#### Abstract

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale , which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for and within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

#### 1. Introduction

##### 1.1. Differential Equations

The study of spectral problems for differential operators has played an important role not only in theoretical but also in practical aspects. The study of spectral theory of differential equations has a long history. For this, we refer to [112] and references therein along this line.

Spectral problems of differential operators fall into two classifications. First, those defined over finite intervals with well-behaved coefficients are called regular. Fine spectral properties can be expected. For example, the spectral set is discrete, infinite, and unbounded, and the eigenfunction basis is complete for a corresponding space.

Spectral problems that are not regular are called singular. These are considerably more difficult to discuss because the spectral set can be much more complicated and, as a result, have only been examined closely during the last century. The Weyl-Titchmarsh theory is an important milestone in the study of spectral problems for linear ordinary differential equations [13]. It has started with the celebrated work by H. Weyl in 1910 [14]. He gave a dichotomy of the limit-point and limit-circle cases for singular spectral problems of second-order formally self-adjoint linear differential equations. He was followed by Titchmarsh [12] and many others. From 1910 until 1945, these mathematicians developed and polished the theory of self-adjoint differential operators of the second order to a high degree. Their work was continued by Coddington and Levinson [3], and so forth. in the late 1940s and 1950s. Not only were additional results found for operators of the second order, but operators of higher orders were also examined. At the same time, the Russian school, led by Kreĭn, Naĭmark, Akhiezer, and Glazman, also made major contributions. For a far more comprehensive survey of this work, we recommend the second volume of Dunford and Schwartz [4], where numerous contributions made by many mathematicians are summarized. Further study continued in the 1960s and 1970s with the work of Atkinson [1] on regular Hamiltonian systems

and Everitt and Kumar [15, 16] on higher-order scalar problems. The work for this period is summarized by Atkinson [1], and Everitt and Kumar [15, 16]. Again, there were many other contributors. One contribution, perhaps, deserves special mention. Walker [17] showed that every scalar self-adjoint problem of an arbitrary order can be reformulated as an equivalent self-adjoint Hamiltonian system. This removed the need to discuss scalar problems and systems separately.

In the 1980s and 1990s, Hinton and Shaw [59, 11], Krall [1820], and Remling [21] have made great progress by considering singular spectral problems in the Hamiltonian system format, following the lead of Atkinson [1]. In the 2000s, Brown and Evans [22], Clark and Gesztesy [23], Qi and Chen [24], Qi [25], Remling [21], Shi [26], Sun et al. [27], Zheng and Chen [28] have made progress by considering spectral problems for Hamiltonian differential systems.

##### 1.2. Difference Equations

Spectral problems of discrete linear Hamiltonian systems

are also divided into two groups: regular and singular problems. Singular spectral problems of second-order self-adjoint scalar difference equations over infinite intervals were first studied by Atkinson [1]. His work was followed by Agarwal et al. [29], Bohner [30], Bohner et al. [31], Clark and Gesztesy [32], Shi [33, 34], and Sun et al. [35]. In [1], Atkinson first studied the Weyl-Titchmarsh theory and the spectral theory for the system (1.2). Following him, Hinton and Shaw have made great progress by considering Weyl-Titchmarsh theory and spectral theory for the system (1.2). Shi studied Weyl-Titchmarsh theory and spectral theory for the system (1.2) in [33, 34]; Clark and Gesztesy established the Weyl-Titchmarsh theory for a class of discrete Hamiltonian systems that include system (1.2) [23]. Sun et al. established the GKN-theory for the system (1.2) [35].

##### 1.3. Dynamic Equations

A time scale is an arbitrary nonempty closed subset of the real numbers. The theory of time scales was introduced by Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis [36]. Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [37, 38] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [39], summarizes and organizes much of the time scale calculus. We refer also to the book by Bohner and Peterson [40] for advances in dynamic equations on time scales and to the book by Lakshmikantham et al. [41].

This paper is devoted to the Weyl-Titchmarsh theory for linear Hamiltonian dynamic systems

where takes values in a time scale , is the forward jump operator on , , and denotes the Hilger derivative. A universal method we provided here allows one to treat both continuous and discrete linear Hamiltonian systems as special cases within one theory and to explain the discrepancies between them. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems on time scales. Some ideas in this paper are motivated by some works in [59, 11, 1820, 34, 35, 42].

The paper is organized as follows. Some fundamental theory for Hamiltonian systems is given in Section 2. Some regular spectral problems are considered in Section 3. The Weyl matrix disks are constructed and their properties are studied in Section 4. These matrix disks are nested and converge to a limiting set of the matrix circle. The results are some generalizations of the Weyl-Titchmarsh theory for both Hamiltonian differential systems [6, 9, 18, 20, 26] and discrete Hamiltonian systems [34]. These investigations are part of a larger program which includes the following: (i) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

#### 2. Assumptions and Preliminary Results

Throughout we use the following assumption.

Assumption 1. is a time scale that is unbounded above, that is, is a closed subset of such that . We let and define .

In this section, we shall study the fundamental theory and properties of solutions for the Hamiltonian dynamic system (1.3), that is,

where

are subject to the following assumptions.

Assumption 2. , , , , are complex-valued matrix functions belonging to , is the complex conjugate transpose of , and , , , are Hermitian,, are nonnegative definite, and is the identity matrix, and is the graininess of defined by .

Remark 2.1. If , then and all points in satisfy , and (1.3) becomes (1.1). If , then and all points in satisfy , and (1.3) turns into (1.2).

Assumption 3. We always assume that the following definiteness condition holds: for any nontrivial solution of (2.1), we have

Remark 2.2. If , then the condition (2.4) is just the same as Atkinson's definiteness condition. In the case of , the condition is the one used in [34, 35].

By a solution of (2.1), we mean an vector-valued function satisfying (2.1) on .

Now we consider the existence of solutions to (2.1).

Theorem 2.3 (Existence and Uniqueness Theorem). For arbitrary initial data , the initial value problem of (2.1) with has a unique solution on .

Proof. By [43, Proposition 1.1], we can rewrite (2.1) as where and is symplectic with respect to , that is, and hence is symplectic. So is invertible and thus (2.1) has a unique solution by [39, Theorem 5.24]. This completes the proof.

Now we consider the structure of solutions for the system (2.1).

Proposition 2.4. If are solutions of (2.1), then any linear combination of and is also a solution of (2.1).

Proposition 2.5. There exist linearly independent solutions of the system (2.1), and every solution of the system (2.1) can be expressed in the form , where are constants.

Every such set of linearly independent solutions is called a fundamental solution set. The matrix-valued function is called a fundamental matrix for the system (2.1).

Corollary 2.6. Let be a fundamental matrix for (2.1). Then every solution of (2.1) can be expressed by for some .

Lemma 2.7. Let be a fundamental matrix for the system (2.1). Then

Proof. From (2.5) and (2.7), we have on , and so by [39, Corollary 1.68] there exists a constant matrix with on . This completes the proof.

In the rest of the paper, we use the following notation for the imaginary part of a complex number or matrix:

#### 3. Eigenvalue Problems

Let be defined on with and let . We consider the boundary condition

Definition 3.1. The boundary condition (3.1) is called formally self adjoint if

Lemma 3.2. Let and be matrices such that . Then the boundary condition (3.1) is formally self adjoint if and only if .

Proof. Let be any matrix with . Then , , and so if and only if and for some . This yields that the boundary condition (3.1) is formally self adjoint if and only if that is,
First, assume that . From , we can conclude that
Hence, the matrix above can be taken to be and then yields , which means that the boundary condition (3.1) is formally self adjoint.
Next, assume that the boundary condition is self adjoint, that is, . Then , and imply that
Hence and for some invertible matrix , and it follows that which completes the proof.

Now we consider the system (2.1) with the formally self-adjoint boundary conditions

where and are matrices satisfying the self-adjoint conditions

Since (3.9) can be written as , , where

we have and . Hence, by Lemma 3.2, the boundary condition (3.8) is self adjoint.

Let and be the matrix-valued solutions of (2.1) satisfying

It is clear that and . Set . Then is the fundamental matrix for (2.1) satisfying

Lemma 3.3. Let be the fundamental matrix for (2.1) satisfying . Then

Proof. From Lemma 2.7, Furthermore, on implies on . It follows that on . This completes the proof.

Theorem 3.4. Assume (3.9). Then is an eigenvalue of the problem (2.1), (3.8) if and only if , and is a corresponding eigenfunction if and only if there exists a vector such that on , where is a nonzero solution of the equation .

Proof. Assume (3.9). Let be an eigenvalue of the eigenvalue problem (2.1), (3.8) with corresponding eigenfunction . Then there exists a unique constant vector such that Then, using (3.8) and (3.9), with . Thus , and (3.8) implies that . Clearly, , since . Hence is a nonzero solution of Thus
Conversely, if satisfies , then has a nonzero solution . Let . Then . Moreover, by (3.9). Taking into account , we get that is a nontrivial solution of (2.1). This completes the proof.

Lemma 3.5. Let and be any solutions of (2.1) corresponding to the parameters , . Then In particular,

Proof. Set Then from [44, Lemma 2] we have This completes the proof.

Theorem 3.6. Assume (3.9). Then all eigenvalues of (2.1), (3.8) are real, and eigenvectors corresponding to different eigenvalues are orthogonal.

Proof. Assume (3.9) and let be an eigenvalue of (2.1), (3.8) with corresponding eigenfunction . Hence satisfies (2.1) and which follows from (3.9) and [10, Corollary 3.1.3]. Thus there exist such that Using Lemma 3.5 and (3.9), we have so that and . Now let and be eigenfunctions corresponding to the eigenvalues . Then, using Lemma 3.5 and proceeding as above, we have so that and are orthogonal.

#### 4. Weyl-Titchmarsh Circles and Disks

In this section, we consider the construction of Weyl-Titchmarsh disks and circles for Hamiltonian dynamic systems (2.1). Assume (3.9) and let and be defined as in Section 3. Suppose and set

(observe Theorems 3.4 and 3.6). For any matrix , define

It is clear that

Definition 4.1. Let . The sets are called a Weyl disk and a Weyl circle, respectively.

Theorem 4.2. Let . Then

Proof. Let . Assume that satisfies (3.9). Let . Then so that (use again (3.9) and [10, Corollary 3.1.3]) and thus there exists such that . Hence by (3.9). So , that is, .
Conversely, if , then
Then and . Since we can define . Then satisfies (3.9) and . It follows that .

Let

Then is a Hermitian matrix and

Lemma 4.3. For and , we have

Proof. From Lemma 3.5, we obtain and so (4.12) follows from (4.10). From (4.12), we obtain This completes the proof.

Theorem 4.4. Let . Then

Proof. Let and . Assume . Then . By Lemma 3.5 and Assumption 2, which implies that . From this, we have . Thus .

Now we study convergence of the disks. For this purpose, we denote

where , and are matrices.

Lemma 4.5. For , and are positive definite and nondecreasing in .

Proof. From (4.10), (4.12), and (4.13), we have Employing Assumption 2 completes the proof.

Using the notation of (4.13), we find that (4.11) can be rewritten as

Lemma 4.6. For , .

Proof. By applying Lemma 3.3 twice, we find Hence From the second relation in (4.22), we have (observe Lemma 4.3) and hence, using also the first relation in (4.22), we obtain which completes the proof.

From Lemma 4.6, (4.11), and hence (4.20), can be rewritten in the form

where

Definition 4.7. is called the center of the Weyl disk or the Weyl circle , while and are called the matrix radii of or .

Theorem 4.8. Define the unit matrix circle and the unit matrix disk by respectively. Then

Proof. We only prove the first statement as the second one can be shown similarly. From (4.25), First, let and put . Then and (4.29) yields . Conversely, let be unitary and define . Then , so that and hence (4.29) yields .

Theorem 4.9. For all , exists and

Proof. From Lemma 4.5, is Hermitian and nondecreasing in . Thus is Hermitian and nonincreasing in . Hence exists and is nonnegative definite.

Theorem 4.10. For all , exists.

Proof. Let with . Let and define By Theorem 4.8, . Hence, by Theorem 4.4, . Again by Theorem 4.8, there exists with Thus satisfies for all . This implies for all . Thus is continuous and hence has a fixed point by Brouwer's fixed point theorem. Letting in (4.33), we have where is a matrix norm. Using Theorem 4.9 completes the proof.

Definition 4.11. Let and define Then is called the center, and and are called the matrix radii of the limiting set

The following result gives another expression for .

Theorem 4.12. The set is given by .

Proof. If , then there exists such that . Hence , where . Let . Then for all by Theorem 4.4 and thus . Therefore .
Conversely, if , then for all , there exists such that . Since is compact, there exist a sequence and such that as . Thus .

Theorem 4.13. For all and for , we have .

Proof. Assume that and let . Fix an arbitrary . From Theorem 4.12, . Hence , and thus . Therefore, (4.13) and Assumption 2 yield The proof is complete.

Definition 4.14. Let be an matrix and . We say that () lies in the limit circle if ;() lies on the boundary of the limit circle if and there exists a sequence as such that .

Theorem 4.15. Let and . Then () lies in the limit circle if and only if () lies on the boundary of the limit circle if and only if

Proof. Assume . Let . Then by Theorem 4.12. Hence . From (4.13), we have that Letting , we arrive at (4.39). Conversely, assume that (4.39) holds. Let . By Assumption 2, So by (4.13). This shows that . Using Theorem 4.12 yields . This proves (), and () can be concluded immediately by result () and (4.13).

Theorem 4.16. Let . Then lies on the boundary of the limit circle if and only if .

Proof. From Lemma 3.5, for any , we have that Since we get From Theorem 4.15, is on the boundary of the limit circle if and only if So by (4.45), we have that is on the boundary of the limit circle if and only if This completes the proof.

#### Acknowledgments

This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Foundation (200802018), the Natural Science Foundation of Shandong (Y2008A28), and the Fund of Doctoral Program Research of the University of Jinan (B0621, XBS0843).