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Sufficient and Necessary Conditions for the Classification of Sturm-Liouville Differential Equations with Complex Coefficients
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.
Consider the Sturm-Liouville differential expression where are both complex valued, is a positive weight function, , and is the so-called spectral parameter. We call a formally symmetric differential expression if are both real valued; otherwise is called formally nonsymmetric. In all cases, we call a formally differential expression or operator.
Let denote the Hilbert space with inner product and the norm for . We call a solution of (1.1) an -solution or square integrable solution if . Set where is the set of complex valued functions that are absolutely continuous on each compact subinterval of . We call the natural (or maximal) domain associated with the formally differential operator .
The aim of the present paper is to study the asymptotic behavior of elements of . 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 . When and 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 -functions, where we say that or (1.1) is in the limit point case at if there exists exactly one -solution (up to constant multiple) for with and is in the limit circle case if all solutions belong to for with . 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 is complex valued . He considered the case where and is semibounded and classified (1.1) into three cases. Recently, this work has been extensively generalized by Brown et al.  under mild assumptions on weighted function and the complex valued coefficients , . They proved that there exist three distinct possible cases for (1.1).
For formally symmetric , it is well known (see [13, 14]) that (1.1) is in the limit point case at if and only if for . 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 at the end point . In this case every self-adjoint extension associated with the differential expression needs not a boundary condition at . 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 , 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  and well studied in . For real valued functions and , we say that (1.1) is in the strong limit point case at the end point if, for ,
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 and . In the classification of Brown et al. in , 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 also depends on the admissible rotation angles. So we first study the properties of the admissible angle set (defined in (2.10)) and prove that 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 ). 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 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 . The main results are given in Section 4.
2. Preliminary Knowledge
Throughout this paper, we always assume that(i), a.e. on and are all locally integrable on ,(ii) and are complex valued, and where denotes the closed convex hull (i.e., the smallest closed convex set containing the exhibited set). Then, for each point on the boundary , there exists a line through this point such that every point of either lies in the same side of this line or is on it. That is, there exists a supporting line through this point. Let be a point on . Denote by an arbitrary supporting line touching at , which may be the tangent to at if it exists. We then perform a transformation of the complex plane and a rotation through an appropriate angle so that the image of coincides with the imaginary axis now and the set is contained in the new right nonnegative half-plane.
For this purpose we introduce the set defined by where is the interior of , and define the corresponding half-plane Then, . From the definition of , for all and ,
The definition of is different from the corresponding one given by Brown et al. , but they are equivalent in describing square integrable solutions.
Using a nesting circle method based on that of both Weyl  and Sims, Brown et al.  divided (1.1) into three cases with respect to the corresponding half-planes 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 , the following three distinct cases are possible. Case I. For all , equation (1.1) has unique solution satisfying and this is the only solution satisfying .Case II. For all , all solutions of (1.1) belong to , and there exists unique solution of (1.1) satisfying (2.6).Case III. For all , all solutions of (1.1) satisfy (2.6).
Lemma 2.2. For every and , there exists such that on .
Using variation of parameters method, we can verify that, if all solutions of (1.1) belong to for some , then it is true for all . This also means the following.
This indicates that Case I is independent of the choice of . But Cases II and III depend on the choice of in general, that is, there may exist such that (1.1) is in Case II with respect to and is in Case III with respect to . In order to make clear the dependence, we introduce the admissible angle set defined by
Remark 2.4. For given , there exist many such that . In fact, if with for some , then for all ,
Theorem 2.5 (cf. [23, Theorem 2.1]). If there exists a such that (1.1) is in Case II with respect to , then (1.1) is in Case II with respect to for all except for at most one such that (1.1) is in Case III with respect to .
3. Properties of the Angel Set
This section gives some properties of the set , which will be used in the proof of our main results in Section 4. In what follows, we say that has more than one point if there exist with (.
Lemma 3.1. Let be defined as in (2.10). (i)The set is connected in the sense of . (ii)If has more than one point, then, for every , there exist with such that for .
Proof. (i) Suppose that has more than one point. Let with (; then ( or (.
If () and , , then we claim that (). Let be the line similarly defined as with and replaced by and , . That is, Let be the intersection point of and . Set It follows from (2.5) that
By () and (3.3), we can get for () on , which means and .
According to the similar method, we can verify that, if () and , , then (), that is, ().
(ii) For , choose and such that and Since has more than one point, we can choose with (). Without loss of generality, we suppose that (). Let be defined as in the proof of (i).
If , then it follows from (2.3) that By () and (3.5), we can get for (), which means for .
Suppose that . Let be the unique point such that . Let ; then by , and by the definition of (see Figure 1).
So, we can get that for by , 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 defined on the interval 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 , where . Recall that (1.1) on is said to be regular at if , and are integrable on for some (and hence any) and singular at otherwise; the regularity and singularity at are defined similarly (cf. ). 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 in (1.3). We also define where The first result of this section is as follows.
Remark 4.2. Clearly , by the definition of . It is easy to see that (4.4) is equivalent to for .
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 where are valued functions, is the transpose of , , and are locally integrable, complex valued matrices on , , and are Hermit matrices and on , and is the spectral parameter. Assume that the definiteness condition (see, e.g., [13, Chapter 9, page 253]) holds: where . Let denote the space of Lebesgue measurable -dimensional functions satisfying . We say that (4.6) is in the limit point case at infinity if there exists exactly 's solutions of (4.6) belonging to for with .
Let be the maximal domain associated with (4.6), that is, if and only if , and there exists an element such that It is well known (cf. [5, 7]) that (4.6) is in the limit point case at infinity if and only if for , and for every with there exists a Green function such that, for , where .
Let and choose . Then from (2.9), one sees that for some . Set Consider the Hamiltonian differential system (4.6) with , and that is, the 2-dimensional Hamiltonian differential system It follows from (4.11) that and it is easy to verify that the definiteness condition holds for the system (4.14). In fact, is a solution of (1.1) if and only if is a solution of (4.14) with This fact immediately yields the following result which is frequently used in the proof of Theorems 4.1 and 4.11.
Lemma 4.4. If has more than one point, then on , the interior of , where
Proof. Let be fixed. There exist such that by Lemma 3.1. Choose . Letting be defined as in (4.12) and solving from the equations we have that with by (4.18). Since for , we have that and hence for . The same proof as the above with replaced by also proves for , where is defined as in (4.12). Therefore, for , Set and . It follows from and (4.15) that Then (4.24) and (4.22) yield that, for , Note that . Then (4.25) gives , or . Clearly, . Thus .
Lemma 4.4 indicates the following.
Proof. If is in Case III with respect to some with , then is restricted in the solution space of (1.1) by the definition of Case III. Since by Lemma 4.4, we have that restricted in the solution space of (1.1). This means that all solutions of (1.1) with satisfy Using variation of parameters method we can prove that it is true for all , and hence is in Case III with respect to , a contradiction.
Lemma 4.6. If is in Case I and , then with , where is the maximal domain associated with (4.14).
Proof. Suppose that is in Case I with respect to . We claim that . Set
for and .
Set , . Then satisfies Conversely, if satisfies (4.28), then solves (4.27). Note that , or , and implies .
Considering (4.28), we get from (4.10) that (4.28) has a solution such that and . Set . Then satisfies (4.27), and hence . Note that , and implies that . Thus, is an -solution of . Since is in Case I with respect to , it follows from (2.6) that and . This together with and gives and . In fact, we have proved that, for , or where and are defined in (4.12) or (4.13). Since , (4.30) means that or , and hence . Recall that . Then (4.30) and (4.28) imply that, if , then .
Corollary 4.7. If is in Case I and , then with .
Proof. For , by . So with by Lemma 4.6. Clearly since is symmetrical. Then we have that with .
Proof of Theorem 4.1. The proof of (i): suppose that 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 . For , since with , , by Lemma 4.6, it follows from (4.9) that
Conversely, assume that (4.3) holds for all elements of . 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 for . Choose , and let be a nontrivial solution of (1.1) satisfying . Then by . Furthermore, it follows from that Integrating (4.33) on we have that Multiplying and to the first and second equalities in (4.34), respectively, and adding them together, we have that since , where and . Note that as by assumption (4.3) and by (4.15). Then letting in (4.35), we have a contradiction. This proves the first part of this theorem.
The proof of (ii): suppose that is in Case I. Set , for , . Then, we can get by Lemma 4.6 and by Corollary 4.7. Hence as by (4.9), that is, as .
Conversely, if is not in Case I, then all solutions of (1.1) belong to for . Let , be the solution of such that Since . Then the Wronskian which contradicts condition (4.5). See Remark 4.2.
Remark 4.8. If and are real valued, then and with . 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 for , that is, (1.4). Clearly, if is real valued and , then (4.3) reduces to (1.4). Therefore, (4.3) is a generalization of (1.4).
Corollary 4.9. If has more than one point, then is in Case I if and only if, for , That is is in the strong limit point case at .
Proof. Suppose that has more than one point and is in Case I. Choose , , with (). Then (4.3) holds for , . This gives that for
and hence (4.42) holds since ().
Conversely, assume that (4.42) holds for all , . Since (4.42) implies (4.3), we conclude from (i) of Theorem 4.1 that is in Case I.
Corollary 4.10. If is symmetric and on , then is in the limit point case at if and only if it is in the strong limit point case at .
Proof. Suppose that is in Case II with respect to some . By the definition of Case II we know that is nonempty and . 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 by Lemma 4.3 and (4.32).
Conversely, suppose that for some and (4.3) holds for . By the proof of Lemma 4.6, we know that is not in Case I with respect to . We only need to prove that is not in Case III with respect to this . If it is not true, then all solutions of (1.1) with satisfy (2.6) and so belong to . Let be a nontrivial solution of (1.1) with . Then , and hence the same proof as in (4.33)–(4.35) yields a contradiction.
Proof. If has more than one point and is in Case II with respect to some , then there exists such that is in Case II with respect to by Theorem 2.5. Since by Theorem 4.11 and by Lemma 4.4, one sees that .
Choose with () such that is in Case II with respect to for by Theorem 2.5. Then (4.3) holds for , by Theorem 4.11. Since , the same proof as in (4.43) gives that (4.42) holds for .
Conversely, suppose that and (4.42) holds for . Since on by Lemma 4.4, we conclude that on and (4.42) holds for . So (4.3) holds for by (4.42). Then, we have that is in Case II with respect to with by Theorem 4.11.
This work was supported by the NSF of Shandong Province (Grant Y2008A02) and the IIFSDU (Grant 2010ZRJQ002).
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