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 τy:=w-1[-(py′)′+qy]=λyon[a,b),
where p,q are both complex valued, w(x) is a positive weight function, -∞<a<b≤+∞, and λ is the so-called spectral parameter. We call τ a formally symmetric differential expression if p,q are both real valued; otherwise τ is called formally nonsymmetric. In all cases, we call τ a formally differential expression or operator.

Let Lw2 denote the Hilbert space Lw2:={yismeasurable:[a,b)⟶C:∫abw(x)|y(x)|2dx<∞}
with inner product 〈y,z〉:=∫abz̅(x)w(x)y(x)dx and the norm ∥y∥2=〈y,y〉 for y,z∈Lw2. We call a solution y of (1.1) an Lw2-solution or square integrable solution if y∈Lw2. Set D(τ)={y∈Lw2:y,py′∈ACloc,τy∈Lw2},
where ACloc=ACloc([a,b),ℂ) is the set of complex valued functions that are absolutely continuous on each compact subinterval of [a,b). We call 𝒟(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 𝒟(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(x) and q(x) 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 m(λ)-functions, where we say that τ or (1.1) is in the limit point case at b if there exists exactly one Lw2-solution (up to constant multiple) for λ∈ℂ with Imλ≠0 and is in the limit circle case if all solutions belong to Lw2 for λ∈ℂ with Imλ≠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 q(x) is complex valued [11]. He considered the case where p(x)=w(x)≡1 and Imq(x) 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 w(x) and the complex valued coefficients p(x), q(x). 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 p(x)[y2(x)y1′¯(x)-y1¯(x)y2′(x)]⟶0asx⟶b
for y1,y2∈𝒟(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 𝒟(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 𝒟(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(x) and q(x), we say that (1.1) is in the strong limit point case at the end point b if, for y1,y2∈𝒟(τ), p(x)y1(x)y2′(x)⟶0asx⟶b.

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 q. 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 𝒟(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 𝒟(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

p(x)≠0, w(x)>0 a.e. on [a,b) and 1/p,q,w are all locally integrable on [a,b),

p and q are complex valued, and
Ω=co¯{q(x)w(x)+rp(x):r>0,x∈[a,b)}≠C,
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 ∂Ω, there exists a line through this point such that every point of Ω 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 ∂Ω. Denote by L an arbitrary supporting line touching Ω at K, which may be the tangent to Ω at K if it exists. We then perform a transformation of the complex plane z↦z-K and a rotation through an appropriate angle θ so that the image of L coincides with the imaginary axis now and the set Ω is contained in the new right nonnegative half-plane.

For this purpose we introduce the set S defined by S={(θ,K):K∉Ω∘,Re{eiθ(μ-K)}≥0∀μ∈Ω},
where Ω∘ is the interior of Ω, and define the corresponding half-plane Λθ,K={μ∈C:Re{eiθ(μ-K)}<0}.
Then, Λθ,K⊂ℂ∖Ω. From the definition of S, for all x∈[a,b) and 0<r<∞, Re{eiθ[q(x)w(x)+rp(x)-K]}≥0.

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 (θ,K)∈SRe{eiθ(μ-K)}≥0⟺cos(θ+γ)≥0whereμ-K=|μ-K|eiγ.

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 Λθ,K as follows. The uniqueness referred to in the theorem and the following sections is only up to constant multiple.

Given a (θ,K)∈S, the following three distinct cases are possible. Case I.

For all λ∈Λθ,K, equation (1.1) has unique solution y satisfying
∫ab[Re{eiθp}|y′|2+Re{eiθ(q-Kw)}|y|2]+∫abw|y|2<∞
and this is the only solution satisfying y∈Lw2.

Case II.

For all λ∈Λθ,K, all solutions of (1.1) belong to Lw2, and there exists unique solution of (1.1) satisfying (2.6).

Case III.

For all λ∈Λθ,K, all solutions of (1.1) satisfy (2.6).

Since every Λθ,K is a half-plane, it holds that Λθ1,K1⋂Λθ2,K2≠∅
for (θj,Kj)∈S, j=1,2, with θ1≠θ2 (modπ). Note that (2.4) implies that, for 0<r<∞ and x∈[a,b), Re{eiθ(q(x)rw(x)+p(x)-Kr)}≥0.
Letting r→0 and r→∞ in (2.4) and (2.8), respectively, we have the following.

Lemma 2.2.

For every (θ,K)∈S and λ∈Λθ,K, there exists δλ(θ)>0 such that
Re{eiθ(q-Kw)}≥0,Re{eiθ(q-λw)}≥δλ(θ)w,Re{eiθp}≥0
on [a,b).

Using variation of parameters method, we can verify that, if all solutions of (1.1) belong to Lw2 for some λ0∈ℂ, then it is true for all λ∈ℂ. This also means the following.

Lemma 2.3.

If there exists a (θ0,K0)∈S such that (1.1) is in Case I with respect to (with respect to for short) Λθ0,K0, then (1.1) is in Case I with respect to Λθ,K for every (θ,K)∈S.

This indicates that Case I is independent of the choice of (θ,K)∈S. But Cases II and III depend on the choice of (θ,K)∈S in general, that is, there may exist (θ1,K1),(θ2,K2)∈S such that (1.1) is in Case II with respect to Λθ1,K1 and is in Case III with respect to Λθ2,K2. In order to make clear the dependence, we introduce the admissible angle set E defined by E={θ:∃K∉Ω∘,(θ,K)∈S}.

Remark 2.4.

For given θ∈E, there exist many K such that (θ,K)∈S. In fact, if θ0∈E with (θ0,K0)∈S for some K0∉Ω∘, then for all K∈L0, (θ0,K)∈SL0={λ∈C:Re{eiθ0(λ-K0)}=0}.

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

If there exists a (θ0,K0)∈S such that (1.1) is in Case II with respect to Λθ0,K0, then (1.1) is in Case II with respect to Λθ,K for all (θ,K)∈S except for at most one θ1∈E(modπ) such that (1.1) is in Case III with respect to Λθ1,K1.

Remark 2.6.

Theorem 2.5 means that, if there exist θj∈E, j=1,2, such that θ1≠θ2 (modπ) and (1.1) is in Case III with respect to Λθj,Kj for j=1,2, then (1.1) is in Case III with respect to Λθ,K for all (θ,K)∈S.

3. Properties of the Angel Set <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M160"><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:math></inline-formula>

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 θ1,θ2∈E with θ1≠θ2 (modπ).

Lemma 3.1.

Let E be defined as in (2.10).

The set E is connected in the sense of mod2π.

If E has more than one point, then, for every λ∈ℂ∖Ω, there exist θ1,θ2∈E with θ1<θ2 such that λ∈Λθ,K for θ∈(θ1,θ2)⊂E.

Proof.

(i) Suppose that E has more than one point. Let θ1,θ2∈E with θ1≠θ2 (modπ); then 0<θ2-θ1<π (mod2π) or π<θ2-θ1<2π (mod2π).

If 0<θ2-θ1<π (mod2π) and (θj,Kj)∈S, j=1,2, then we claim that [θ1,θ2]⊂E (mod2π). Let Lj be the line similarly defined as L0 with K0 and θ0 replaced by Kj and θj, j=1,2. That is,
Lj={λ∈C:Re{eiθj(λ-Kj)}=0},Kj∉Ω∘,j=1,2.
Let K be the intersection point of L1 and L2. Set
μ-K=|μ-K|eiγ(μ,K),μ∈Ω.
It follows from (2.5) that
cos(γ(μ,K)+θj)≥0,μ∈Ω,j=1,2.

By 0<θ2-θ1<π (mod2π) and (3.3), we can get cos(γ(μ,K)+θ)≥0 for θ∈[θ1,θ2] (mod2π) on Ω, which means (θ,K)∈S and θ∈E.

According to the similar method, we can verify that, if π<θ2-θ1<2π (mod2π) and (θj,Kj)∈S, j=1,2, then [0,θ1]∪[θ2,2π]⊂E (mod2π), that is, [θ2,θ1]⊂E (mod2π).

(ii) For λ0∈ℂ∖Ω, choose (θ1,K1)∈S and δ0>0 such that λ0∈Λθ1,K1 and
Re{eiθ1(K1-λ0)}=δ0>0.
Since E has more than one point, we can choose θ2∈E with θ2≠θ1 (modπ). Without loss of generality, we suppose that 0<θ2-θ1<π (mod2π). Let K be defined as in the proof of (i).

If λ0∈Λθ1,K∩Λθ2,K, then it follows from (2.3) that
cos(γ+θj)<0,j=1,2,whereλ0-K=|λ0-K|eiγ.
By 0<θ2-θ1<π (mod2π) and (3.5), we can get cos(γ+θ)<0 for θ∈[θ1,θ2] (mod2π), which means λ0∈Λθ,K for θ∈[θ1,θ2].

Suppose that λ0∉Λθ2,K. Let λ1∈L1 be the unique point such that δ0=dist(λ0,L1)=dist(λ1,λ0). Let α=arctan|λ0-λ1|/2|K-λ1|; then α+θ1∈(θ1,θ2)⊂E by λ0∉Λθ2,K, and λ0∈Λα+θ1,K by the definition of α (see Figure 1).

So, we can get that λ0∈Λθ,K for θ∈[θ1,θ1+α] by λ0∈Λθ1,K∩Λθ1+α,K, and the lemma is proved.

Figure of Lemma 3.1(ii).

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 (a,b), where -∞≤a<b≤+∞. Recall that (1.1) on (a,b) is said to be regular at a if 1/p, q and w are integrable on (a,c) for some (and hence any) c∈(a,b) and singular at a 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 𝒟(τ) in (1.3). We also define D(τ¯)={y∈Lw2:y,p¯y′∈ACloc,τ¯y∈Lw2},
where τ¯y:=w-1[-(p¯y′)′+q¯y]on[0,∞).
The first result of this section is as follows.

Theorem 4.1.

(i) τ is in Case I if and only if for y1,y2∈𝒟(τ) and θ∈Ep(x)¯y2(x)y1′(x)¯+e2iθp(x)y1(x)¯y2′(x)⟶0asx⟶∞.

(ii) τ is in Case I if and only if for y1∈𝒟(τ),y2∈𝒟(τ¯)p(x)[y2(x)y1′(x)¯-y1(x)¯y2′(x)]⟶0asx⟶∞.

Remark 4.2.

Clearly 𝒟(τ¯)=𝒟(τ)¯, by the definition of 𝒟(τ¯). It is easy to see that (4.4) is equivalent to
p(x)[y2(x)y1′(x)-y1(x)y2′(x)]⟶0asx⟶∞
for y1,y2∈𝒟(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 u′=Au+Bv+ξW2v,v′=Cu-A*v-ξW1u,on[0,∞),
where u,v are ℂn valued functions, uT is the transpose of u, A,B,C,W1, and W2 are locally integrable, complex valued n×n matrices on [0,∞), B,C,W1, and W2 are Hermit matrices and W1(t)>0,W2(t)≥0 on [0,∞), and ξ is the spectral parameter. Assume that the definiteness condition (see, e.g., [13, Chapter 9, page 253]) holds: ∫0∞y*Wy>0foreachnontrivialsolutionyof(4.6),
where W=diag(W1,W2). Let LW2:=LW2[0,∞) denote the space of Lebesgue measurable 2n-dimensional functions f satisfying ∫0∞f*(s)W(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 LW2 for ξ∈ℂ with Imξ≠0.

Let 𝒟 be the maximal domain associated with (4.6), that is, (uT,vT)T∈𝒟 if and only if (uT,vT)T∈ACloc∩LW2, and there exists an element (fT,gT)T∈LW2 such that u′=Au+Bv+ξW2v+W2g,v′=Cu-A*v-ξW1u-W1f,on[0,∞).
It is well known (cf. [5, 7]) that (4.6) is in the limit point case at infinity if and only if Y1*(x)JY2(x)⟶0asx⟶∞,J=(0-InIn0)
for Y1,Y2∈𝒟, and for every ξ∈ℂ with Imξ≠0 there exists a Green function G(t,s,ξ) such that, for F=(fT,gT)T∈LW2, Y=(uv)=TξF∈LW2,satisfies(4.8),
where (TξF)(x)=∫0∞G(x,s,ξ)W(s)F(s)ds.

Let (θ,K)∈S and choose λ0∈Λθ,K. Then from (2.9), one sees that Re{eiθ(q-λ0ω)}≥δ0ω>0,Re{eiθ(q-K0w)}≥0,Re{eiθp}≥0
for some δ0>0. Set r1(x)=|q(x)-λ0w(x)|,q(x)-λ0w(x)=r1(x)eiα(x),α1(x)=θ+α(x),r2(x)=|p(x)|,p(x)=r2(x)eiβ(x),β1(x)=θ+β(x).
Consider the Hamiltonian differential system (4.6) with n=1, A(x)≡0 and C(x)=r1(x)sinα1(x),W1(x):=w1(x)=r1(x)cosα1(x),B(x)=sinβ1(x)r2(x),W2(x):=w2(x)=cosβ1(x)r2(x),
that is, the 2-dimensional Hamiltonian differential system H(θ):u′=Bv+ξw2v,v′=Cu-ξw1u.
It follows from (4.11) that w1=Re{eiθ(q-λ0w)}≥δ0w>0,w2=Re{eiθp(t)}r22≥0,
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 (u,v)T is a solution of (4.14) with u=y,v=-ieiθpy′.
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 (θ,K)∈S if and only if H(θ) is in the limit point case at ∞.

(ii) τ is in Case III with respect to (θ,K)∈S if and only if H(θ) is in the limit circle case at ∞.

Lemma 4.4.

If E has more than one point, then 𝒟θ(τ)≡𝒟s(τ) on Eo, the interior of E, where
Dθ(τ)={y∈D(τ):∫0∞[Re{eiθp}|y′|2+Re{eiθq}|y|2]<∞},Ds(τ)={y∈D(τ):∫0∞[|p||y′|2+|q||y|2]<∞}.

Proof.

Let θ1∈Eo be fixed. There exist θ2,θ3∈Eo such that
θ3<θ1<θ2mod(2π),0<θ2-θ3<π2mod(2π),⋂j=13Λθj,Kj≠∅
by Lemma 3.1. Choose λ0∈⋂j=13Λθj,Kj. Letting β:=β(x) be defined as in (4.12) and solving cos(θ1+β) from the equations
cos(θj+β)=cos(θ1+β)cos(θj-θ1)-sin(θ1+β)sin(θj-θ1),j=2,3,
we have that cos(θ1+β)=C1cos(θ2+β)+C2cos(θ3+β) with
C1=sin(θ1-θ3)sin(θ2-θ3)>0,C2=sin(θ2-θ1)sin(θ2-θ3)>0
by (4.18). Since ∫0∞Re{eiθ1p}|y′|2<∞ for y∈𝒟θ1(τ), we have that
C1∫0∞Re{eiθ2p}|y′|2+C2∫0∞Re{eiθ3p}|y′|2=∫0∞Re{eiθ1p}|y′|2<∞,
and hence ∫0∞Re{eiθ2p}|y′|2<∞ for y∈𝒟θ1(τ). The same proof as the above with β replaced by α also proves ∫0∞Re{eiθ2(q-λ0w)}|y|2<∞ for y∈𝒟θ1(τ), where α:=α(x) is defined as in (4.12). Therefore, for y∈𝒟θ1(τ),
∫0∞[Re{eiθjp}|y′|2],∫0∞Re{eiθj(q-λ0w)}|y|2<∞,j=1,2.
Set pθ=eiθp and qθ=eiθ(q-λ0w). It follows from
sin2(θ2-θ1)=cos2θ2+cos2θ1-2cosθ2cosθ1cos(θ1-θ2)≤(cosθ2+cosθ1)2
and (4.15) that
Re(pθ1+pθ2)≥ε0|p|,Re(qθ1+qθ2)≥ε0|q-λw|,ε0=sin(θ2-θ1).
Then (4.24) and (4.22) yield that, for y∈𝒟θ1(τ),
∫0∞|p||y′|2,∫0∞|q-λ0w||y|2<∞.
Note that y∈Lw2. Then (4.25) gives y∈𝒟s(τ), or 𝒟θ1(τ)⊂𝒟s(τ). Clearly, 𝒟s(τ)⊂𝒟θ1(τ). Thus 𝒟θ1(τ)=𝒟s(τ).

Lemma 4.4 indicates the following.

Corollary 4.5.

If τ is in Case II with respect to some (θ0,K0)∈S 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 (θ1,K1)∈S with θ1∈Eo, then 𝒟(τ)=𝒟θ1(τ) is restricted in the solution space of (1.1) by the definition of Case III. Since 𝒟θ1(τ)=𝒟s(τ) by Lemma 4.4, we have that 𝒟(τ)=𝒟s(τ) restricted in the solution space of (1.1). This means that all solutions of (1.1) with λ∈Λθ1,K1 satisfy
∫0∞(|p||y′|2+|q||y|2)<∞.
Using variation of parameters method we can prove that it is true for all λ∈ℂ, and hence τ is in Case III with respect to (θ0,K0), a contradiction.

Lemma 4.6.

If τ is in Case I and y∈𝒟(τ), then (y,v)T∈𝒟(θ) with v=-ieiθpy′, where 𝒟(θ) is the maximal domain associated with (4.14).

Proof.

Suppose that τ is in Case I with respect to (θ,K)∈S. We claim that 𝒟(τ)=𝒟θ(τ). Set
(τ-λ0)y0=w-1[-(py0′)′+(q-λ0w)y0]=g0,
for y0∈𝒟(τ) and λ0∈Λθ,K.

Set u0=y0, v0=-ieiθpy0′. Then (u0,v0) satisfies
u′=Bv+iw2v,v′=Cu-iw1u-w1f1,f1=ww1(-ieiθg0).
Conversely, if (u,v) satisfies (4.28), then y=u solves (4.27). Note that g0∈Lw2, or -ieiθ0g0∈Lw2, and w1≥δw implies f1∈Lw12.

Considering (4.28), we get from (4.10) that (4.28) has a solution (u1,v1)T such that u1∈Lw12,v1∈Lw22 and v1=-ieiθ0pu1′. Set y1=u1. Then y1 satisfies (4.27), and hence (τ-λ0)(y0-y1)=0. Note that y1=u1∈Lw12, and w1≥δw implies that y1∈Lw2. Thus, y1-y0 is an Lw2-solution of τy=λ0y. Since τ is in Case I with respect to (θ0,K0), it follows from (2.6) that y1-y0∈Lw12 and v1-v0∈Lw22. This together with y1∈Lw12 and v1∈Lw22 gives y0∈Lw12 and v0∈Lw22. In fact, we have proved that, for y∈𝒟(τ),
∫0∞|q-λ0w|cosα1|y|2<∞,∫0∞|p|cosβ1|y′|2<∞,
or
∫0∞[Re{eiθp}|y′|2+Re{eiθ(q-λ0w)}|y|2]<∞,
where α1 and β1 are defined in (4.12) or (4.13). Since y∈Lw2, (4.30) means that
∫0∞[Re{eiθp}|y′|2+Re{eiθq}|y|2]<∞
or y∈𝒟θ(τ), and hence 𝒟(τ)=𝒟θ(τ). Recall that f1∈Lw12. Then (4.30) and (4.28) imply that, if y∈𝒟(τ), then (y,v)T∈𝒟(θ).

Corollary 4.7.

If τ is in Case I and y∈𝒟(τ¯), then (y,v)T∈𝒟(θ) with v=ie-iθp¯y′.

Proof.

For y∈𝒟(τ¯), y¯∈𝒟(τ) by 𝒟(τ¯)=𝒟(τ)¯. So (y¯,v¯)T∈𝒟(θ) with v¯=-ieiθpy¯′ by Lemma 4.6. Clearly 𝒟(θ)=𝒟(θ)¯ since H(θ) is symmetrical. Then we have that (y,v)T∈𝒟(θ) with v=ie-iθp¯y′.

Proof of Theorem <xref ref-type="statement" rid="thm4.1">4.1</xref>.

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 Y1,Y2∈𝒟(θ). For y1,y2∈𝒟(τ), since (yj,vj)T∈𝒟(θ) with vj=-ieiθpyj′, j=1,2, by Lemma 4.6, it follows from (4.9) that
(y¯1,v¯1)(0-110)(y2v2)=ie-iθ(p¯y¯1′y2+e2iθpy¯1y2′)⟶0
as x→∞.

Conversely, assume that (4.3) holds for all elements of 𝒟(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 Lw2 for λ∈ℂ. Choose λ0∈Λθ,K, and let y0 be a nontrivial solution of (1.1) satisfying y0(0)=0. Then y0∈𝒟(τ) by y0∈Lw2. Furthermore, it follows from (τ-λ0)y0=0 that
-(py0′)′y̅0+(q-λ0w)|y0|2=0,-(p̅y̅0′)′y0+(q̅-λ̅0w)|y0|2=0.
Integrating (4.33) on [0,x] we have that
-(py0′)y̅0|0x+∫0x[p|y0′|2+(q-λ0w)|y0|2]=0,-(p̅y̅0′)y0|0x+∫0x[p̅|y0′|2+(q̅-λ̅0w)|y0|2]=0.
Multiplying eiθ and e-iθ to the first and second equalities in (4.34), respectively, and adding them together, we have that
-[eiθ(py0′)y̅0+e-iθ(p̅y̅0′)y0](x)+∫0x[(pθ+p̅θ)|y0′|2+(qθ+q̅θ)|y0|2]=0
since y0(0)=0, where pθ=eiθp and qθ=eiθ(q-λ0w). Note that
[eiθ(py0′)y̅0+e-iθ(p̅y̅0′)y0](x)=e-iθ[p̅y̅0′y0+e2iθpy0′y̅0](x)⟶0
as x→∞ by assumption (4.3) and
Repθ=Re{eiθp}≥0,Reqθ=Re{eiθ(q-λ0w)}≥δ0w
by (4.15). Then letting x→∞ 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=-ieiθpy1′, v2=ie-iθp¯y2′ for y1∈𝒟(τ), y2∈𝒟(τ¯). Then, we can get (y1,v1)T∈𝒟(θ) by Lemma 4.6 and (y2,v2)T∈𝒟(θ) by Corollary 4.7. Hence
(y¯1,v¯1)(0-110)(y2v2)=ie-iθ(p¯y¯1′y2-p¯y¯1y2)⟶0
as x→∞ by (4.9), that is, p(x)[y2(x)y1′(x)¯-y1(x)¯y2′(x)]→0 as x→∞.

Conversely, if τ is not in Case I, then all solutions of (1.1) belong to Lw2 for λ∈ℂ. Let yi,i=1,2, be the solution of (τ-λ0)y=0 such that
(py1′(0)y1(0))=(10),(py2′(0)y2(0))=(01).
Since yi∈Lw2,yi∈D(τ),i=1,2. Then the Wronskian
|py1′y1py2′y2|=p(y1′y2-y1y2′)≡1,
which contradicts condition (4.5). See Remark 4.2.

Remark 4.8.

If q(x) and p(x) are real valued, then Ω⊂ℝ and (θ,K)=(±π/2,0)∈S with Re{eiθp(x)}=Re{eiθ(q(x)-Kw(x))}≡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
p(x)[y2(x)y1′¯(x)-y1¯(x)y2′(x)]⟶0asx⟶∞
for y1,y2∈𝒟(τ), that is, (1.4). Clearly, if p is real valued and π/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,y2∈𝒟(τ),
p(x)y1(x)y2′(x)⟶0asx⟶∞.
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 θj∈E, j=1,2, with θ1≠θ2 (modπ). Then (4.3) holds for θ=θj, j=1,2. This gives that for y1,y2∈𝒟(τ)(e2iθ1-e2iθ2)py¯1y2′⟶0asx⟶∞,
and hence (4.42) holds since θ1≠θ2 (modπ).

Conversely, assume that (4.42) holds for all yi∈𝒟(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 q(x)≥q0w(x) 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, θ∈[-π/2,π/2], (θ,q0)∈S. Then [-π/2,π/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 (θ0,K0)∈S if and only if 𝒟(τ)≠𝒟θ0(τ)≠∅ and (4.3) holds for y1,y2∈𝒟θ0(τ).

Proof.

Suppose that τ is in Case II with respect to some (θ,K)∈S. By the definition of Case II we know that 𝒟θ(τ) is nonempty and 𝒟(τ)≠𝒟θ(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,y2∈𝒟θ0(τ) by Lemma 4.3 and (4.32).

Conversely, suppose that 𝒟(τ)≠𝒟θ(τ) for some (θ,K)∈S and (4.3) holds for y∈𝒟θ(τ). By the proof of Lemma 4.6, we know that τ is not in Case I with respect to (θ,K). We only need to prove that τ is not in Case III with respect to this (θ,K). If it is not true, then all solutions of (1.1) with λ∈Λθ,K satisfy (2.6) and so belong to 𝒟θ(τ). Let y0 be a nontrivial solution of (1.1) with y(0)=0. Then y0∈𝒟θ(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 (θ,K)∈S if and only if 𝒟(τ)≠𝒟s(τ) and (4.42) holds for y1,y2∈𝒟θ(τ) with θ∈Eo.

Proof.

If E has more than one point and τ is in Case II with respect to some (θ0,K0)∈S, then there exists θ1∈Eo such that τ is in Case II with respect to (θ1,K1)∈S by Theorem 2.5. Since 𝒟θ1(τ)⊊𝒟(τ) by Theorem 4.11 and 𝒟s(τ)=𝒟θ1(τ) by Lemma 4.4, one sees that 𝒟(τ)≠𝒟s(τ).

Choose θ2∈Eo with θ1≠θ2 (modπ) such that τ is in Case II with respect to (θj,Kj)∈S for j=1,2 by Theorem 2.5. Then (4.3) holds for θ=θj, j=1,2 by Theorem 4.11. Since 𝒟θj(τ)=𝒟s(τ), the same proof as in (4.43) gives that (4.42) holds for y1,y2∈𝒟s(τ).

Conversely, suppose that 𝒟(τ)≠𝒟s(τ) and (4.42) holds for y∈𝒟s(τ). Since 𝒟θ(τ)≡𝒟s(τ) on Eo by Lemma 4.4, we conclude that 𝒟θ(τ)≠𝒟(τ) on Eo and (4.42) holds for y∈𝒟θ(τ). So (4.3) holds for y∈𝒟θ(τ) by (4.42). Then, we have that τ is in Case II with respect to (θ,K)∈S with θ∈Eo 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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