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 πœπ‘¦βˆΆ=π‘€βˆ’1ξ‚ƒβˆ’ξ€·π‘π‘¦ξ…žξ€Έξ…žξ‚„[+π‘žπ‘¦=πœ†π‘¦onπ‘Ž,𝑏),(1.1) where 𝑝,π‘ž are both complex valued, 𝑀(π‘₯) is a positive weight function, βˆ’βˆž<π‘Ž<𝑏≀+∞, and πœ† is the so-called spectral parameter. We call 𝜏 a formally symmetric differential expression if 𝑝,π‘ž are both real valued; otherwise 𝜏 is called formally nonsymmetric. In all cases, we call 𝜏 a formally differential expression or operator.

Let 𝐿2𝑀 denote the Hilbert space 𝐿2𝑀[ξ€œβˆΆ=𝑦ismeasurableβˆΆπ‘Ž,𝑏)βŸΆβ„‚βˆΆπ‘π‘Ž||||𝑀(π‘₯)𝑦(π‘₯)2ξ‚Όdπ‘₯<∞(1.2) with inner product βˆ«βŸ¨π‘¦,π‘§βŸ©βˆΆ=π‘π‘Žπ‘§(π‘₯)𝑀(π‘₯)𝑦(π‘₯)dπ‘₯ and the norm ‖𝑦‖2=βŸ¨π‘¦,π‘¦βŸ© for 𝑦,π‘§βˆˆπΏ2𝑀. We call a solution 𝑦 of (1.1) an 𝐿2𝑀-solution or square integrable solution if π‘¦βˆˆπΏ2𝑀. Set ξ€½π’Ÿ(𝜏)=π‘¦βˆˆπΏ2π‘€βˆΆπ‘¦,π‘π‘¦ξ…žβˆˆACloc,πœπ‘¦βˆˆπΏ2𝑀,(1.3) where ACloc=ACloc([π‘Ž,𝑏),β„‚) is the set of complex valued functions that are absolutely continuous on each compact subinterval of [π‘Ž,𝑏). We call π’Ÿ(𝜏) the natural (or maximal) domain associated with the formally differential operator 𝜏.

The aim of the present paper is to study the asymptotic behavior of elements of π’Ÿ(𝜏). 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 𝑝(π‘₯) and π‘ž(π‘₯) 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 π‘š(πœ†)-functions, where we say that 𝜏 or (1.1) is in the limit point case at 𝑏 if there exists exactly one 𝐿2𝑀-solution (up to constant multiple) for πœ†βˆˆβ„‚ with Imπœ†β‰ 0 and is in the limit circle case if all solutions belong to 𝐿2𝑀 for πœ†βˆˆβ„‚ with Imπœ†β‰ 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 [2–10] and references therein.

The same problem was also studied by Sims in 1957 for the case where π‘ž(π‘₯) is complex valued [11]. He considered the case where 𝑝(π‘₯)=𝑀(π‘₯)≑1 and Imπ‘ž(π‘₯) 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 𝑀(π‘₯) and the complex valued coefficients 𝑝(π‘₯), π‘ž(π‘₯). They proved that there exist three distinct possible cases for (1.1).

For formally symmetric 𝜏, it is well known (see [13, 14]) that (1.1) is in the limit point case at 𝑏 if and only if 𝑦𝑝(π‘₯)2(π‘₯)π‘¦ξ…ž1(π‘₯)βˆ’π‘¦1(π‘₯)π‘¦ξ…ž2ξ‚„(π‘₯)⟢0asπ‘₯βŸΆπ‘(1.4) for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏). 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 π’Ÿ(𝜏) at the end point 𝑏. In this case every self-adjoint extension associated with the differential expression needs not a boundary condition at 𝑏. 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 π’Ÿ(𝜏), 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 [17–19] 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 𝑝(π‘₯) and π‘ž(π‘₯), we say that (1.1) is in the strong limit point case at the end point 𝑏 if, for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏), 𝑝(π‘₯)𝑦1(π‘₯)π‘¦ξ…ž2(π‘₯)⟢0asπ‘₯βŸΆπ‘.(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 𝑝 and π‘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 π’Ÿ(𝜏) also depends on the admissible rotation angles. So we first study the properties of the admissible angle set 𝐸 (defined in (2.10)) and prove that 𝐸 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 π’Ÿ(𝜏). 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 𝐸 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 𝐸. The main results are given in Section 4.

2. Preliminary Knowledge

Throughout this paper, we always assume that(i)𝑝(π‘₯)β‰ 0, 𝑀(π‘₯)>0 a.e. on [π‘Ž,𝑏) and 1/𝑝,π‘ž,𝑀 are all locally integrable on [π‘Ž,𝑏),(ii)𝑝 and π‘ž are complex valued, and Ξ©=ξ‚»coπ‘ž(π‘₯)[𝑀(π‘₯)+π‘Ÿπ‘(π‘₯)βˆΆπ‘Ÿ>0,π‘₯βˆˆπ‘Ž,𝑏)β‰ β„‚,(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 πœ•Ξ©, there exists a line through this point such that every point of Ξ© either lies in the same side of this line or is on it. That is, there exists a supporting line through this point. Let 𝐾 be a point on πœ•Ξ©. Denote by 𝐿 an arbitrary supporting line touching Ξ© at 𝐾, which may be the tangent to Ξ© at 𝐾 if it exists. We then perform a transformation of the complex plane π‘§β†¦π‘§βˆ’πΎ and a rotation through an appropriate angle πœƒ so that the image of 𝐿 coincides with the imaginary axis now and the set Ξ© is contained in the new right nonnegative half-plane.

For this purpose we introduce the set 𝑆 defined by 𝑆=(πœƒ,𝐾)βˆΆπΎβˆ‰Ξ©βˆ˜ξ€½π‘’,Reπ‘–πœƒξ€Ύξ€Ύ(πœ‡βˆ’πΎ)β‰₯0βˆ€πœ‡βˆˆΞ©,(2.2) where Ω∘ is the interior of Ξ©, and define the corresponding half-plane Ξ›πœƒ,𝐾=ξ€½ξ€½π‘’πœ‡βˆˆβ„‚βˆΆReπ‘–πœƒξ€Ύξ€Ύ(πœ‡βˆ’πΎ)<0.(2.3) Then, Ξ›πœƒ,πΎβŠ‚β„‚β§΅Ξ©. From the definition of 𝑆, for all π‘₯∈[π‘Ž,𝑏) and 0<π‘Ÿ<∞, 𝑒Reπ‘–πœƒξ‚Έπ‘ž(π‘₯)𝑀(π‘₯)+π‘Ÿπ‘(π‘₯)βˆ’πΎξ‚Ήξ‚Όβ‰₯0.(2.4)

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

Besides, for (πœƒ,𝐾)βˆˆπ‘†ξ€½π‘’Reπ‘–πœƒξ€Ύ||||𝑒(πœ‡βˆ’πΎ)β‰₯0⟺cos(πœƒ+𝛾)β‰₯0whereπœ‡βˆ’πΎ=πœ‡βˆ’πΎπ‘–π›Ύ.(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 Ξ›πœƒ,𝐾 as follows. The uniqueness referred to in the theorem and the following sections is only up to constant multiple.

Theorem 2.1 (cf. [12, Theorem  2.1]). Given a (πœƒ,𝐾)βˆˆπ‘†, the following three distinct cases are possible. Case I. For all πœ†βˆˆΞ›πœƒ,𝐾, equation (1.1) has unique solution 𝑦 satisfying ξ€œπ‘π‘Žξ‚ƒξ€½π‘’Reπ‘–πœƒπ‘ξ€Ύ||π‘¦ξ…ž||2𝑒+Reπ‘–πœƒξ€Ύ||𝑦||(π‘žβˆ’πΎπ‘€)2ξ‚„+ξ€œπ‘π‘Žπ‘€||𝑦||2<∞(2.6) and this is the only solution satisfying π‘¦βˆˆπΏ2𝑀.Case II. For all πœ†βˆˆΞ›πœƒ,𝐾, all solutions of (1.1) belong to 𝐿2𝑀, and there exists unique solution of (1.1) satisfying (2.6).Case III. For all πœ†βˆˆΞ›πœƒ,𝐾, all solutions of (1.1) satisfy (2.6).

Since every Ξ›πœƒ,𝐾 is a half-plane, it holds that Ξ›πœƒ1,𝐾1ξ™Ξ›πœƒ2,𝐾2β‰ βˆ…(2.7) for (πœƒπ‘—,𝐾𝑗)βˆˆπ‘†, 𝑗=1,2, with πœƒ1β‰ πœƒ2 (modπœ‹). Note that (2.4) implies that, for 0<π‘Ÿ<∞ and π‘₯∈[π‘Ž,𝑏), 𝑒Reπ‘–πœƒξ‚΅π‘ž(π‘₯)πΎπ‘Ÿπ‘€(π‘₯)+𝑝(π‘₯)βˆ’π‘Ÿξ‚Άξ‚Όβ‰₯0.(2.8) Letting π‘Ÿβ†’0 and π‘Ÿβ†’βˆž in (2.4) and (2.8), respectively, we have the following.

Lemma 2.2. For every (πœƒ,𝐾)βˆˆπ‘† and πœ†βˆˆΞ›πœƒ,𝐾, there exists π›Ώπœ†(πœƒ)>0 such that 𝑒Reπ‘–πœƒξ€Ύξ€½π‘’(π‘žβˆ’πΎπ‘€)β‰₯0,Reπ‘–πœƒξ€Ύ(π‘žβˆ’πœ†π‘€)β‰₯π›Ώπœ†ξ€½π‘’(πœƒ)𝑀,Reπ‘–πœƒπ‘ξ€Ύβ‰₯0(2.9) on [π‘Ž,𝑏).

Using variation of parameters method, we can verify that, if all solutions of (1.1) belong to 𝐿2𝑀 for some πœ†0βˆˆβ„‚, then it is true for all πœ†βˆˆβ„‚. This also means the following.

Lemma 2.3. If there exists a (πœƒ0,𝐾0)βˆˆπ‘† such that (1.1) is in Case I with respect to (with respect to for short) Ξ›πœƒ0,𝐾0, then (1.1) is in Case I with respect to Ξ›πœƒ,𝐾 for every (πœƒ,𝐾)βˆˆπ‘†.

This indicates that Case I is independent of the choice of (πœƒ,𝐾)βˆˆπ‘†. But Cases II and III depend on the choice of (πœƒ,𝐾)βˆˆπ‘† in general, that is, there may exist (πœƒ1,𝐾1),(πœƒ2,𝐾2)βˆˆπ‘† such that (1.1) is in Case II with respect to Ξ›πœƒ1,𝐾1 and is in Case III with respect to Ξ›πœƒ2,𝐾2. In order to make clear the dependence, we introduce the admissible angle set 𝐸 defined by 𝐸={πœƒβˆΆβˆƒπΎβˆ‰Ξ©βˆ˜,(πœƒ,𝐾)βˆˆπ‘†}.(2.10)

Remark 2.4. For given πœƒβˆˆπΈ, there exist many 𝐾 such that (πœƒ,𝐾)βˆˆπ‘†. In fact, if πœƒ0∈𝐸 with (πœƒ0,𝐾0)βˆˆπ‘† for some 𝐾0βˆ‰Ξ©βˆ˜, then for all 𝐾∈𝐿0, (πœƒ0,𝐾)βˆˆπ‘†πΏ0=ξ€½ξ€½π‘’πœ†βˆˆβ„‚βˆΆReπ‘–πœƒ0ξ€·πœ†βˆ’πΎ0ξ€Ύξ€Έξ€Ύ=0.(2.11)

The exact dependence of Cases II and III on (πœƒ,𝐾) can be given with the similar proof in [23, Theorem  2.1].

Theorem 2.5 (cf. [23, Theorem  2.1]). If there exists a (πœƒ0,𝐾0)βˆˆπ‘† such that (1.1) is in Case II with respect to Ξ›πœƒ0,𝐾0, then (1.1) is in Case II with respect to Ξ›πœƒ,𝐾 for all (πœƒ,𝐾)βˆˆπ‘† except for at most one πœƒ1∈𝐸(modπœ‹) such that (1.1) is in Case III with respect to Ξ›πœƒ1,𝐾1.

Remark 2.6. Theorem 2.5 means that, if there exist πœƒπ‘—βˆˆπΈ, 𝑗=1,2, such that πœƒ1β‰ πœƒ2 (modπœ‹) and (1.1) is in Case III with respect to Ξ›πœƒπ‘—,𝐾𝑗 for 𝑗=1,2, then (1.1) is in Case III with respect to Ξ›πœƒ,𝐾 for all (πœƒ,𝐾)βˆˆπ‘†.

3. Properties of the Angel Set 𝐸

This section gives some properties of the set 𝐸, which will be used in the proof of our main results in Section 4. In what follows, we say that 𝐸 has more than one point if there exist πœƒ1,πœƒ2∈𝐸 with πœƒ1β‰ πœƒ2 (modπœ‹).

Lemma 3.1. Let 𝐸 be defined as in (2.10). (i)The set 𝐸 is connected in the sense of mod2πœ‹. (ii)If 𝐸 has more than one point, then, for every πœ†βˆˆβ„‚β§΅Ξ©, there exist πœƒ1,πœƒ2∈𝐸 with πœƒ1<πœƒ2 such that πœ†βˆˆΞ›πœƒ,𝐾 for πœƒβˆˆ(πœƒ1,πœƒ2)βŠ‚πΈ.

Proof. (i) Suppose that 𝐸 has more than one point. Let πœƒ1,πœƒ2∈𝐸 with πœƒ1β‰ πœƒ2 (modπœ‹); then 0<πœƒ2βˆ’πœƒ1<πœ‹ (mod2πœ‹) or πœ‹<πœƒ2βˆ’πœƒ1<2πœ‹ (mod2πœ‹).
If 0<πœƒ2βˆ’πœƒ1<πœ‹ (mod2πœ‹) and (πœƒπ‘—,𝐾𝑗)βˆˆπ‘†, 𝑗=1,2, then we claim that [πœƒ1,πœƒ2]βŠ‚πΈ (mod2πœ‹). Let 𝐿𝑗 be the line similarly defined as 𝐿0 with 𝐾0 and πœƒ0 replaced by 𝐾𝑗 and πœƒπ‘—, 𝑗=1,2. That is, 𝐿𝑗=ξ€½ξ€½π‘’πœ†βˆˆβ„‚βˆΆReπ‘–πœƒπ‘—ξ€·πœ†βˆ’πΎπ‘—ξ€Ύξ€Έξ€Ύ=0,πΎπ‘—βˆ‰Ξ©βˆ˜,𝑗=1,2.(3.1) Let 𝐾 be the intersection point of 𝐿1 and 𝐿2. Set ||||π‘’πœ‡βˆ’πΎ=πœ‡βˆ’πΎπ‘–π›Ύ(πœ‡,𝐾),πœ‡βˆˆΞ©.(3.2) It follows from (2.5) that 𝛾cos(πœ‡,𝐾)+πœƒπ‘—ξ€Έβ‰₯0,πœ‡βˆˆΞ©,𝑗=1,2.(3.3)
By 0<πœƒ2βˆ’πœƒ1<πœ‹ (mod2πœ‹) and (3.3), we can get cos(𝛾(πœ‡,𝐾)+πœƒ)β‰₯0 for πœƒβˆˆ[πœƒ1,πœƒ2] (mod2πœ‹) on Ξ©, which means (πœƒ,𝐾)βˆˆπ‘† and πœƒβˆˆπΈ.
According to the similar method, we can verify that, if πœ‹<πœƒ2βˆ’πœƒ1<2πœ‹ (mod2πœ‹) and (πœƒπ‘—,𝐾𝑗)βˆˆπ‘†, 𝑗=1,2, then [0,πœƒ1]βˆͺ[πœƒ2,2πœ‹]βŠ‚πΈ (mod2πœ‹), that is, [πœƒ2,πœƒ1]βŠ‚πΈ (mod2πœ‹).
(ii) For πœ†0βˆˆβ„‚β§΅Ξ©, choose (πœƒ1,𝐾1)βˆˆπ‘† and 𝛿0>0 such that πœ†0βˆˆΞ›πœƒ1,𝐾1 and 𝑒Reπ‘–πœƒ1𝐾1βˆ’πœ†0ξ€Έξ€Ύ=𝛿0>0.(3.4) Since 𝐸 has more than one point, we can choose πœƒ2∈𝐸 with πœƒ2β‰ πœƒ1 (modπœ‹). Without loss of generality, we suppose that 0<πœƒ2βˆ’πœƒ1<πœ‹ (mod2πœ‹). Let 𝐾 be defined as in the proof of (i).
If πœ†0βˆˆΞ›πœƒ1,πΎβˆ©Ξ›πœƒ2,𝐾, then it follows from (2.3) that ξ€·cos𝛾+πœƒπ‘—ξ€Έ<0,𝑗=1,2,whereπœ†0||πœ†βˆ’πΎ=0||π‘’βˆ’πΎπ‘–π›Ύ.(3.5) By 0<πœƒ2βˆ’πœƒ1<πœ‹ (mod2πœ‹) and (3.5), we can get cos(𝛾+πœƒ)<0 for πœƒβˆˆ[πœƒ1,πœƒ2] (mod2πœ‹), which means πœ†0βˆˆΞ›πœƒ,𝐾 for πœƒβˆˆ[πœƒ1,πœƒ2].
Suppose that πœ†0βˆ‰Ξ›πœƒ2,𝐾. Let πœ†1∈𝐿1 be the unique point such that 𝛿0=dist(πœ†0,𝐿1)=dist(πœ†1,πœ†0). Let 𝛼=arctan|πœ†0βˆ’πœ†1|/2|πΎβˆ’πœ†1|; then 𝛼+πœƒ1∈(πœƒ1,πœƒ2)βŠ‚πΈ by πœ†0βˆ‰Ξ›πœƒ2,𝐾, and πœ†0βˆˆΞ›π›Ό+πœƒ1,𝐾 by the definition of 𝛼 (see Figure 1).
So, we can get that πœ†0βˆˆΞ›πœƒ,𝐾 for πœƒβˆˆ[πœƒ1,πœƒ1+𝛼] by πœ†0βˆˆΞ›πœƒ1,πΎβˆ©Ξ›πœƒ1+𝛼,𝐾, 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 𝜏 defined on the interval [0,∞) with 0 being a regular end point and +∞ 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 (π‘Ž,𝑏), where βˆ’βˆžβ‰€π‘Ž<𝑏≀+∞. Recall that (1.1) on (π‘Ž,𝑏) is said to be regular at π‘Ž if 1/𝑝, π‘ž and 𝑀 are integrable on (π‘Ž,𝑐) for some (and hence any) π‘βˆˆ(π‘Ž,𝑏) and singular at π‘Ž otherwise; the regularity and singularity at 𝑏 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, Section  9.1]. See also [10, Theorem 2.3.1]. Recall the definition of π’Ÿ(𝜏) in (1.3). We also define π’Ÿξ€·πœξ€Έ=ξ€½π‘¦βˆˆπΏ2π‘€βˆΆπ‘¦,π‘π‘¦ξ…žβˆˆACloc,πœπ‘¦βˆˆπΏ2𝑀,(4.1) where πœπ‘¦βˆΆ=π‘€βˆ’1ξ€Ίβˆ’ξ€·ξ€Έπ‘π‘¦β€²β€²+ξ€»[π‘žπ‘¦on0,∞).(4.2) The first result of this section is as follows.

Theorem 4.1. (i) 𝜏 is in Case I if and only if for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏) and πœƒβˆˆπΈπ‘(π‘₯)𝑦2(π‘₯)π‘¦ξ…ž1(π‘₯)+𝑒2π‘–πœƒπ‘(π‘₯)𝑦1(π‘₯)π‘¦ξ…ž2(π‘₯)⟢0asπ‘₯⟢∞.(4.3)
(ii) 𝜏 is in Case I if and only if for 𝑦1βˆˆπ’Ÿ(𝜏),𝑦2βˆˆπ’Ÿ(𝜏)𝑦𝑝(π‘₯)2(π‘₯)π‘¦ξ…ž1(π‘₯)βˆ’π‘¦1(π‘₯)π‘¦ξ…ž2ξ‚„(π‘₯)⟢0asπ‘₯⟢∞.(4.4)

Remark 4.2. Clearly π’Ÿ(𝜏)=π’Ÿ(𝜏), by the definition of π’Ÿ(𝜏). It is easy to see that (4.4) is equivalent to 𝑝𝑦(π‘₯)2(π‘₯)π‘¦ξ…ž1(π‘₯)βˆ’π‘¦1(π‘₯)π‘¦ξ…ž2ξ€»(π‘₯)⟢0asπ‘₯⟢∞(4.5) for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏).

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 π‘’ξ…ž=𝐴𝑒+𝐡𝑣+πœ‰π‘Š2𝑣,𝑣′=πΆπ‘’βˆ’π΄βˆ—π‘£βˆ’πœ‰π‘Š1[𝑒,on0,∞),(4.6) where 𝑒,𝑣 are ℂ𝑛 valued functions, 𝑒𝑇 is the transpose of 𝑒, 𝐴,𝐡,𝐢,π‘Š1, and π‘Š2 are locally integrable, complex valued 𝑛×𝑛 matrices on [0,∞), 𝐡,𝐢,π‘Š1, and π‘Š2 are Hermit matrices and π‘Š1(𝑑)>0,π‘Š2(𝑑)β‰₯0 on [0,∞), and πœ‰ is the spectral parameter. Assume that the definiteness condition (see, e.g., [13, Chapter  9, page 253]) holds: ξ€œβˆž0π‘¦βˆ—π‘Šπ‘¦>0foreachnontrivialsolution𝑦of(4.6),(4.7) where π‘Š=diag(π‘Š1,π‘Š2). Let 𝐿2π‘ŠβˆΆ=𝐿2π‘Š[0,∞) denote the space of Lebesgue measurable 2𝑛-dimensional functions 𝑓 satisfying ∫∞0π‘“βˆ—(𝑠)π‘Š(𝑠)𝑓(𝑠)𝑑𝑠<∞. We say that (4.6) is in the limit point case at infinity if there exists exactly 𝑛's solutions of (4.6) belonging to 𝐿2π‘Š for πœ‰βˆˆβ„‚ with Imπœ‰β‰ 0.

Let π’Ÿ be the maximal domain associated with (4.6), that is, (𝑒𝑇,𝑣𝑇)π‘‡βˆˆπ’Ÿ if and only if (𝑒𝑇,𝑣𝑇)π‘‡βˆˆACloc∩𝐿2π‘Š, and there exists an element (𝑓𝑇,𝑔𝑇)π‘‡βˆˆπΏ2π‘Š such that π‘’ξ…ž=𝐴𝑒+𝐡𝑣+πœ‰π‘Š2𝑣+π‘Š2𝑔,𝑣′=πΆπ‘’βˆ’π΄βˆ—π‘£βˆ’πœ‰π‘Š1π‘’βˆ’π‘Š1[𝑓,on0,∞).(4.8) It is well known (cf. [5, 7]) that (4.6) is in the limit point case at infinity if and only if π‘Œβˆ—1(π‘₯)π½π‘Œ2(π‘₯)⟢0asπ‘₯⟢∞,𝐽=0βˆ’πΌπ‘›πΌπ‘›0ξƒͺ(4.9) for π‘Œ1,π‘Œ2βˆˆπ’Ÿ, and for every πœ‰βˆˆβ„‚ with Imπœ‰β‰ 0 there exists a Green function 𝐺(𝑑,𝑠,πœ‰) such that, for 𝐹=(𝑓𝑇,𝑔𝑇)π‘‡βˆˆπΏ2π‘Š, 𝑒𝑣ξƒͺπ‘Œ==π‘‡πœ‰πΉβˆˆπΏ2π‘Š,satisfies(4.8),(4.10) where (π‘‡πœ‰βˆ«πΉ)(π‘₯)=∞0𝐺(π‘₯,𝑠,πœ‰)π‘Š(𝑠)𝐹(𝑠)d𝑠.

Let (πœƒ,𝐾)βˆˆπ‘† and choose πœ†0βˆˆΞ›πœƒ,𝐾. Then from (2.9), one sees that 𝑒Reπ‘–πœƒξ€·π‘žβˆ’πœ†0πœ”ξ€Έξ€Ύβ‰₯𝛿0ξ€½π‘’πœ”>0,Reπ‘–πœƒξ€·π‘žβˆ’πΎ0𝑀𝑒β‰₯0,Reπ‘–πœƒπ‘ξ€Ύβ‰₯0(4.11) for some 𝛿0>0. Set π‘Ÿ1||(π‘₯)=π‘ž(π‘₯)βˆ’πœ†0||𝑀(π‘₯),π‘ž(π‘₯)βˆ’πœ†0𝑀(π‘₯)=π‘Ÿ1(π‘₯)𝑒𝑖𝛼(π‘₯),𝛼1π‘Ÿ(π‘₯)=πœƒ+𝛼(π‘₯),2(||||π‘₯)=𝑝(π‘₯),𝑝(π‘₯)=π‘Ÿ2(π‘₯)𝑒𝑖𝛽(π‘₯),𝛽1(π‘₯)=πœƒ+𝛽(π‘₯).(4.12) Consider the Hamiltonian differential system (4.6) with 𝑛=1, 𝐴(π‘₯)≑0 and 𝐢(π‘₯)=π‘Ÿ1(π‘₯)sin𝛼1(π‘₯),π‘Š1(π‘₯)∢=𝑀1(π‘₯)=π‘Ÿ1(π‘₯)cos𝛼1(π‘₯),𝐡(π‘₯)=sin𝛽1(π‘₯)π‘Ÿ2(π‘₯),π‘Š2(π‘₯)∢=𝑀2(π‘₯)=cos𝛽1(π‘₯)π‘Ÿ2(,π‘₯)(4.13) that is, the 2-dimensional Hamiltonian differential system 𝐻(πœƒ)βˆΆπ‘’β€²=𝐡𝑣+πœ‰π‘€2𝑣,π‘£ξ…ž=πΆπ‘’βˆ’πœ‰π‘€1𝑒.(4.14) It follows from (4.11) that 𝑀1𝑒=Reπ‘–πœƒξ€·π‘žβˆ’πœ†0𝑀β‰₯𝛿0𝑀>0,𝑀2=𝑒Reπ‘–πœƒξ€Ύπ‘(𝑑)π‘Ÿ22β‰₯0,(4.15) and it is easy to verify that the definiteness condition holds for the system (4.14). In fact, 𝑦 is a solution of (1.1) if and only if (𝑒,𝑣)𝑇 is a solution of (4.14) with 𝑒=𝑦,𝑣=βˆ’π‘–π‘’π‘–πœƒπ‘π‘¦β€².(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) 𝜏 is in Case I or Case II with respect to (πœƒ,𝐾)βˆˆπ‘† if and only if 𝐻(πœƒ) is in the limit point case at ∞.
(ii) 𝜏 is in Case III with respect to (πœƒ,𝐾)βˆˆπ‘† if and only if 𝐻(πœƒ) is in the limit circle case at ∞.

Lemma 4.4. If 𝐸 has more than one point, then π’Ÿπœƒ(𝜏)β‰‘π’Ÿπ‘ (𝜏) on πΈπ‘œ, the interior of 𝐸, where π’Ÿπœƒξ‚»ξ€œ(𝜏)=π‘¦βˆˆπ’Ÿ(𝜏)∢∞0𝑒Reπ‘–πœƒπ‘ξ€Ύ||π‘¦ξ…ž||2𝑒+Reπ‘–πœƒπ‘žξ€Ύ||𝑦||2ξ‚„ξ‚Ό,π’Ÿ<βˆžπ‘ ξ‚»ξ€œ(𝜏)=π‘¦βˆˆπ’Ÿ(𝜏)∢∞0||𝑝||||π‘¦ξ…ž||2+||π‘ž||||𝑦||2ξ‚„ξ‚Ό.<∞(4.17)

Proof. Let πœƒ1βˆˆπΈπ‘œ be fixed. There exist πœƒ2,πœƒ3βˆˆπΈπ‘œ such that πœƒ3<πœƒ1<πœƒ2mod(2πœ‹),0<πœƒ2βˆ’πœƒ3<πœ‹,2mod(2πœ‹)3𝑗=1Ξ›πœƒπ‘—,πΎπ‘—β‰ βˆ…(4.18) by Lemma 3.1. Choose πœ†0βˆˆβ‹‚3𝑗=1Ξ›πœƒπ‘—,𝐾𝑗. Letting π›½βˆΆ=𝛽(π‘₯) be defined as in (4.12) and solving cos(πœƒ1+𝛽) from the equations ξ€·πœƒcosπ‘—ξ€Έξ€·πœƒ+𝛽=cos1ξ€Έξ€·πœƒ+𝛽cosπ‘—βˆ’πœƒ1ξ€Έξ€·πœƒβˆ’sin1ξ€Έξ€·πœƒ+𝛽sinπ‘—βˆ’πœƒ1ξ€Έ,𝑗=2,3,(4.19) we have that cos(πœƒ1+𝛽)=𝐢1cos(πœƒ2+𝛽)+𝐢2cos(πœƒ3+𝛽) with 𝐢1=ξ€·πœƒsin1βˆ’πœƒ3ξ€Έξ€·πœƒsin2βˆ’πœƒ3ξ€Έ>0,𝐢2=ξ€·πœƒsin2βˆ’πœƒ1ξ€Έξ€·πœƒsin2βˆ’πœƒ3ξ€Έ>0(4.20) by (4.18). Since ∫∞0Re{π‘’π‘–πœƒ1𝑝}|π‘¦ξ…ž|2<∞ for π‘¦βˆˆπ’Ÿπœƒ1(𝜏), we have that 𝐢1ξ€œβˆž0𝑒Reπ‘–πœƒ2𝑝||π‘¦ξ…ž||2+𝐢2ξ€œβˆž0𝑒Reπ‘–πœƒ3𝑝||π‘¦ξ…ž||2=ξ€œβˆž0𝑒Reπ‘–πœƒ1𝑝||π‘¦ξ…ž||2<∞,(4.21) and hence ∫∞0Re{π‘’π‘–πœƒ2𝑝}|π‘¦ξ…ž|2<∞ for π‘¦βˆˆπ’Ÿπœƒ1(𝜏). The same proof as the above with 𝛽 replaced by 𝛼 also proves ∫∞0Re{π‘’π‘–πœƒ2(π‘žβˆ’πœ†0𝑀)}|𝑦|2<∞ for π‘¦βˆˆπ’Ÿπœƒ1(𝜏), where π›ΌβˆΆ=𝛼(π‘₯) is defined as in (4.12). Therefore, for π‘¦βˆˆπ’Ÿπœƒ1(𝜏), ξ€œβˆž0𝑒Reπ‘–πœƒπ‘—π‘ξ€Ύ||π‘¦ξ…ž||2ξ‚„,ξ€œβˆž0𝑒Reπ‘–πœƒπ‘—ξ€·π‘žβˆ’πœ†0𝑀||𝑦||ξ€Έξ€Ύ2<∞,𝑗=1,2.(4.22) Set π‘πœƒ=π‘’π‘–πœƒπ‘ and π‘žπœƒ=π‘’π‘–πœƒ(π‘žβˆ’πœ†0𝑀). It follows from sin2ξ€·πœƒ2βˆ’πœƒ1ξ€Έ=cos2πœƒ2+cos2πœƒ1βˆ’2cosπœƒ2cosπœƒ1ξ€·πœƒcos1βˆ’πœƒ2≀cosπœƒ2+cosπœƒ1ξ€Έ2(4.23) and (4.15) that 𝑝Reπœƒ1+π‘πœƒ2ξ€Έβ‰₯πœ€0||𝑝||ξ€·π‘ž,Reπœƒ1+π‘žπœƒ2ξ€Έβ‰₯πœ€0||||π‘žβˆ’πœ†π‘€,πœ€0ξ€·πœƒ=sin2βˆ’πœƒ1ξ€Έ.(4.24) Then (4.24) and (4.22) yield that, for π‘¦βˆˆπ’Ÿπœƒ1(𝜏), ξ€œβˆž0||𝑝||||π‘¦ξ…ž||2,ξ€œβˆž0||π‘žβˆ’πœ†0𝑀||||𝑦||2<∞.(4.25) Note that π‘¦βˆˆπΏ2𝑀. Then (4.25) gives π‘¦βˆˆπ’Ÿπ‘ (𝜏), or π’Ÿπœƒ1(𝜏)βŠ‚π’Ÿπ‘ (𝜏). Clearly, π’Ÿπ‘ (𝜏)βŠ‚π’Ÿπœƒ1(𝜏). Thus π’Ÿπœƒ1(𝜏)=π’Ÿπ‘ (𝜏).

Lemma 4.4 indicates the following.

Corollary 4.5. If 𝜏 is in Case II with respect to some (πœƒ0,𝐾0)βˆˆπ‘† and 𝐸 has more than one point, then Case III only occurs at the end point of 𝐸.

Proof. If 𝜏 is in Case III with respect to some (πœƒ1,𝐾1)βˆˆπ‘† with πœƒ1βˆˆπΈπ‘œ, then π’Ÿ(𝜏)=π’Ÿπœƒ1(𝜏) is restricted in the solution space of (1.1) by the definition of Case III. Since π’Ÿπœƒ1(𝜏)=π’Ÿπ‘ (𝜏) by Lemma 4.4, we have that π’Ÿ(𝜏)=π’Ÿπ‘ (𝜏) restricted in the solution space of (1.1). This means that all solutions of (1.1) with πœ†βˆˆΞ›πœƒ1,𝐾1 satisfy ξ€œβˆž0ξ‚€||𝑝||||π‘¦ξ…ž||2+||π‘ž||||𝑦||2<∞.(4.26) Using variation of parameters method we can prove that it is true for all πœ†βˆˆβ„‚, and hence 𝜏 is in Case III with respect to (πœƒ0,𝐾0), a contradiction.

Lemma 4.6. If 𝜏 is in Case I and π‘¦βˆˆπ’Ÿ(𝜏), then (𝑦,𝑣)π‘‡βˆˆπ’Ÿ(πœƒ) with 𝑣=βˆ’π‘–π‘’π‘–πœƒπ‘π‘¦ξ…ž, where π’Ÿ(πœƒ) is the maximal domain associated with (4.14).

Proof. Suppose that 𝜏 is in Case I with respect to (πœƒ,𝐾)βˆˆπ‘†. We claim that π’Ÿ(𝜏)=π’Ÿπœƒ(𝜏). Set ξ€·πœβˆ’πœ†0𝑦0=π‘€βˆ’1ξ‚ƒβˆ’ξ€·π‘π‘¦ξ…ž0ξ€Έξ…ž+ξ€·π‘žβˆ’πœ†0𝑀𝑦0ξ‚„=𝑔0,(4.27) for 𝑦0βˆˆπ’Ÿ(𝜏) and πœ†0βˆˆΞ›πœƒ,𝐾.
Set 𝑒0=𝑦0, 𝑣0=βˆ’π‘–π‘’π‘–πœƒπ‘π‘¦ξ…ž0. Then (𝑒0,𝑣0) satisfies 𝑒′=𝐡𝑣+𝑖𝑀2𝑣,π‘£ξ…ž=πΆπ‘’βˆ’π‘–π‘€1π‘’βˆ’π‘€1𝑓1,𝑓1=𝑀𝑀1ξ€·βˆ’π‘–π‘’π‘–πœƒπ‘”0ξ€Έ.(4.28) Conversely, if (𝑒,𝑣) satisfies (4.28), then 𝑦=𝑒 solves (4.27). Note that 𝑔0∈𝐿2𝑀, or βˆ’π‘–π‘’π‘–πœƒ0𝑔0∈𝐿2𝑀, and 𝑀1β‰₯𝛿𝑀 implies 𝑓1∈𝐿2𝑀1.
Considering (4.28), we get from (4.10) that (4.28) has a solution (𝑒1,𝑣1)𝑇 such that 𝑒1∈𝐿2𝑀1,𝑣1∈𝐿2𝑀2 and 𝑣1=βˆ’π‘–π‘’π‘–πœƒ0π‘π‘’ξ…ž1. Set 𝑦1=𝑒1. Then 𝑦1 satisfies (4.27), and hence (πœβˆ’πœ†0)(𝑦0βˆ’π‘¦1)=0. Note that 𝑦1=𝑒1∈𝐿2𝑀1, and 𝑀1β‰₯𝛿𝑀 implies that 𝑦1∈𝐿2𝑀. Thus, 𝑦1βˆ’π‘¦0 is an 𝐿2𝑀-solution of πœπ‘¦=πœ†0𝑦. Since 𝜏 is in Case I with respect to (πœƒ0,𝐾0), it follows from (2.6) that 𝑦1βˆ’π‘¦0∈𝐿2𝑀1 and 𝑣1βˆ’π‘£0∈𝐿2𝑀2. This together with 𝑦1∈𝐿2𝑀1 and 𝑣1∈𝐿2𝑀2 gives 𝑦0∈𝐿2𝑀1 and 𝑣0∈𝐿2𝑀2. In fact, we have proved that, for π‘¦βˆˆπ’Ÿ(𝜏), ξ€œβˆž0||π‘žβˆ’πœ†0𝑀||cos𝛼1||𝑦||2ξ€œ<∞,∞0||𝑝||cos𝛽1||π‘¦ξ…ž||2<∞,(4.29) or ξ€œβˆž0𝑒Reπ‘–πœƒπ‘ξ€Ύ||π‘¦ξ…ž||2𝑒+Reπ‘–πœƒξ€·π‘žβˆ’πœ†0𝑀||𝑦||ξ€Έξ€Ύ2ξ‚„<∞,(4.30) where 𝛼1 and 𝛽1 are defined in (4.12) or (4.13). Since π‘¦βˆˆπΏ2𝑀, (4.30) means that ξ€œβˆž0𝑒Reπ‘–πœƒπ‘ξ€Ύ||π‘¦ξ…ž||2𝑒+Reπ‘–πœƒπ‘žξ€Ύ||𝑦||2ξ‚„<∞(4.31) or π‘¦βˆˆπ’Ÿπœƒ(𝜏), and hence π’Ÿ(𝜏)=π’Ÿπœƒ(𝜏). Recall that 𝑓1∈𝐿2𝑀1. Then (4.30) and (4.28) imply that, if π‘¦βˆˆπ’Ÿ(𝜏), then (𝑦,𝑣)π‘‡βˆˆπ’Ÿ(πœƒ).

Corollary 4.7. If 𝜏 is in Case I and π‘¦βˆˆπ’Ÿ(𝜏), then (𝑦,𝑣)π‘‡βˆˆπ’Ÿ(πœƒ) with 𝑣=π‘–π‘’βˆ’π‘–πœƒπ‘π‘¦ξ…ž.

Proof. For π‘¦βˆˆπ’Ÿ(𝜏), π‘¦βˆˆπ’Ÿ(𝜏) by π’Ÿ(𝜏)=π’Ÿ(𝜏). So (𝑦,𝑣)π‘‡βˆˆπ’Ÿ(πœƒ) with 𝑣=βˆ’π‘–π‘’π‘–πœƒπ‘π‘¦ξ…ž by Lemma 4.6. Clearly π’Ÿ(πœƒ)=π’Ÿ(πœƒ) since 𝐻(πœƒ) is symmetrical. Then we have that (𝑦,𝑣)π‘‡βˆˆπ’Ÿ(πœƒ) with 𝑣=π‘–π‘’βˆ’π‘–πœƒπ‘π‘¦ξ…ž.

Proof of Theorem 4.1. The proof of (i): suppose that 𝜏 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 π‘Œ1,π‘Œ2βˆˆπ’Ÿ(πœƒ). For 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏), since (𝑦𝑗,𝑣𝑗)π‘‡βˆˆπ’Ÿ(πœƒ) with 𝑣𝑗=βˆ’π‘–π‘’π‘–πœƒπ‘π‘¦ξ…žπ‘—, 𝑗=1,2, by Lemma 4.6, it follows from (4.9) that 𝑦1,𝑣1𝑦0βˆ’110ξƒͺ2𝑣2ξƒͺ=π‘–π‘’βˆ’π‘–πœƒξ€·π‘π‘¦ξ…ž1𝑦2+𝑒2π‘–πœƒπ‘π‘¦1π‘¦ξ…ž2ξ€ΈβŸΆ0(4.32) as π‘₯β†’βˆž.
Conversely, assume that (4.3) holds for all elements of π’Ÿ(𝜏). 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 𝐿2𝑀 for πœ†βˆˆβ„‚. Choose πœ†0βˆˆΞ›πœƒ,𝐾, and let 𝑦0 be a nontrivial solution of (1.1) satisfying 𝑦0(0)=0. Then 𝑦0βˆˆπ’Ÿ(𝜏) by 𝑦0∈𝐿2𝑀. Furthermore, it follows from (πœβˆ’πœ†0)𝑦0=0 that βˆ’ξ€·π‘π‘¦ξ…ž0ξ€Έξ…žπ‘¦0+ξ€·π‘žβˆ’πœ†0𝑀||𝑦0||2ξ€·=0,βˆ’π‘π‘¦ξ…ž0ξ€Έξ…žπ‘¦0+ξ‚€π‘žβˆ’πœ†0𝑀||𝑦0||2=0.(4.33) Integrating (4.33) on [0,π‘₯] we have that βˆ’(π‘π‘¦ξ…ž0)𝑦0||π‘₯0+ξ€œπ‘₯0𝑝||π‘¦ξ…ž0||2+ξ€·π‘žβˆ’πœ†0𝑀||𝑦0||2ξ‚„=0,βˆ’(π‘π‘¦ξ…ž0)𝑦0||π‘₯0+ξ€œπ‘₯0𝑝||π‘¦ξ…ž0||2+ξ‚€π‘žβˆ’πœ†0𝑀||𝑦0||2ξ‚„=0.(4.34) Multiplying π‘’π‘–πœƒ and π‘’βˆ’π‘–πœƒ to the first and second equalities in (4.34), respectively, and adding them together, we have that βˆ’ξ€Ίπ‘’π‘–πœƒξ€·π‘π‘¦ξ…ž0𝑦0+π‘’βˆ’π‘–πœƒξ€·π‘π‘¦ξ…ž0𝑦0ξ€»ξ€œ(π‘₯)+π‘₯0ξ‚ƒξ€·π‘πœƒ+π‘πœƒξ€Έ||π‘¦ξ…ž0||2+ξ€·π‘žπœƒ+π‘žπœƒξ€Έ||𝑦0||2ξ‚„=0(4.35) since 𝑦0(0)=0, where π‘πœƒ=π‘’π‘–πœƒπ‘ and π‘žπœƒ=π‘’π‘–πœƒ(π‘žβˆ’πœ†0𝑀). Note that ξ€Ίπ‘’π‘–πœƒξ€·π‘π‘¦ξ…ž0𝑦0+π‘’βˆ’π‘–πœƒξ€·π‘π‘¦ξ…ž0𝑦0ξ€»(π‘₯)=π‘’βˆ’π‘–πœƒξ€Ίπ‘π‘¦ξ…ž0𝑦0+𝑒2π‘–πœƒπ‘π‘¦ξ…ž0𝑦0ξ€»(π‘₯)⟢0(4.36) as π‘₯β†’βˆž by assumption (4.3) and Reπ‘πœƒξ€½π‘’=Reπ‘–πœƒπ‘ξ€Ύβ‰₯0,Reπ‘žπœƒξ€½π‘’=Reπ‘–πœƒξ€·π‘žβˆ’πœ†0𝑀β‰₯𝛿0𝑀(4.37) by (4.15). Then letting π‘₯β†’βˆž in (4.35), we have a contradiction. This proves the first part of this theorem.
The proof of (ii): suppose that 𝜏 is in Case I. Set 𝑣1=βˆ’π‘–π‘’π‘–πœƒπ‘π‘¦ξ…ž1, 𝑣2=π‘–π‘’βˆ’π‘–πœƒπ‘π‘¦ξ…ž2 for 𝑦1βˆˆπ’Ÿ(𝜏), 𝑦2βˆˆπ’Ÿ(𝜏). Then, we can get (𝑦1,𝑣1)π‘‡βˆˆπ’Ÿ(πœƒ) by Lemma 4.6 and (𝑦2,𝑣2)π‘‡βˆˆπ’Ÿ(πœƒ) by Corollary 4.7. Hence 𝑦1,𝑣1𝑦0βˆ’110ξƒͺ2𝑣2ξƒͺ=π‘–π‘’βˆ’π‘–πœƒξ€·π‘π‘¦ξ…ž1𝑦2βˆ’π‘π‘¦1𝑦2ξ€ΈβŸΆ0(4.38) as π‘₯β†’βˆž by (4.9), that is, 𝑝(π‘₯)[𝑦2(π‘₯)π‘¦ξ…ž1(π‘₯)βˆ’π‘¦1(π‘₯)π‘¦ξ…ž2(π‘₯)]β†’0 as π‘₯β†’βˆž.
Conversely, if 𝜏 is not in Case I, then all solutions of (1.1) belong to 𝐿2𝑀 for πœ†βˆˆβ„‚. Let 𝑦𝑖,𝑖=1,2, be the solution of (πœβˆ’πœ†0)𝑦=0 such that ξƒ©π‘π‘¦ξ…ž1𝑦(0)1ξƒͺ=10ξƒͺ,(0)π‘π‘¦ξ…ž2𝑦(0)2ξƒͺ=01ξƒͺ(0).(4.39) Since π‘¦π‘–βˆˆπΏ2𝑀,π‘¦π‘–βˆˆπ·(𝜏),𝑖=1,2. Then the Wronskian |||||π‘π‘¦ξ…ž1𝑦1π‘π‘¦ξ…ž2𝑦2|||||𝑦=π‘ξ…ž1𝑦2βˆ’π‘¦1π‘¦ξ…ž2≑1,(4.40) which contradicts condition (4.5). See Remark 4.2.

Remark 4.8. If π‘ž(π‘₯) and 𝑝(π‘₯) are real valued, then Ξ©βŠ‚β„ and (πœƒ,𝐾)=(Β±πœ‹/2,0)βˆˆπ‘† with Re{π‘’π‘–πœƒπ‘(π‘₯)}=Re{π‘’π‘–πœƒ(π‘ž(π‘₯)βˆ’πΎπ‘€(π‘₯))}≑0. 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 ∞ if and only if 𝑦𝑝(π‘₯)2(π‘₯)π‘¦ξ…ž1(π‘₯)βˆ’π‘¦1(π‘₯)π‘¦ξ…ž2ξ‚„(π‘₯)⟢0asπ‘₯⟢∞(4.41) for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏), that is, (1.4). Clearly, if 𝑝 is real valued and πœ‹/2∈𝐸, then (4.3) reduces to (1.4). Therefore, (4.3) is a generalization of (1.4).

Corollary 4.9. If 𝐸 has more than one point, then 𝜏 is in Case I if and only if, for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏), 𝑝(π‘₯)𝑦1(π‘₯)π‘¦ξ…ž2(π‘₯)⟢0asπ‘₯⟢∞.(4.42) That is 𝜏 is in the strong limit point case at ∞.

Proof. Suppose that 𝐸 has more than one point and 𝜏 is in Case I. Choose πœƒπ‘—βˆˆπΈ, 𝑗=1,2, with πœƒ1β‰ πœƒ2 (modπœ‹). Then (4.3) holds for πœƒ=πœƒπ‘—, 𝑗=1,2. This gives that for 𝑦1,𝑦2βˆˆπ’Ÿ(𝜏)𝑒2π‘–πœƒ1βˆ’π‘’2π‘–πœƒ2𝑝𝑦1π‘¦ξ…ž2⟢0asπ‘₯⟢∞,(4.43) and hence (4.42) holds since πœƒ1β‰ πœƒ2 (modπœ‹).
Conversely, assume that (4.42) holds for all π‘¦π‘–βˆˆπ’Ÿ(𝜏), 𝑖=1,2. Since (4.42) implies (4.3), we conclude from (i) of Theorem 4.1 that 𝜏 is in Case I.

Corollary 4.10. If 𝜏 is symmetric and π‘ž(π‘₯)β‰₯π‘ž0𝑀(π‘₯) on [0,∞), then 𝜏 is in the limit point case at ∞ if and only if it is in the strong limit point case at ∞.

Proof. Note that for, πœƒβˆˆ[βˆ’πœ‹/2,πœ‹/2], (πœƒ,π‘ž0)βˆˆπ‘†. Then [βˆ’πœ‹/2,πœ‹/2]∈𝐸. Therefore, (4.3) holds if and only if (4.42) holds by Corollary 4.9.

Theorem 4.11. 𝜏 is in Case II with respect to (πœƒ0,𝐾0)βˆˆπ‘† if and only if π’Ÿ(𝜏)β‰ π’Ÿπœƒ0(𝜏)β‰ βˆ… and (4.3) holds for 𝑦1,𝑦2βˆˆπ’Ÿπœƒ0(𝜏).

Proof. Suppose that 𝜏 is in Case II with respect to some (πœƒ,𝐾)βˆˆπ‘†. By the definition of Case II we know that π’Ÿπœƒ(𝜏) is nonempty and π’Ÿ(𝜏)β‰ π’Ÿπœƒ(𝜏). 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 𝑦1,𝑦2βˆˆπ’Ÿπœƒ0(𝜏) by Lemma 4.3 and (4.32).
Conversely, suppose that π’Ÿ(𝜏)β‰ π’Ÿπœƒ(𝜏) for some (πœƒ,𝐾)βˆˆπ‘† and (4.3) holds for π‘¦βˆˆπ’Ÿπœƒ(𝜏). By the proof of Lemma 4.6, we know that 𝜏 is not in Case I with respect to (πœƒ,𝐾). We only need to prove that 𝜏 is not in Case III with respect to this (πœƒ,𝐾). If it is not true, then all solutions of (1.1) with πœ†βˆˆΞ›πœƒ,𝐾 satisfy (2.6) and so belong to π’Ÿπœƒ(𝜏). Let 𝑦0 be a nontrivial solution of (1.1) with 𝑦(0)=0. Then 𝑦0βˆˆπ’Ÿπœƒ(𝜏), and hence the same proof as in (4.33)–(4.35) yields a contradiction.

Corollary 4.12. If 𝐸 has more than one point, then 𝜏 is in case II with respect to some (πœƒ,𝐾)βˆˆπ‘† if and only if π’Ÿ(𝜏)β‰ π’Ÿπ‘ (𝜏) and (4.42) holds for 𝑦1,𝑦2βˆˆπ’Ÿπœƒ(𝜏) with πœƒβˆˆπΈπ‘œ.

Proof. If 𝐸 has more than one point and 𝜏 is in Case II with respect to some (πœƒ0,𝐾0)βˆˆπ‘†, then there exists πœƒ1βˆˆπΈπ‘œ such that 𝜏 is in Case II with respect to (πœƒ1,𝐾1)βˆˆπ‘† by Theorem 2.5. Since π’Ÿπœƒ1(𝜏)βŠŠπ’Ÿ(𝜏) by Theorem 4.11 and π’Ÿπ‘ (𝜏)=π’Ÿπœƒ1(𝜏) by Lemma 4.4, one sees that π’Ÿ(𝜏)β‰ π’Ÿπ‘ (𝜏).
Choose πœƒ2βˆˆπΈπ‘œ with πœƒ1β‰ πœƒ2 (modπœ‹) such that 𝜏 is in Case II with respect to (πœƒπ‘—,𝐾𝑗)βˆˆπ‘† for 𝑗=1,2 by Theorem 2.5. Then (4.3) holds for πœƒ=πœƒπ‘—, 𝑗=1,2 by Theorem 4.11. Since π’Ÿπœƒπ‘—(𝜏)=π’Ÿπ‘ (𝜏), the same proof as in (4.43) gives that (4.42) holds for 𝑦1,𝑦2βˆˆπ’Ÿπ‘ (𝜏).
Conversely, suppose that π’Ÿ(𝜏)β‰ π’Ÿπ‘ (𝜏) and (4.42) holds for π‘¦βˆˆπ’Ÿπ‘ (𝜏). Since π’Ÿπœƒ(𝜏)β‰‘π’Ÿπ‘ (𝜏) on πΈπ‘œ by Lemma 4.4, we conclude that π’Ÿπœƒ(𝜏)β‰ π’Ÿ(𝜏) on πΈπ‘œ and (4.42) holds for π‘¦βˆˆπ’Ÿπœƒ(𝜏). So (4.3) holds for π‘¦βˆˆπ’Ÿπœƒ(𝜏) by (4.42). Then, we have that 𝜏 is in Case II with respect to (πœƒ,𝐾)βˆˆπ‘† with πœƒβˆˆπΈπ‘œ by Theorem 4.11.


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