐‘—=1,2.(3.3)
By 0<๐œƒ2โˆ’๐œƒ1<๐œ‹ (mod2๐œ‹) and (3.3), we can get cos(๐›พ(๐œ‡,๐พ)+๐œƒ)โ‰ฅ0 for ๐œƒโˆˆ[๐œƒ1,๐œƒ2] (mod2๐œ‹) on ฮฉ, which means (๐œƒ,๐พ)โˆˆ๐‘† and ๐œƒโˆˆ๐ธ.
According to the similar method, we can verify that, if ๐œ‹<๐œƒ2โˆ’๐œƒ1<2๐œ‹ (mod2๐œ‹) and (๐œƒ๐‘—,๐พ๐‘—)โˆˆ๐‘†, ๐‘—=1,2, then [0,๐œƒ1]โˆช[๐œƒ2,2๐œ‹]โŠ‚๐ธ (mod2๐œ‹), that is, [๐œƒ2,๐œƒ1]โŠ‚๐ธ (mod2๐œ‹).
(ii) For ๐œ†0โˆˆโ„‚โงตฮฉ, choose (๐œƒ1,๐พ1)โˆˆ๐‘† and ๐›ฟ0>0 such that ๐œ†0โˆˆฮ›๐œƒ1,๐พ1 and ๎€ฝ๐‘’Re๐‘–๐œƒ1๎€ท๐พ1โˆ’๐œ†0๎€ธ๎€พ=๐›ฟ0>0.(3.4) Since ๐ธ has more than one point, we can choose ๐œƒ2โˆˆ๐ธ with ๐œƒ2โ‰ ๐œƒ1 (mod๐œ‹). Without loss of generality, we suppose that 0<๐œƒ2โˆ’๐œƒ1<๐œ‹ (mod2๐œ‹). Let ๐พ be defined as in the proof of (i).
If ๐œ†0โˆˆฮ›๐œƒ1,๐พโˆฉฮ›๐œƒ2,๐พ, then it follows from (2.3) that ๎€ทcos๐›พ+๐œƒ๐‘—๎€ธ<0,๐‘—=1,2,where๐œ†0||๐œ†โˆ’๐พ=0||๐‘’โˆ’๐พ๐‘–๐›พ.(3.5) By 0<๐œƒ2โˆ’๐œƒ1<๐œ‹ (mod2๐œ‹) and (3.5), we can get cos(๐›พ+๐œƒ)<0 for ๐œƒโˆˆ[๐œƒ1,๐œƒ2] (mod2๐œ‹), which means ๐œ†0โˆˆฮ›๐œƒ,๐พ for ๐œƒโˆˆ[๐œƒ1,๐œƒ2].
Suppose that ๐œ†0โˆ‰ฮ›๐œƒ2,๐พ. Let ๐œ†1โˆˆ๐ฟ1 be the unique point such that ๐›ฟ0=dist(๐œ†0,๐ฟ1)=dist(๐œ†1,๐œ†0). Let ๐›ผ=arctan|๐œ†0โˆ’๐œ†1|/2|๐พโˆ’๐œ†1|; then ๐›ผ+๐œƒ1โˆˆ(๐œƒ1,๐œƒ2)โŠ‚๐ธ by ๐œ†0โˆ‰ฮ›๐œƒ2,๐พ, and ๐œ†0โˆˆฮ›๐›ผ+๐œƒ1,๐พ by the definition of ๐›ผ (see Figure 1).
So, we can get that ๐œ†0โˆˆฮ›๐œƒ,๐พ for ๐œƒโˆˆ[๐œƒ1,๐œƒ1+๐›ผ] by ๐œ†0โˆˆฮ›๐œƒ1,๐พโˆฉฮ›๐œƒ1+๐›ผ,๐พ, and the lemma is proved.

4. Asymptotic Behavior

In this section, we will give asymptotic behavior of elements in the natural domain of the formally differential operator ๐œ defined on the interval [0,โˆž) with 0 being a regular end point and +โˆž being implicitly a singular end point. All results in this section can be stated for any singular end point, left or right on an arbitrary interval (๐‘Ž,๐‘), where โˆ’โˆžโ‰ค๐‘Ž<๐‘โ‰ค+โˆž. Recall that (1.1) on (๐‘Ž,๐‘) is said to be regular at ๐‘Ž if 1/๐‘, ๐‘ž and ๐‘ค are integrable on (๐‘Ž,๐‘) for some (and hence any) ๐‘โˆˆ(๐‘Ž,๐‘) and singular at ๐‘Ž otherwise; the regularity and singularity at ๐‘ are defined similarly (cf. [24]). Note that the regularity (resp., singularity) of an end point is solely determined by the integrability (resp., nonintegrability) of the coefficients in (1.1) at the end point, not the finiteness (resp., infiniteness) of the end point, as already remarked by Atkinson at the end of [13, Sectionโ€‰โ€‰9.1]. See also [10, Theoremโ€‰2.3.1]. Recall the definition of ๐’Ÿ(๐œ) in (1.3). We also define ๐’Ÿ๎€ท๐œ๎€ธ=๎€ฝ๐‘ฆโˆˆ๐ฟ2๐‘คโˆถ๐‘ฆ,๐‘๐‘ฆ๎…žโˆˆACloc,๐œ๐‘ฆโˆˆ๐ฟ2๐‘ค๎€พ,(4.1) where ๐œ๐‘ฆโˆถ=๐‘คโˆ’1๎€บโˆ’๎€ท๎€ธ๐‘๐‘ฆโ€ฒโ€ฒ+๎€ป[๐‘ž๐‘ฆon0,โˆž).(4.2) The first result of this section is as follows.

Theorem 4.1. (i) ๐œ is in Case I if and only if for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ) and ๐œƒโˆˆ๐ธ๐‘(๐‘ฅ)๐‘ฆ2(๐‘ฅ)๐‘ฆ๎…ž1(๐‘ฅ)+๐‘’2๐‘–๐œƒ๐‘(๐‘ฅ)๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2(๐‘ฅ)โŸถ0as๐‘ฅโŸถโˆž.(4.3)
(ii) ๐œ is in Case I if and only if for ๐‘ฆ1โˆˆ๐’Ÿ(๐œ),๐‘ฆ2โˆˆ๐’Ÿ(๐œ)๎‚ƒ๐‘ฆ๐‘(๐‘ฅ)2(๐‘ฅ)๐‘ฆ๎…ž1(๐‘ฅ)โˆ’๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2๎‚„(๐‘ฅ)โŸถ0as๐‘ฅโŸถโˆž.(4.4)

Remark 4.2. Clearly ๐’Ÿ(๐œ)=๐’Ÿ(๐œ), by the definition of ๐’Ÿ(๐œ). It is easy to see that (4.4) is equivalent to ๐‘๎€บ๐‘ฆ(๐‘ฅ)2(๐‘ฅ)๐‘ฆ๎…ž1(๐‘ฅ)โˆ’๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2๎€ป(๐‘ฅ)โŸถ0as๐‘ฅโŸถโˆž(4.5) for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ).

We will use spectral theory of Hamiltonian differential systems to prove Theorem 4.1, so that we first prepare some known results for the Hamiltonian differential system ๐‘ข๎…ž=๐ด๐‘ข+๐ต๐‘ฃ+๐œ‰๐‘Š2๐‘ฃ,๐‘ฃโ€ฒ=๐ถ๐‘ขโˆ’๐ดโˆ—๐‘ฃโˆ’๐œ‰๐‘Š1[๐‘ข,on0,โˆž),(4.6) where ๐‘ข,๐‘ฃ are โ„‚๐‘› valued functions, ๐‘ข๐‘‡ is the transpose of ๐‘ข, ๐ด,๐ต,๐ถ,๐‘Š1, and ๐‘Š2 are locally integrable, complex valued ๐‘›ร—๐‘› matrices on [0,โˆž), ๐ต,๐ถ,๐‘Š1, and ๐‘Š2 are Hermit matrices and ๐‘Š1(๐‘ก)>0,๐‘Š2(๐‘ก)โ‰ฅ0 on [0,โˆž), and ๐œ‰ is the spectral parameter. Assume that the definiteness condition (see, e.g., [13, Chapterโ€‰โ€‰9, page 253]) holds: ๎€œโˆž0๐‘ฆโˆ—๐‘Š๐‘ฆ>0foreachnontrivialsolution๐‘ฆof(4.6),(4.7) where ๐‘Š=diag(๐‘Š1,๐‘Š2). Let ๐ฟ2๐‘Šโˆถ=๐ฟ2๐‘Š[0,โˆž) denote the space of Lebesgue measurable 2๐‘›-dimensional functions ๐‘“ satisfying โˆซโˆž0๐‘“โˆ—(๐‘ )๐‘Š(๐‘ )๐‘“(๐‘ )๐‘‘๐‘ <โˆž. We say that (4.6) is in the limit point case at infinity if there exists exactly ๐‘›'s solutions of (4.6) belonging to ๐ฟ2๐‘Š for ๐œ‰โˆˆโ„‚ with Im๐œ‰โ‰ 0.

Let ๐’Ÿ be the maximal domain associated with (4.6), that is, (๐‘ข๐‘‡,๐‘ฃ๐‘‡)๐‘‡โˆˆ๐’Ÿ if and only if (๐‘ข๐‘‡,๐‘ฃ๐‘‡)๐‘‡โˆˆAClocโˆฉ๐ฟ2๐‘Š, and there exists an element (๐‘“๐‘‡,๐‘”๐‘‡)๐‘‡โˆˆ๐ฟ2๐‘Š such that ๐‘ข๎…ž=๐ด๐‘ข+๐ต๐‘ฃ+๐œ‰๐‘Š2๐‘ฃ+๐‘Š2๐‘”,๐‘ฃโ€ฒ=๐ถ๐‘ขโˆ’๐ดโˆ—๐‘ฃโˆ’๐œ‰๐‘Š1๐‘ขโˆ’๐‘Š1[๐‘“,on0,โˆž).(4.8) It is well known (cf. [5, 7]) that (4.6) is in the limit point case at infinity if and only if ๐‘Œโˆ—1(๐‘ฅ)๐ฝ๐‘Œ2๎ƒฉ(๐‘ฅ)โŸถ0as๐‘ฅโŸถโˆž,๐ฝ=0โˆ’๐ผ๐‘›๐ผ๐‘›0๎ƒช(4.9) for ๐‘Œ1,๐‘Œ2โˆˆ๐’Ÿ, and for every ๐œ‰โˆˆโ„‚ with Im๐œ‰โ‰ 0 there exists a Green function ๐บ(๐‘ก,๐‘ ,๐œ‰) such that, for ๐น=(๐‘“๐‘‡,๐‘”๐‘‡)๐‘‡โˆˆ๐ฟ2๐‘Š, ๎ƒฉ๐‘ข๐‘ฃ๎ƒช๐‘Œ==๐‘‡๐œ‰๐นโˆˆ๐ฟ2๐‘Š,satis๏ฌes(4.8),(4.10) where (๐‘‡๐œ‰โˆซ๐น)(๐‘ฅ)=โˆž0๐บ(๐‘ฅ,๐‘ ,๐œ‰)๐‘Š(๐‘ )๐น(๐‘ )d๐‘ .

Let (๐œƒ,๐พ)โˆˆ๐‘† and choose ๐œ†0โˆˆฮ›๐œƒ,๐พ. Then from (2.9), one sees that ๎€ฝ๐‘’Re๐‘–๐œƒ๎€ท๐‘žโˆ’๐œ†0๐œ”๎€ธ๎€พโ‰ฅ๐›ฟ0๎€ฝ๐‘’๐œ”>0,Re๐‘–๐œƒ๎€ท๐‘žโˆ’๐พ0๐‘ค๎€ฝ๐‘’๎€ธ๎€พโ‰ฅ0,Re๐‘–๐œƒ๐‘๎€พโ‰ฅ0(4.11) for some ๐›ฟ0>0. Set ๐‘Ÿ1||(๐‘ฅ)=๐‘ž(๐‘ฅ)โˆ’๐œ†0||๐‘ค(๐‘ฅ),๐‘ž(๐‘ฅ)โˆ’๐œ†0๐‘ค(๐‘ฅ)=๐‘Ÿ1(๐‘ฅ)๐‘’๐‘–๐›ผ(๐‘ฅ),๐›ผ1๐‘Ÿ(๐‘ฅ)=๐œƒ+๐›ผ(๐‘ฅ),2(||||๐‘ฅ)=๐‘(๐‘ฅ),๐‘(๐‘ฅ)=๐‘Ÿ2(๐‘ฅ)๐‘’๐‘–๐›ฝ(๐‘ฅ),๐›ฝ1(๐‘ฅ)=๐œƒ+๐›ฝ(๐‘ฅ).(4.12) Consider the Hamiltonian differential system (4.6) with ๐‘›=1, ๐ด(๐‘ฅ)โ‰ก0 and ๐ถ(๐‘ฅ)=๐‘Ÿ1(๐‘ฅ)sin๐›ผ1(๐‘ฅ),๐‘Š1(๐‘ฅ)โˆถ=๐‘ค1(๐‘ฅ)=๐‘Ÿ1(๐‘ฅ)cos๐›ผ1(๐‘ฅ),๐ต(๐‘ฅ)=sin๐›ฝ1(๐‘ฅ)๐‘Ÿ2(๐‘ฅ),๐‘Š2(๐‘ฅ)โˆถ=๐‘ค2(๐‘ฅ)=cos๐›ฝ1(๐‘ฅ)๐‘Ÿ2(,๐‘ฅ)(4.13) that is, the 2-dimensional Hamiltonian differential system ๐ป(๐œƒ)โˆถ๐‘ขโ€ฒ=๐ต๐‘ฃ+๐œ‰๐‘ค2๐‘ฃ,๐‘ฃ๎…ž=๐ถ๐‘ขโˆ’๐œ‰๐‘ค1๐‘ข.(4.14) It follows from (4.11) that ๐‘ค1๎€ฝ๐‘’=Re๐‘–๐œƒ๎€ท๐‘žโˆ’๐œ†0๐‘ค๎€ธ๎€พโ‰ฅ๐›ฟ0๐‘ค>0,๐‘ค2=๎€ฝ๐‘’Re๐‘–๐œƒ๎€พ๐‘(๐‘ก)๐‘Ÿ22โ‰ฅ0,(4.15) and it is easy to verify that the definiteness condition holds for the system (4.14). In fact, ๐‘ฆ is a solution of (1.1) if and only if (๐‘ข,๐‘ฃ)๐‘‡ is a solution of (4.14) with ๐‘ข=๐‘ฆ,๐‘ฃ=โˆ’๐‘–๐‘’๐‘–๐œƒ๐‘๐‘ฆโ€ฒ.(4.16) This fact immediately yields the following result which is frequently used in the proof of Theorems 4.1 and 4.11.

Lemma 4.3. (i) ๐œ is in Case I or Case II with respect to (๐œƒ,๐พ)โˆˆ๐‘† if and only if ๐ป(๐œƒ) is in the limit point case at โˆž.
(ii) ๐œ is in Case III with respect to (๐œƒ,๐พ)โˆˆ๐‘† if and only if ๐ป(๐œƒ) is in the limit circle case at โˆž.

Lemma 4.4. If ๐ธ has more than one point, then ๐’Ÿ๐œƒ(๐œ)โ‰ก๐’Ÿ๐‘ (๐œ) on ๐ธ๐‘œ, the interior of ๐ธ, where ๐’Ÿ๐œƒ๎‚ป๎€œ(๐œ)=๐‘ฆโˆˆ๐’Ÿ(๐œ)โˆถโˆž0๎‚ƒ๎€ฝ๐‘’Re๐‘–๐œƒ๐‘๎€พ||๐‘ฆ๎…ž||2๎€ฝ๐‘’+Re๐‘–๐œƒ๐‘ž๎€พ||๐‘ฆ||2๎‚„๎‚ผ,๐’Ÿ<โˆž๐‘ ๎‚ป๎€œ(๐œ)=๐‘ฆโˆˆ๐’Ÿ(๐œ)โˆถโˆž0๎‚ƒ||๐‘||||๐‘ฆ๎…ž||2+||๐‘ž||||๐‘ฆ||2๎‚„๎‚ผ.<โˆž(4.17)

Proof. Let ๐œƒ1โˆˆ๐ธ๐‘œ be fixed. There exist ๐œƒ2,๐œƒ3โˆˆ๐ธ๐‘œ such that ๐œƒ3<๐œƒ1<๐œƒ2mod(2๐œ‹),0<๐œƒ2โˆ’๐œƒ3<๐œ‹,2mod(2๐œ‹)3๎™๐‘—=1ฮ›๐œƒ๐‘—,๐พ๐‘—โ‰ โˆ…(4.18) by Lemma 3.1. Choose ๐œ†0โˆˆโ‹‚3๐‘—=1ฮ›๐œƒ๐‘—,๐พ๐‘—. Letting ๐›ฝโˆถ=๐›ฝ(๐‘ฅ) be defined as in (4.12) and solving cos(๐œƒ1+๐›ฝ) from the equations ๎€ท๐œƒcos๐‘—๎€ธ๎€ท๐œƒ+๐›ฝ=cos1๎€ธ๎€ท๐œƒ+๐›ฝcos๐‘—โˆ’๐œƒ1๎€ธ๎€ท๐œƒโˆ’sin1๎€ธ๎€ท๐œƒ+๐›ฝsin๐‘—โˆ’๐œƒ1๎€ธ,๐‘—=2,3,(4.19) we have that cos(๐œƒ1+๐›ฝ)=๐ถ1cos(๐œƒ2+๐›ฝ)+๐ถ2cos(๐œƒ3+๐›ฝ) with ๐ถ1=๎€ท๐œƒsin1โˆ’๐œƒ3๎€ธ๎€ท๐œƒsin2โˆ’๐œƒ3๎€ธ>0,๐ถ2=๎€ท๐œƒsin2โˆ’๐œƒ1๎€ธ๎€ท๐œƒsin2โˆ’๐œƒ3๎€ธ>0(4.20) by (4.18). Since โˆซโˆž0Re{๐‘’๐‘–๐œƒ1๐‘}|๐‘ฆ๎…ž|2<โˆž for ๐‘ฆโˆˆ๐’Ÿ๐œƒ1(๐œ), we have that ๐ถ1๎€œโˆž0๎€ฝ๐‘’Re๐‘–๐œƒ2๐‘๎€พ||๐‘ฆ๎…ž||2+๐ถ2๎€œโˆž0๎€ฝ๐‘’Re๐‘–๐œƒ3๐‘๎€พ||๐‘ฆ๎…ž||2=๎€œโˆž0๎€ฝ๐‘’Re๐‘–๐œƒ1๐‘๎€พ||๐‘ฆ๎…ž||2<โˆž,(4.21) and hence โˆซโˆž0Re{๐‘’๐‘–๐œƒ2๐‘}|๐‘ฆ๎…ž|2<โˆž for ๐‘ฆโˆˆ๐’Ÿ๐œƒ1(๐œ). The same proof as the above with ๐›ฝ replaced by ๐›ผ also proves โˆซโˆž0Re{๐‘’๐‘–๐œƒ2(๐‘žโˆ’๐œ†0๐‘ค)}|๐‘ฆ|2<โˆž for ๐‘ฆโˆˆ๐’Ÿ๐œƒ1(๐œ), where ๐›ผโˆถ=๐›ผ(๐‘ฅ) is defined as in (4.12). Therefore, for ๐‘ฆโˆˆ๐’Ÿ๐œƒ1(๐œ), ๎€œโˆž0๎‚ƒ๎€ฝ๐‘’Re๐‘–๐œƒ๐‘—๐‘๎€พ||๐‘ฆ๎…ž||2๎‚„,๎€œโˆž0๎€ฝ๐‘’Re๐‘–๐œƒ๐‘—๎€ท๐‘žโˆ’๐œ†0๐‘ค||๐‘ฆ||๎€ธ๎€พ2<โˆž,๐‘—=1,2.(4.22) Set ๐‘๐œƒ=๐‘’๐‘–๐œƒ๐‘ and ๐‘ž๐œƒ=๐‘’๐‘–๐œƒ(๐‘žโˆ’๐œ†0๐‘ค). It follows from sin2๎€ท๐œƒ2โˆ’๐œƒ1๎€ธ=cos2๐œƒ2+cos2๐œƒ1โˆ’2cos๐œƒ2cos๐œƒ1๎€ท๐œƒcos1โˆ’๐œƒ2๎€ธโ‰ค๎€ทcos๐œƒ2+cos๐œƒ1๎€ธ2(4.23) and (4.15) that ๎€ท๐‘Re๐œƒ1+๐‘๐œƒ2๎€ธโ‰ฅ๐œ€0||๐‘||๎€ท๐‘ž,Re๐œƒ1+๐‘ž๐œƒ2๎€ธโ‰ฅ๐œ€0||||๐‘žโˆ’๐œ†๐‘ค,๐œ€0๎€ท๐œƒ=sin2โˆ’๐œƒ1๎€ธ.(4.24) Then (4.24) and (4.22) yield that, for ๐‘ฆโˆˆ๐’Ÿ๐œƒ1(๐œ), ๎€œโˆž0||๐‘||||๐‘ฆ๎…ž||2,๎€œโˆž0||๐‘žโˆ’๐œ†0๐‘ค||||๐‘ฆ||2<โˆž.(4.25) Note that ๐‘ฆโˆˆ๐ฟ2๐‘ค. Then (4.25) gives ๐‘ฆโˆˆ๐’Ÿ๐‘ (๐œ), or ๐’Ÿ๐œƒ1(๐œ)โŠ‚๐’Ÿ๐‘ (๐œ). Clearly, ๐’Ÿ๐‘ (๐œ)โŠ‚๐’Ÿ๐œƒ1(๐œ). Thus ๐’Ÿ๐œƒ1(๐œ)=๐’Ÿ๐‘ (๐œ).

Lemma 4.4 indicates the following.

Corollary 4.5. If ๐œ is in Case II with respect to some (๐œƒ0,๐พ0)โˆˆ๐‘† and ๐ธ has more than one point, then Case III only occurs at the end point of ๐ธ.

Proof. If ๐œ is in Case III with respect to some (๐œƒ1,๐พ1)โˆˆ๐‘† with ๐œƒ1โˆˆ๐ธ๐‘œ, then ๐’Ÿ(๐œ)=๐’Ÿ๐œƒ1(๐œ) is restricted in the solution space of (1.1) by the definition of Case III. Since ๐’Ÿ๐œƒ1(๐œ)=๐’Ÿ๐‘ (๐œ) by Lemma 4.4, we have that ๐’Ÿ(๐œ)=๐’Ÿ๐‘ (๐œ) restricted in the solution space of (1.1). This means that all solutions of (1.1) with ๐œ†โˆˆฮ›๐œƒ1,๐พ1 satisfy ๎€œโˆž0๎‚€||๐‘||||๐‘ฆ๎…ž||2+||๐‘ž||||๐‘ฆ||2๎‚<โˆž.(4.26) Using variation of parameters method we can prove that it is true for all ๐œ†โˆˆโ„‚, and hence ๐œ is in Case III with respect to (๐œƒ0,๐พ0), a contradiction.

Lemma 4.6. If ๐œ is in Case I and ๐‘ฆโˆˆ๐’Ÿ(๐œ), then (๐‘ฆ,๐‘ฃ)๐‘‡โˆˆ๐’Ÿ(๐œƒ) with ๐‘ฃ=โˆ’๐‘–๐‘’๐‘–๐œƒ๐‘๐‘ฆ๎…ž, where ๐’Ÿ(๐œƒ) is the maximal domain associated with (4.14).

Proof. Suppose that ๐œ is in Case I with respect to (๐œƒ,๐พ)โˆˆ๐‘†. We claim that ๐’Ÿ(๐œ)=๐’Ÿ๐œƒ(๐œ). Set ๎€ท๐œโˆ’๐œ†0๎€ธ๐‘ฆ0=๐‘คโˆ’1๎‚ƒโˆ’๎€ท๐‘๐‘ฆ๎…ž0๎€ธ๎…ž+๎€ท๐‘žโˆ’๐œ†0๐‘ค๎€ธ๐‘ฆ0๎‚„=๐‘”0,(4.27) for ๐‘ฆ0โˆˆ๐’Ÿ(๐œ) and ๐œ†0โˆˆฮ›๐œƒ,๐พ.
Set ๐‘ข0=๐‘ฆ0, ๐‘ฃ0=โˆ’๐‘–๐‘’๐‘–๐œƒ๐‘๐‘ฆ๎…ž0. Then (๐‘ข0,๐‘ฃ0) satisfies ๐‘ขโ€ฒ=๐ต๐‘ฃ+๐‘–๐‘ค2๐‘ฃ,๐‘ฃ๎…ž=๐ถ๐‘ขโˆ’๐‘–๐‘ค1๐‘ขโˆ’๐‘ค1๐‘“1,๐‘“1=๐‘ค๐‘ค1๎€ทโˆ’๐‘–๐‘’๐‘–๐œƒ๐‘”0๎€ธ.(4.28) Conversely, if (๐‘ข,๐‘ฃ) satisfies (4.28), then ๐‘ฆ=๐‘ข solves (4.27). Note that ๐‘”0โˆˆ๐ฟ2๐‘ค, or โˆ’๐‘–๐‘’๐‘–๐œƒ0๐‘”0โˆˆ๐ฟ2๐‘ค, and ๐‘ค1โ‰ฅ๐›ฟ๐‘ค implies ๐‘“1โˆˆ๐ฟ2๐‘ค1.
Considering (4.28), we get from (4.10) that (4.28) has a solution (๐‘ข1,๐‘ฃ1)๐‘‡ such that ๐‘ข1โˆˆ๐ฟ2๐‘ค1,๐‘ฃ1โˆˆ๐ฟ2๐‘ค2 and ๐‘ฃ1=โˆ’๐‘–๐‘’๐‘–๐œƒ0๐‘๐‘ข๎…ž1. Set ๐‘ฆ1=๐‘ข1. Then ๐‘ฆ1 satisfies (4.27), and hence (๐œโˆ’๐œ†0)(๐‘ฆ0โˆ’๐‘ฆ1)=0. Note that ๐‘ฆ1=๐‘ข1โˆˆ๐ฟ2๐‘ค1, and ๐‘ค1โ‰ฅ๐›ฟ๐‘ค implies that ๐‘ฆ1โˆˆ๐ฟ2๐‘ค. Thus, ๐‘ฆ1โˆ’๐‘ฆ0 is an ๐ฟ2๐‘ค-solution of ๐œ๐‘ฆ=๐œ†0๐‘ฆ. Since ๐œ is in Case I with respect to (๐œƒ0,๐พ0), it follows from (2.6) that ๐‘ฆ1โˆ’๐‘ฆ0โˆˆ๐ฟ2๐‘ค1 and ๐‘ฃ1โˆ’๐‘ฃ0โˆˆ๐ฟ2๐‘ค2. This together with ๐‘ฆ1โˆˆ๐ฟ2๐‘ค1 and ๐‘ฃ1โˆˆ๐ฟ2๐‘ค2 gives ๐‘ฆ0โˆˆ๐ฟ2๐‘ค1 and ๐‘ฃ0โˆˆ๐ฟ2๐‘ค2. In fact, we have proved that, for ๐‘ฆโˆˆ๐’Ÿ(๐œ), ๎€œโˆž0||๐‘žโˆ’๐œ†0๐‘ค||cos๐›ผ1||๐‘ฆ||2๎€œ<โˆž,โˆž0||๐‘||cos๐›ฝ1||๐‘ฆ๎…ž||2<โˆž,(4.29) or ๎€œโˆž0๎‚ƒ๎€ฝ๐‘’Re๐‘–๐œƒ๐‘๎€พ||๐‘ฆ๎…ž||2๎€ฝ๐‘’+Re๐‘–๐œƒ๎€ท๐‘žโˆ’๐œ†0๐‘ค||๐‘ฆ||๎€ธ๎€พ2๎‚„<โˆž,(4.30) where ๐›ผ1 and ๐›ฝ1 are defined in (4.12) or (4.13). Since ๐‘ฆโˆˆ๐ฟ2๐‘ค, (4.30) means that ๎€œโˆž0๎‚ƒ๎€ฝ๐‘’Re๐‘–๐œƒ๐‘๎€พ||๐‘ฆ๎…ž||2๎€ฝ๐‘’+Re๐‘–๐œƒ๐‘ž๎€พ||๐‘ฆ||2๎‚„<โˆž(4.31) or ๐‘ฆโˆˆ๐’Ÿ๐œƒ(๐œ), and hence ๐’Ÿ(๐œ)=๐’Ÿ๐œƒ(๐œ). Recall that ๐‘“1โˆˆ๐ฟ2๐‘ค1. Then (4.30) and (4.28) imply that, if ๐‘ฆโˆˆ๐’Ÿ(๐œ), then (๐‘ฆ,๐‘ฃ)๐‘‡โˆˆ๐’Ÿ(๐œƒ).

Corollary 4.7. If ๐œ is in Case I and ๐‘ฆโˆˆ๐’Ÿ(๐œ), then (๐‘ฆ,๐‘ฃ)๐‘‡โˆˆ๐’Ÿ(๐œƒ) with ๐‘ฃ=๐‘–๐‘’โˆ’๐‘–๐œƒ๐‘๐‘ฆ๎…ž.

Proof. For ๐‘ฆโˆˆ๐’Ÿ(๐œ), ๐‘ฆโˆˆ๐’Ÿ(๐œ) by ๐’Ÿ(๐œ)=๐’Ÿ(๐œ). So (๐‘ฆ,๐‘ฃ)๐‘‡โˆˆ๐’Ÿ(๐œƒ) with ๐‘ฃ=โˆ’๐‘–๐‘’๐‘–๐œƒ๐‘๐‘ฆ๎…ž by Lemma 4.6. Clearly ๐’Ÿ(๐œƒ)=๐’Ÿ(๐œƒ) since ๐ป(๐œƒ) is symmetrical. Then we have that (๐‘ฆ,๐‘ฃ)๐‘‡โˆˆ๐’Ÿ(๐œƒ) with ๐‘ฃ=๐‘–๐‘’โˆ’๐‘–๐œƒ๐‘๐‘ฆ๎…ž.

Proof of Theorem 4.1. The proof of (i): suppose that ๐œ is in Case I. Since (4.14) is in the limit point case at infinity by Lemma 4.3, we know that (4.9) holds for all ๐‘Œ1,๐‘Œ2โˆˆ๐’Ÿ(๐œƒ). For ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ), since (๐‘ฆ๐‘—,๐‘ฃ๐‘—)๐‘‡โˆˆ๐’Ÿ(๐œƒ) with ๐‘ฃ๐‘—=โˆ’๐‘–๐‘’๐‘–๐œƒ๐‘๐‘ฆ๎…ž๐‘—, ๐‘—=1,2, by Lemma 4.6, it follows from (4.9) that ๎€ท๐‘ฆ1,๐‘ฃ1๎€ธ๎ƒฉ๐‘ฆ0โˆ’110๎ƒช๎ƒฉ2๐‘ฃ2๎ƒช=๐‘–๐‘’โˆ’๐‘–๐œƒ๎€ท๐‘๐‘ฆ๎…ž1๐‘ฆ2+๐‘’2๐‘–๐œƒ๐‘๐‘ฆ1๐‘ฆ๎…ž2๎€ธโŸถ0(4.32) as ๐‘ฅโ†’โˆž.
Conversely, assume that (4.3) holds for all elements of ๐’Ÿ(๐œ). We claim that (1.1) must be in Case I. Suppose on the contrary that (1.1) is not in Case I. Then all solutions of (1.1) belong to ๐ฟ2๐‘ค for ๐œ†โˆˆโ„‚. Choose ๐œ†0โˆˆฮ›๐œƒ,๐พ, and let ๐‘ฆ0 be a nontrivial solution of (1.1) satisfying ๐‘ฆ0(0)=0. Then ๐‘ฆ0โˆˆ๐’Ÿ(๐œ) by ๐‘ฆ0โˆˆ๐ฟ2๐‘ค. Furthermore, it follows from (๐œโˆ’๐œ†0)๐‘ฆ0=0 that โˆ’๎€ท๐‘๐‘ฆ๎…ž0๎€ธ๎…ž๐‘ฆ0+๎€ท๐‘žโˆ’๐œ†0๐‘ค๎€ธ||๐‘ฆ0||2๎€ท=0,โˆ’๐‘๐‘ฆ๎…ž0๎€ธ๎…ž๐‘ฆ0+๎‚€๐‘žโˆ’๐œ†0๐‘ค๎‚||๐‘ฆ0||2=0.(4.33) Integrating (4.33) on [0,๐‘ฅ] we have that โˆ’(๐‘๐‘ฆ๎…ž0)๐‘ฆ0||๐‘ฅ0+๎€œ๐‘ฅ0๎‚ƒ๐‘||๐‘ฆ๎…ž0||2+๎€ท๐‘žโˆ’๐œ†0๐‘ค๎€ธ||๐‘ฆ0||2๎‚„=0,โˆ’(๐‘๐‘ฆ๎…ž0)๐‘ฆ0||๐‘ฅ0+๎€œ๐‘ฅ0๎‚ƒ๐‘||๐‘ฆ๎…ž0||2+๎‚€๐‘žโˆ’๐œ†0๐‘ค๎‚||๐‘ฆ0||2๎‚„=0.(4.34) Multiplying ๐‘’๐‘–๐œƒ and ๐‘’โˆ’๐‘–๐œƒ to the first and second equalities in (4.34), respectively, and adding them together, we have that โˆ’๎€บ๐‘’๐‘–๐œƒ๎€ท๐‘๐‘ฆ๎…ž0๎€ธ๐‘ฆ0+๐‘’โˆ’๐‘–๐œƒ๎€ท๐‘๐‘ฆ๎…ž0๎€ธ๐‘ฆ0๎€ป๎€œ(๐‘ฅ)+๐‘ฅ0๎‚ƒ๎€ท๐‘๐œƒ+๐‘๐œƒ๎€ธ||๐‘ฆ๎…ž0||2+๎€ท๐‘ž๐œƒ+๐‘ž๐œƒ๎€ธ||๐‘ฆ0||2๎‚„=0(4.35) since ๐‘ฆ0(0)=0, where ๐‘๐œƒ=๐‘’๐‘–๐œƒ๐‘ and ๐‘ž๐œƒ=๐‘’๐‘–๐œƒ(๐‘žโˆ’๐œ†0๐‘ค). Note that ๎€บ๐‘’๐‘–๐œƒ๎€ท๐‘๐‘ฆ๎…ž0๎€ธ๐‘ฆ0+๐‘’โˆ’๐‘–๐œƒ๎€ท๐‘๐‘ฆ๎…ž0๎€ธ๐‘ฆ0๎€ป(๐‘ฅ)=๐‘’โˆ’๐‘–๐œƒ๎€บ๐‘๐‘ฆ๎…ž0๐‘ฆ0+๐‘’2๐‘–๐œƒ๐‘๐‘ฆ๎…ž0๐‘ฆ0๎€ป(๐‘ฅ)โŸถ0(4.36) as ๐‘ฅโ†’โˆž by assumption (4.3) and Re๐‘๐œƒ๎€ฝ๐‘’=Re๐‘–๐œƒ๐‘๎€พโ‰ฅ0,Re๐‘ž๐œƒ๎€ฝ๐‘’=Re๐‘–๐œƒ๎€ท๐‘žโˆ’๐œ†0๐‘ค๎€ธ๎€พโ‰ฅ๐›ฟ0๐‘ค(4.37) by (4.15). Then letting ๐‘ฅโ†’โˆž in (4.35), we have a contradiction. This proves the first part of this theorem.
The proof of (ii): suppose that ๐œ is in Case I. Set ๐‘ฃ1=โˆ’๐‘–๐‘’๐‘–๐œƒ๐‘๐‘ฆ๎…ž1, ๐‘ฃ2=๐‘–๐‘’โˆ’๐‘–๐œƒ๐‘๐‘ฆ๎…ž2 for ๐‘ฆ1โˆˆ๐’Ÿ(๐œ), ๐‘ฆ2โˆˆ๐’Ÿ(๐œ). Then, we can get (๐‘ฆ1,๐‘ฃ1)๐‘‡โˆˆ๐’Ÿ(๐œƒ) by Lemma 4.6 and (๐‘ฆ2,๐‘ฃ2)๐‘‡โˆˆ๐’Ÿ(๐œƒ) by Corollary 4.7. Hence ๎€ท๐‘ฆ1,๐‘ฃ1๎€ธ๎ƒฉ๐‘ฆ0โˆ’110๎ƒช๎ƒฉ2๐‘ฃ2๎ƒช=๐‘–๐‘’โˆ’๐‘–๐œƒ๎€ท๐‘๐‘ฆ๎…ž1๐‘ฆ2โˆ’๐‘๐‘ฆ1๐‘ฆ2๎€ธโŸถ0(4.38) as ๐‘ฅโ†’โˆž by (4.9), that is, ๐‘(๐‘ฅ)[๐‘ฆ2(๐‘ฅ)๐‘ฆ๎…ž1(๐‘ฅ)โˆ’๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2(๐‘ฅ)]โ†’0 as ๐‘ฅโ†’โˆž.
Conversely, if ๐œ is not in Case I, then all solutions of (1.1) belong to ๐ฟ2๐‘ค for ๐œ†โˆˆโ„‚. Let ๐‘ฆ๐‘–,๐‘–=1,2, be the solution of (๐œโˆ’๐œ†0)๐‘ฆ=0 such that ๎ƒฉ๐‘๐‘ฆ๎…ž1๐‘ฆ(0)1๎ƒช=๎ƒฉ10๎ƒช,๎ƒฉ(0)๐‘๐‘ฆ๎…ž2๐‘ฆ(0)2๎ƒช=๎ƒฉ01๎ƒช(0).(4.39) Since ๐‘ฆ๐‘–โˆˆ๐ฟ2๐‘ค,๐‘ฆ๐‘–โˆˆ๐ท(๐œ),๐‘–=1,2. Then the Wronskian |||||๐‘๐‘ฆ๎…ž1๐‘ฆ1๐‘๐‘ฆ๎…ž2๐‘ฆ2|||||๎€ท๐‘ฆ=๐‘๎…ž1๐‘ฆ2โˆ’๐‘ฆ1๐‘ฆ๎…ž2๎€ธโ‰ก1,(4.40) which contradicts condition (4.5). See Remark 4.2.

Remark 4.8. If ๐‘ž(๐‘ฅ) and ๐‘(๐‘ฅ) are real valued, then ฮฉโŠ‚โ„ and (๐œƒ,๐พ)=(ยฑ๐œ‹/2,0)โˆˆ๐‘† with Re{๐‘’๐‘–๐œƒ๐‘(๐‘ฅ)}=Re{๐‘’๐‘–๐œƒ(๐‘ž(๐‘ฅ)โˆ’๐พ๐‘ค(๐‘ฅ))}โ‰ก0. This means that Case I, Cases II and III reduce to Weyl's limit point, limit-circle cases, respectively. For this case, we know that (1.1) is in the limit point case at โˆž if and only if ๎‚ƒ๐‘ฆ๐‘(๐‘ฅ)2(๐‘ฅ)๐‘ฆ๎…ž1(๐‘ฅ)โˆ’๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2๎‚„(๐‘ฅ)โŸถ0as๐‘ฅโŸถโˆž(4.41) for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ), that is, (1.4). Clearly, if ๐‘ is real valued and ๐œ‹/2โˆˆ๐ธ, then (4.3) reduces to (1.4). Therefore, (4.3) is a generalization of (1.4).

Corollary 4.9. If ๐ธ has more than one point, then ๐œ is in Case I if and only if, for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ), ๐‘(๐‘ฅ)๐‘ฆ1(๐‘ฅ)๐‘ฆ๎…ž2(๐‘ฅ)โŸถ0as๐‘ฅโŸถโˆž.(4.42) That is ๐œ is in the strong limit point case at โˆž.

Proof. Suppose that ๐ธ has more than one point and ๐œ is in Case I. Choose ๐œƒ๐‘—โˆˆ๐ธ, ๐‘—=1,2, with ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹). Then (4.3) holds for ๐œƒ=๐œƒ๐‘—, ๐‘—=1,2. This gives that for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ(๐œ)๎€ท๐‘’2๐‘–๐œƒ1โˆ’๐‘’2๐‘–๐œƒ2๎€ธ๐‘๐‘ฆ1๐‘ฆ๎…ž2โŸถ0as๐‘ฅโŸถโˆž,(4.43) and hence (4.42) holds since ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹).
Conversely, assume that (4.42) holds for all ๐‘ฆ๐‘–โˆˆ๐’Ÿ(๐œ), ๐‘–=1,2. Since (4.42) implies (4.3), we conclude from (i) of Theorem 4.1 that ๐œ is in Case I.

Corollary 4.10. If ๐œ is symmetric and ๐‘ž(๐‘ฅ)โ‰ฅ๐‘ž0๐‘ค(๐‘ฅ) on [0,โˆž), then ๐œ is in the limit point case at โˆž if and only if it is in the strong limit point case at โˆž.

Proof. Note that for, ๐œƒโˆˆ[โˆ’๐œ‹/2,๐œ‹/2], (๐œƒ,๐‘ž0)โˆˆ๐‘†. Then [โˆ’๐œ‹/2,๐œ‹/2]โˆˆ๐ธ. Therefore, (4.3) holds if and only if (4.42) holds by Corollary 4.9.

Theorem 4.11. ๐œ is in Case II with respect to (๐œƒ0,๐พ0)โˆˆ๐‘† if and only if ๐’Ÿ(๐œ)โ‰ ๐’Ÿ๐œƒ0(๐œ)โ‰ โˆ… and (4.3) holds for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ๐œƒ0(๐œ).

Proof. Suppose that ๐œ is in Case II with respect to some (๐œƒ,๐พ)โˆˆ๐‘†. By the definition of Case II we know that ๐’Ÿ๐œƒ(๐œ) is nonempty and ๐’Ÿ(๐œ)โ‰ ๐’Ÿ๐œƒ(๐œ). With a similar proof to that one in the first part of (i) in Theorem 4.1, we can get that (4.3) holds for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ๐œƒ0(๐œ) by Lemma 4.3 and (4.32).
Conversely, suppose that ๐’Ÿ(๐œ)โ‰ ๐’Ÿ๐œƒ(๐œ) for some (๐œƒ,๐พ)โˆˆ๐‘† and (4.3) holds for ๐‘ฆโˆˆ๐’Ÿ๐œƒ(๐œ). By the proof of Lemma 4.6, we know that ๐œ is not in Case I with respect to (๐œƒ,๐พ). We only need to prove that ๐œ is not in Case III with respect to this (๐œƒ,๐พ). If it is not true, then all solutions of (1.1) with ๐œ†โˆˆฮ›๐œƒ,๐พ satisfy (2.6) and so belong to ๐’Ÿ๐œƒ(๐œ). Let ๐‘ฆ0 be a nontrivial solution of (1.1) with ๐‘ฆ(0)=0. Then ๐‘ฆ0โˆˆ๐’Ÿ๐œƒ(๐œ), and hence the same proof as in (4.33)โ€“(4.35) yields a contradiction.

Corollary 4.12. If ๐ธ has more than one point, then ๐œ is in case II with respect to some (๐œƒ,๐พ)โˆˆ๐‘† if and only if ๐’Ÿ(๐œ)โ‰ ๐’Ÿ๐‘ (๐œ) and (4.42) holds for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ๐œƒ(๐œ) with ๐œƒโˆˆ๐ธ๐‘œ.

Proof. If ๐ธ has more than one point and ๐œ is in Case II with respect to some (๐œƒ0,๐พ0)โˆˆ๐‘†, then there exists ๐œƒ1โˆˆ๐ธ๐‘œ such that ๐œ is in Case II with respect to (๐œƒ1,๐พ1)โˆˆ๐‘† by Theorem 2.5. Since ๐’Ÿ๐œƒ1(๐œ)โŠŠ๐’Ÿ(๐œ) by Theorem 4.11 and ๐’Ÿ๐‘ (๐œ)=๐’Ÿ๐œƒ1(๐œ) by Lemma 4.4, one sees that ๐’Ÿ(๐œ)โ‰ ๐’Ÿ๐‘ (๐œ).
Choose ๐œƒ2โˆˆ๐ธ๐‘œ with ๐œƒ1โ‰ ๐œƒ2 (mod๐œ‹) such that ๐œ is in Case II with respect to (๐œƒ๐‘—,๐พ๐‘—)โˆˆ๐‘† for ๐‘—=1,2 by Theorem 2.5. Then (4.3) holds for ๐œƒ=๐œƒ๐‘—, ๐‘—=1,2 by Theorem 4.11. Since ๐’Ÿ๐œƒ๐‘—(๐œ)=๐’Ÿ๐‘ (๐œ), the same proof as in (4.43) gives that (4.42) holds for ๐‘ฆ1,๐‘ฆ2โˆˆ๐’Ÿ๐‘ (๐œ).
Conversely, suppose that ๐’Ÿ(๐œ)โ‰ ๐’Ÿ๐‘ (๐œ) and (4.42) holds for ๐‘ฆโˆˆ๐’Ÿ๐‘ (๐œ). Since ๐’Ÿ๐œƒ(๐œ)โ‰ก๐’Ÿ๐‘ (๐œ) on ๐ธ๐‘œ by Lemma 4.4, we conclude that ๐’Ÿ๐œƒ(๐œ)โ‰ ๐’Ÿ(๐œ) on ๐ธ๐‘œ and (4.42) holds for ๐‘ฆโˆˆ๐’Ÿ๐œƒ(๐œ). So (4.3) holds for ๐‘ฆโˆˆ๐’Ÿ๐œƒ(๐œ) by (4.42). Then, we have that ๐œ is in Case II with respect to (๐œƒ,๐พ)โˆˆ๐‘† with ๐œƒโˆˆ๐ธ๐‘œ by Theorem 4.11.

Acknowledgments

This work was supported by the NSF of Shandong Province (Grant Y2008A02) and the IIFSDU (Grant 2010ZRJQ002).

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Copyright © 2011 Bing Xie and Jian Gang Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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