Classification of Sturm–Liouville differential equations with complex coefficients and operator realizations

Jiangang Qi, Zhaowen Zheng, Huaqing Sun

Abstract

In this paper, a new classification of Sturm–Liouville differential equations with complex coefficients is given. Compared with the corresponding result of Brown et al., this classification reveals the great effects of rotation angle and it is independent of the rotation angles. Moreover, the asymptotic behaviours of functions in the maximal domain are presented and J-self-adjoint extensions associated with the differential equations are characterized.

1. Introduction

Consider the Sturm–Liouville differential equation Embedded Image1.1where p,q are complex functions, p(x)≠0 and w(x)>0 a.e. on (a,b), Embedded Image, 1/p,q,w are all locally integrable on (a,b), λ is the so-called spectral parameter.

One of the aims of the present paper is to study the classification of equation (1.1) according to the number of square integrable solutions of equation (1.1) in suitable weighted integrable spaces. This type classification of differential equations plays an important role in the spectral theory of differential operators as it can tell us how to obtain the operator realizations associated with the differential equations. The study of this problem has a long history since the pioneering work of Weyl (1910). When p(x) and q(x) are all real-valued, Weyl classified equation (1.1) into the limit-point and limit-circle cases by introducing the m(λ)-functions. 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 Dunford & Schwartz (1963), Eastham (1979), Hinton & Shaw (1981, 1983, 1984), Weidmann (1987), Krall (1989a,b), Everitt & Markus (1999), Brown et al. (2003), Zettl (2005) and references therein.

The same problem was also studied by Sims (1957) for the case where q(x) is complex-valued. He considered the case where p(x)=w(x)≡1, Im q(x) is semi-bounded and classified equation (1.1) into three cases. Recently, this work was extensively generalized by Brown et al. (1999, 2003).

In order to state clearly the classification given in Brown et al. (1999), we introduce some notations. Define Embedded Image1.2and assume that Embedded Image, where Embedded Image 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 each point of Ω either lies in the same side of this line or is on it. 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 zzK and a rotation through an appropriate angle θ∈(−π,π], so that the image of L coincides with the new imaginary axis and the set Ω is contained in the new right non-negative half-plane. Therefore, for all x∈(a,b) and Embedded Image, Embedded Image1.3

For such admissible values of K and θ, set Embedded Image1.4and define the corresponding half-plane Embedded Image1.5Then, Embedded Image. Let Embedded Image be the Hilbert space Embedded Image with inner product Embedded Image and the norm ∥y∥=〈y,y1/2 for Embedded Image. We call a solution y of equation (1.1) a Embedded Image-solution if Embedded Image.

With these definitions and using a nesting circle method based on the methods of both Weyl (1910) and Sims (1957), Brown et al. (1999) divided equation (1.1) into three cases with respect to the corresponding half-planes Λθ,K as follows. The uniqueness referred to in the theorem and following sections is only up to constant multiples.

Theorem 1.1

(see Brown et al. (1999, theorem 2.1)). Given a (θ,K)∈S, the following three distinct cases are possible.

  • I: For all λ∈Λθ,K, equation (1.1) has a unique solution y satisfying Embedded Image1.6and this is the only solution satisfying Embedded Image;

  • II: For all λ∈Λθ, K, all solutions of equation (1.1) belong to Embedded Image and there exists a unique non-trivial solution of equation (1.1) that satisfies equation (1.6).

  • III: For all λ∈Λθ, K, all solutions of equation (1.1) satisfy equation (1.6).

Remark 1.2

If q(x) and p(x) are real-valued, then Embedded Image and (θ,K)=(π/2,0)∈S. Hence Re{eiθp(x)}=Re{eiθ(q(x)−Kw)}≡0. So case II is vacuous. This means that the classification mentioned above reduces to Weyl’s limit-point, limit-circle classification.

One sees from theorem 1.1 that, unlike the classification of equation (1.1) with real coefficients, the three cases in theorem 1.1 formally depend on the choice of (θ,K) in S or the half-planes Λθ,K. Indeed, cases II and III depend on the choice of (θ,K). See theorem 2.3 in §2. One of the main results in this paper is to give a new classification of equation (1.1) which is independent of the rotation angles (or the half-planes). See theorem 3.2 in §3. This kind of classification gives detailed properties for Embedded Image-solutions of equation (1.1). Applying theses properties we obtain the asymptotic behaviours of functions in the maximal domain when equation (1.1) is in cases I or II in §4. (theorem 4.1). Furthermore, the asymptotic behaviours offer a simple tool to characterize the J-self-adjoint operator realizations associated with equation (1.1). See theorems 5.1 and 5.2, which are the generalization of the results theorem 4.4 in Brown et al. (1999) and theorem 10.26 in Edmunds & Evans (1987), respectively.

Following this section, §2 gives some preliminary results for equation (1.1) and §3 presents a new classification for equation (1.1). Section 4 studies the asymptotic behaviours and §5 deals with the J-self-adjoint operator realizations associated with equation (1.1).

2. Preliminary results

Let Ω, S and Λθ,K be defined as in §1. Since each Λθ, K is a half-plane, for (θj,Kj)∈S, j=1,2 with θ1θ2(mod π), it holds that Embedded Image2.1Note that equation (1.3) implies that for Embedded Image and x∈(a,b), Embedded Image2.2Let r→0 and Embedded Image in equations (1.3) and (2.2), respectively, we have that:

Lemma 2.1

For each ( θ,K)∈S and λ∈Λθ,K, there exists δλ)>0 such that Embedded Image2.3

Using variation of parameters formula, we can verify that if all solutions of equation (1.1) belong to Embedded Image for some Embedded Image, then it is true for all Embedded Image. This also means that:

Lemma 2.2

If there exists a (θ0,K0)∈S such that equation (1.1) is in case I with respect to (w.r.t.) Λθ0,K0, then equation (1.1) is in case I w.r.t. Λθ, K for each (θ,K)∈S.

This indicates that case I is independent of the choice of (θ,K)∈S. However, 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 equation (1.1) is in case II w.r.t. Λθ1,K1 and is in case III w.r.t. Λθ2,K2. In order to make clear the dependence, we introduce the admissible angle set E defined by Embedded Image2.4Note that for fixed θ0E, the K such that (θ0,K)∈S may be not unique. The exact dependence of cases II and III on (θ,K) is given in the following theorem.

Theorem 2.3

(cf. Sun & Qi (2010, theorem 2.1)) If there exists a (θ0,K0)∈S such that equation (1.1) is in case II w.r.t. Λθ0,K0, then equation (1.1) is in case II w.r.t. Λθ,K for all (θ,K)∈S except for at most one θ1∈E (in the sense of mod π) such that equation (1.1) is in case III w.r.t. Λθ1,K1, where (θ1,K1)∈S.

Remark 2.4

Theorem 2.3 indicates that if there exists θ1,θ2E such that θ1θ2 (mod π) and equation (1.1) is in case III w.r.t. Λθj,Kj for j=1,2, then equation (1.1) is in case III w.r.t. Λθ,K for all (θ,K)∈S.

3. A new classification

Let E be defined as in equation (2.4). We will find that the number of the elements in E determines the dependence of cases II and III on (θ,K)∈S. In fact, if E has only one point, then the classification of Brown et al. in theorem 1.1 is independent of the choice of (θ,K)∈S. In order to give a new classification which is independent of (θ,K)∈S, we need only to consider the case when E has more than one point. In what follows, we assume that E has more than one point, i.e. there exist at least θ1,θ2E with θ1θ2 (mod π). To begin with, we prepare some properties of the set E.

Lemma 3.1

Let E be defined as in equation (2.4).

  1. If E has more than one point, then E is a sub-interval of (−π,π].

  2. If E has more than one point, then for each Embedded Image, there exist θ1θ2E with θ1<θ2 such that for θ∈(θ1,θ2)⊂E, λΛθ, K, where (θ,K)∈S.

Proof.

(i) Let θ1,θ2E with θ1θ2 (mod π), θ1<θ2 and K1,K2 be the points on ∂Ω such that (θj,Kj)∈S, j=1,2. We claim that [θ1,θ2]⊂E. For the case K1=K2=K, we prove that (θ,K)∈S for all θ∈(θ1,θ2). Set Embedded ImageIt follows from the definition of S that for j=1,2 and r≥0, Re{eiθj((q(x)/w(x))+rp(x)−K)}≥0 on (a,b), or equivalently, Embedded Image. Without any confusion, we write γ(x,r,K) (respectively, γj(x,r,K)) as γ (respectively, γj). If we set Embedded Imagethen Embedded Image and Embedded Image can be expressed as Embedded Image3.1by using the formulae Embedded Image for j=1,2. This equality gives that Embedded Image3.2for θ∈(θ1,θ2). Since Embedded Image for j=1,2 implies −π/2≤γ1,γ2π/2(mod 2π), we have from πθ2>θ1>−π, θ2θ1π and θ2θ1=γ2γ1 that 0<θ2θ1<π. Consequently, for θ∈(θ1,θ2), we obtain Embedded ImageTherefore, each term in the right-hand side of equation (3.2) is non-negative, so Embedded Image on (a,b). That is, for r≥0 and x∈(a,b), Embedded Imagewhich implies (θ,K)∈S for θ∈(θ1,θ2).

In case K1K2, we choose μ0Λθ1,K1Λθ2,K2. Then it holds that Embedded Imageon (a,b) for j=1,2 and r≥0 by the definition of Λθ,K, or Embedded Image. Then, the similar proof in equations (3.1) and (3.2) yields that Embedded Image3.3Let L be the line defined by Embedded Image3.4for fixed θ∈(θ1,θ2). One sees from equations (3.3), (3.4) and the definition of Ω that Embedded Image. Set d=dist(L,∂Ω) and let K∈∂Ω be a point such that d=dist(K,L). Since Embedded Image3.5we have that Re{eiθ(μμ0)}=dist(μ,L)≥dist(K,L)=Re{eiθ(Kμ0)} for μΩ, hence Re{eiθ(μK)}=Re{eiθ(μμ0)}−Re{eiθ(Kμ0)}≥0, or Embedded Imageon (a,b) for r≥0 and θ∈(θ1,θ2), which means (θ,K)∈S, or θE. This proves (i) of lemma 3.1.

(ii) For Embedded Image, choose (θ0,K0)∈S and δ0>0 such that λ0Λθ0,K0 and Embedded ImageSince E has more than one point, we can choose Embedded Image such that Embedded Image. Without loss of generality, we suppose that Embedded Image. It follows from the conclusion of (i) in lemma 3.1 that Embedded Image. For each Embedded Image, there exists a point K(θ)∈∂Ω such that (θ,K(θ))∈S. By the definition of S we have that Embedded Image3.6If we set r(θ)=|K(θ)−K0| and K(θ)−K0=r(θ) eiη(θ), then equation (3.6) means that Embedded ImageThis together with θ>θ0 gives that θ+η(θ)≥π/2≥θ0+η(θ) (mod 2π), hence θ0+η(θ)→π/2 (mod 2π) as θθ0+0. We claim that r(θ) is bounded in a right-neighbourhood of θ0. Suppose, on the contrary, there exists a sequence, say, {θn} such that θnθ0+0 and Embedded Image as Embedded Image. Choose Embedded Image such that η0θ0+π/2<π and a corresponding point K(η0)∈∂Ω such that (η0,K(η0))∈S, we have that Embedded Imagewhich is a contradiction. Since r(θ) is bounded and θ+η(θ)→π/2 (mod 2π) as θθ0+0, we have that Embedded Image as θθ0+0. Hence Embedded Imageas θθ0+0. Therefore, there exists Embedded Image such that for all θ∈(θ0,ξ), Re{eiθ(λ0K(θ))}<0. This means that λ0Λθ, K for θ∈(θ0,ξ). This completes the proof. ■

Now, we state the new accurate classification of equation (1.1) under the condition that E has more than one point. This classification is independent on the choice of (θ,K)∈S.

Theorem 3.2

Suppose that E has more than one point. Then, the following three distinct cases are possible.

Case one: For all Embedded Image, equation (1.1) has a unique solution Embedded Image and this solution y also satisfies Embedded Image3.7

Case two: For all Embedded Image, all solutions of equation (1.1) belong to Embedded Image but only a unique solution satisfies equation (3.7);

Case three: For all Embedded Image, all solutions of equation (1.1) satisfy equation (3.7).

Proof.

For fixed Embedded Image, by lemma 2.2 we have two cases to be considered:

  1. There exists a unique solution of equation (1.1) belonging to Embedded Image;

  2. All solutions of equation (1.1) belong to Embedded Image.

Suppose that the case (a) occurs. This means that equation (1.1) is in case I for all (θ,K)∈S. By (ii) of lemma 3.1 there exist (θ1,K1),(θ2,K2)∈S with θ1θ2 (mod π) such that λΛθ1,K1Λθ2,K2, hence there exists a unique solution y of equation (1.1) such that Embedded Image3.8hold simultaneously. Set ρ(x)=|q(x)−λw(x)| and Embedded Image3.9Then lemma 2.1 ensures that Embedded Image on (a,b). It follows that Embedded Image3.10hence Embedded Image3.11Similarly, we have that Embedded Image3.12It follows from equation (3.8) that Embedded ImageSince |q|≤|qλw|+|λ|w| and Embedded Image, we have Embedded Image. With the similar proof as above and using equations (3.8) and (3.12), we have that Embedded Image. This proves that equation (1.1) is in case one.

Suppose that the case (b) holds for Embedded Image. It follows from theorem 2.3 that there are two subcases to be considered:

(b1) There exists at most one θ0E such that equation (1.1) is in case III w.r.t. Λθ0,K0 and equation (1.1) is in case II w.r.t. Λθ,K for all (θ,K)∈S with θθ0 (mod π);

(b2) For all (θ,K)∈S, equation (1.1) is in case III w.r.t. Λθ,K.

For the subcase (b1) we prove first that for given Embedded Image, there exist Embedded Image with Embedded Image (mod π) and there exists a unique solution of equation (1.1) satisfying equation (3.8) simultaneously for θ, Embedded Image. In fact, by (ii) of lemma 3.1 there exist (θj,Kj)∈S, j=1,2,3 such that Embedded Imageθjθ0 (if such θ0 exists) for j=1,2,3 and Embedded Image. Since equation (1.1) is in case II w.r.t. (θ1,K1), there exists a unique solution y of equation (1.1) satisfying equation (3.8) with j=1 (this solution depends on θ1E). Define α=α(x) as in equation (3.9). Then, solving Embedded Image from the equations Embedded Imagewe find that Embedded Image3.13with Embedded Image, Embedded Image. Now for the solution y relate to θ1, we obtain by equation (3.13) that Embedded Imagehence Embedded Image3.14Similarly, we have Embedded Image3.15Therefore, the solution y of equation (1.1) satisfies equation (3.8) simultaneously for θ2,θ3E with θ2θ3 (mod π) by equations (3.14) and (3.15). Next, the similar proof as in the case (a) proves that this solution y satisfies equation (3.7). Clearly, such a solution which satisfies equation (3.7) is unique since equation (1.1) is in case II w.r.t. Λθj,Kj, j=1,2, hence equation (1.1) is in case two.

Finally, we prove that the subcase (b2) implies equation (1.1) is in case three. Using lemma 3.1 we can choose (θj,Kj)∈S such that θ1θ2 (mod π) and λΛθ1K1Λθ2K2. Since equation (1.1) are in case III w.r.t. Λθj,Kj, j=1,2, all solutions of equation (1.1) satisfy equation (3.8) for both j=1 and j=2. Therefore, the similar argument as above yields that all solutions of equation (1.1) satisfy equation (3.7). This gives that equation (1.1) is in case three. ■

Remark 3.3

If E has more than one point, then the relationship between the classifications in theorems 1.1 and 3.2 can be stated clearly as follows:

  1. equation (1.1) is in case I if and only if equation (1.1) is in case one;

  2. equation (1.1) is in case II w.r.t. Λθ,K for some θE if and only if it is in case two; and

  3. equation (1.1) is in case III w.r.t. Λθ,K for all θE if and only if it is in case three.

Many assumptions on coefficients can ensure that E has more than one point. For example, if p(x)>0 and q(x)≥q0w(x), then E has more than one point. E has only one point if and only if the boundary of Ω is one or two straight lines. For this case, the classification of equation (1.1) in theorem 3.2 may be not true.

For example, consider equation (1.1) with Embedded Image and λ=i. We conclude from Hille (1969, theorem 10.1.5, p. 503) that equation (1.1) is in the limit point case at Embedded Image. Denote by y(x,i) the unique solution of equation (1.1) with λ=i such that Embedded Image. It follows from Hille (1969, exercise 10, p. 504) that y′, Embedded Image. So, this equation is not in any one of the three cases in theorem 3.2. We note that E has only one point π/2 for this equation. Examples of complex-valued p and q where cases two and three occur are given in Sun & Qi (2010).

4. Asymptotic behaviours

In this section, we will give asymptotic behaviours of the elements in the maximal domain of the formal differential operator τ defined on the interval Embedded Image with 0 being a regular endpoint and Embedded Image being implicitly a singular endpoint. All results in this section can be proved with the similar argument for any singular endpoint, left or right on an arbitrary interval (a,b), where Embedded Image. The interval Embedded Image considered here is only for the sake of simplicity. Recall that equation (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; and the regularity and singularity at b are defined similarly. Note that the regularity (respectively, singularity) of an endpoint is solely determined by the integrability (respectively, non-integrability) of the coefficients in equation (1.1) at the endpoint, not the finiteness (respectively, infiniteness) of the endpoint, as has already been remarked by Atkinson (1964, §9.1). See also Zettl (2005, theorem 2.3.1). Consider the formal differential operator τ associated with equation (1.1) Embedded Image4.1

Let Embedded Image denote the set of complex-valued functions which are absolutely continuous on each compact sub-interval of Embedded Image. We define the domain of the maximal operator associated with τ as follows: Embedded Image4.2

Theorem 4.1

(i) If equation (1.1) is in case I w.r.t. some (θ,K)∈S, then Embedded Image4.3for all Embedded Image. If, in addition, E has more than one point, then Embedded Image, Embedded Image for all Embedded Image, and for Embedded Image, Embedded Image4.4

(ii) If equation (1.1) is in case II w.r.t. some (θ,K)∈S, then equation (4.3) holds for Embedded Image, Embedded Image4.5If, in addition, E has more than one point, then equation (4.4) holds for y1, Embedded Image.

We will use spectral theory of Hamiltonian differential systems to prove theorem 4.1, so we first prepare some known results for the Hamiltonian differential system Embedded Image4.6on the Embedded Image valued (column) functions Y =(uT,vT)T, where u,v are Embedded Image valued functions, uT is the transpose of u, A,B,C,W1 and W2 are locally integrable, complex-valued n×n matrices on Embedded Image, B,C,W1,W2 are Hermitian matrices and W1(t)>0,W2(t)≥0 on Embedded Image, ξ is the so-called spectral parameter. Assume that the definiteness condition (e.g. Atkinson 1964, ch. 9, p. 253) holds: Embedded Imagewhere W = diag(W1,W2). Let Embedded Image be the space of Lebesgue measurable 2n-dimensional functions f satisfying Embedded Image. We say that equation (4.6) is in the limit-point case at infinity if there exists exactly n’s linearly independent solutions of equation (4.6) belong to Embedded Image for ξi. Particularly, if n=1 and A,B,C are real functions, then equation (4.6) is in the limit point case at infinity if and only if there exists a unique solution of equation (4.6) belonging to Embedded Image for ξ=i or ξ=−i.

Let Embedded Image be the maximal domain associated with equation (4.6), i.e. Embedded Image if and only if Embedded Image and there exists an element Embedded Image such that Embedded Image4.7It is well known (cf. Hinton & Shaw 1983; Krall 1989a) that equation (4.6) is in the limit point case at infinity if and only if Embedded Image4.8for Embedded Image, and for each Embedded Image with Im ξ≠0 there exists a Green function G(t,s,ξ) such that for Embedded Image, equation (4.7) has an Embedded Image-solution Y given by Embedded Image4.9

The proof of theorem 4.1.

The proof of (i). Suppose that τ is in case I w.r.t. (θ0,K0)∈S. Choose λ0Λθ0,K0. Then Embedded Image4.10for some δ0>0. Set Embedded Image4.11Set u=y, v=−ieiθ0py′, then equation (1.1) (with λ=λ0) is transformed into the Hamiltonian differential system Embedded Image4.12with the new spectral parameter ξ=i, where Embedded Image4.13This is the Hamiltonian differential system (4.6) with n=1, A(x)≡0 and ξ=i. Clearly, w1=Re{eiθ0(qλ0w)}≥δ0w>0, Embedded Image by equation (4.10). We note further that the coefficients of the Hamiltonian system (4.12) are real functions. It is easy to verify that the definiteness condition holds for the system (4.12). Therefore, equation (1.1) is in cases I or II w.r.t. (θ0,K0)∈S if and only if equation (4.12) is in the limit point case at Embedded Image.

Let Embedded Image be the maximal domain associated with equation (4.12), i.e. Embedded Image if and only if Embedded Image and there exists an element Embedded Image such that Embedded Image4.14For Embedded Image, set Embedded Image4.15Then y0 satisfies Embedded Image4.16

Set u0=y0, v0=−ieiθ0py0. Then (u0,v0) satisfies Embedded Image4.17Conversely, if (u,v) satisfies equation (4.17), then y=u solves equation (4.16).

Let g0 be given in equation (4.15). Consider the equation (4.17), we get from equation (4.9) that equation (4.17) has a solution (u1,v1)T such that Embedded Image, Embedded Image and v1=−ieiθ0pu1. Set y1=u1. Then y1 satisfies equation (4.16), hence (τλ0)(y0y1)=0. Note that Embedded Image and w1δw implies that Embedded Image. Thus, y1y0 is an Embedded Image-solution of τy=λ0y. Since τ is in case I w.r.t. (θ0,K0), it follows from equation (1.6) that Embedded Image and Embedded Image. This together with Embedded Image and Embedded Image gives that Embedded Image and Embedded Image. In fact, we have proved that for Embedded Image, Embedded Image4.18where α1 and β1 are defined in equation (4.11). Recall that Embedded Image, or Embedded Image and w1δw implies Embedded Image. It follows equation (4.17) that (y0,v0) satisfies equation (4.14) with f=iy0+f1 and g=iv0. This yields that Embedded Image4.19

Note that Embedded Image if and only if Embedded Image. Then for Embedded Image, we have from equation (4.19) that Embedded Image with Embedded Image, hence Embedded Image4.20

Let Embedded Image. Since equation (4.12) is in the limit-point case at infinity and Embedded Image for j=1,2 with v1=−ieiθ0py1 and Embedded Image by equations (4.19) and (4.20), respectively, we get from equation (4.8) that Embedded Image4.21as Embedded Image. This proves that equation (4.3) holds. Furthermore, for y1Embedded Image, since Embedded Image by equation (4.19) with vj=−ieiθ0pyj, j=1,2, equation (4.8) also gives that Embedded Imageas Embedded Image, or Embedded Image4.22as Embedded Image. If, in addition, E has more than one point, then there exists θ1E such that Embedded Image4.23and equation (1.1) is in case I w.r.t. (θ1,K1)∈S by lemmas 2.2 and 3.1, hence, the above argument proves that equation (4.22) holds for Embedded Image with θ0 replaced by θ1. Therefore, we have Embedded Image4.24as Embedded Image. This clearly gives equation (4.4) for Embedded Image since θ1θ0 (mod π).

Now, it remains to prove Embedded Image and Embedded Image for Embedded Image. Let θ1 be defined as in equation (4.23). Then, the similar proof as in equation (4.18) gives that for Embedded Image, Embedded Image4.25where α2 and β2 are defined as in equation (4.11) with θ0 replaced by θ1. Therefore, equations (4.18), (4.25) and (3.10) together yield that Embedded Image and Embedded Image, or Embedded Image and Embedded Image since Embedded Image. This completes the proof of (i) for this theorem.

The proof of (ii). Suppose that τ is in case II w.r.t. (θ0,K0)∈S. In fact, the above proof in equation (4.19) proves that Embedded Image implies that Embedded Image with v=−ieiθ0py′. Since the Hamiltonian system (4.12) is also in the limit-point case at Embedded Image for this case, the similar proof as in equation (4.21) proves that equation (4.3) holds for all Embedded Image.

In addition, if E has more than one point, then equation (4.3) holds for θ=θ1 and Embedded Image, where θ1 is defined as in equation (4.23). The similar argument in equation (4.24) proves equation (4.4). ■

Remark 4.2

If q(x) and p(x) are both real-valued, then Embedded Image if and only if Embedded Image. In this case, case I and cases II and III reduce to Weyl’s limit-point, limit-circle cases, respectively, and it is known that equation (1.1) is in the limit point case at Embedded Image if and only if Embedded Image4.26for Embedded Image. Therefore, equation (4.3) is a generalization of the necessity of equation (4.26).

5. Operator realizations

The J-self-adjoint extension of Sturm–Liouville operators is one of the important research fields in spectral theory of differential operators (see Race (1985), Edmunds & Evans (1987) for details). In this section, we will give the characterization of domains of the J-self-adjoint operators under cases I and II.

A densely defined linear operator T in a Hilbert space H is said to be J-symmetric if JTJT*, and T is said to be J-self-adjoint if JTJ=T* (see Edmunds & Evans (1987, pp. 114–115) for more details).

Let Embedded Image be defined as in equation (4.2), Embedded Image. Let T0(τ), T1(τ) be the operators associated with τ with the domains Embedded Image5.1respectively. We call T0(τ) and T1(τ) the pre-minimal and maximal operators associated to τ. It is known that Embedded Image(and hence Embedded Image) is dense in Embedded Image. This section characterizes the J-self-adjoint extension of T0(τ) associated with equation (1.1), which also is a restriction of the maximal operator T1(τ) on Embedded Image. It is closely related to the famous Glazman–Krein–Naimark theory in this field and has been extensively researched both for formally symmetric differential equations (Everitt & Giertz 1975; Weidmann 1987; Zettl 2005; Wang et al. 2009) and Hamiltonian differential systems (Hinton & Shaw 1984; Krall 1989a,b).

As we know, one can get the J-self-adjoint realizations by imposing boundary conditions for functions in Embedded Image on endpoints of the interval under consideration and the number of independent boundary conditions depends on the cases in the classification of theorem 1.1. Among these cases, J-self-adjoint realizations in case I are comparatively simple since equation (4.3) gives a natural boundary condition of functions in Embedded Image at the endpoint b.

When p(x) is real and Imq(x) is semi-bounded, theorem 10.14 in Edmunds & Evans (1987, p. 150) points out that each J-self-adjoint realization associated with τ has the domain Embedded Image5.2if τ is in case I. This result is an analogue of the corresponding result for the case when q(x) and p(x) are real. When p,q are both complex-valued, with the aid of resolvent operators Brown et al. proves that the set Embedded Image is a J-self-adjoint realization associated with τ if τ is in case I under the condition ReEmbedded Image for some θE (see Brown et al. (1999, theorems 4.4 and 4.5)).

Applying the asymptotic behaviour obtained in theorem 4.1, we will give all J-self-adjoint realizations associated with τ if τ is in case I, in which the restriction ReEmbedded Image is removed. When τ is in case II, we give a class of operator realizations associated with τ not by imposing boundary conditions at the singular endpoint but a restriction of a suitable Sobolev subspace of Embedded Image.

Theorem 5.1

If equation (1.1) is in case I, then each J-self-adjoint realization Tα associated with τ is given by Embedded Image5.3

Proof.

Let Tα be defined as in equation (5.3). We prove that Tα is J-self-adjoint, that is Embedded Image5.4where J is the usual conjugation, i.e. Embedded Image. It is easy to verify that JτJy=τ+y, where τ+ is the so-called Lagrange or formal adjoint of τ defined by Embedded Image

We first prove TαJT*αJ, i.e. Embedded Image5.5For Embedded Image and all Embedded Image, Embedded ImageThis equality together with the boundary condition Embedded Image and Bαy(0)=0 gives Embedded Image5.6It follows from equation (4.3) of theorem 4.1 that Embedded Image, hence Embedded Image and JTαJy0=T*αy0 by equation (5.6). This proves that TαJT*αJ.

Now we prove that T*αJTαJ. Similarly as in equation (5.1), define Embedded Image and Embedded Image associated with τ+. Clearly, Embedded Image5.7

With the similar argument as in Edmunds & Evans (1987, theorem 10.7, pp. 144 and 145) we have that Embedded Image

Let Embedded Image. Since T0(τ)⊂Tα(τ), we have that Embedded Image. So, Embedded Image, or Embedded Image, hence, Embedded Image as Embedded Image for Embedded Image by equation (4.3) of theorem 4.1. This gives that for Embedded Image, Embedded ImageAs y satisfies the boundary condition Bαy(0)=0, we have that Embedded ImageChoose Embedded Image with y(0)≠0, we get Embedded Image, i.e. Embedded Image, which implies Embedded Image, hence JTαJ=T*α.

Conversely, suppose that T is a J-self-adjoint realization associated with τ. Let y0 be fixed such that Embedded Image. Set Embedded Image and Embedded Image. Then the above argument gives that for Embedded Image, Embedded Imagehence TTα, or T*αT*. Since Tα=JT*αJT=JT*J, we have T=Tα. ■

Theorem 5.2

If equation (1.1) is in case II w.r.t. some (θ,K)∈S, then the set Embedded Image5.8is a domain of a J-self-adjoint realization associated with τ. If, in addition, E has more that one point, then Embedded Image5.9

Proof.

Embedded Image for Embedded Image by the first part of (ii) in theorem 4.1. Then the similar argument as in the proof of theorem 5.1 proves that Tα is J-self-adjoint. The second conclusion follows from the second part of (ii) in theorem 4.1 in a similar way. ■

Acknowledgements

This research was partially supported by the NSF of China (grant no. 10801089) and NSF of Shandong Province (grant nos. Y2008A02 and ZR2009AQ010).

  • Received June 1, 2010.
  • Accepted November 18, 2010.

References

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