Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems

Abstract

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale 𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for 𝕋 = ℝ and 𝕋 = ℤ 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) 𝑀(𝜆) 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 [1–12] 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

𝑥^′(𝑡)=𝐴(𝑡)𝑥(𝑡)+(𝐵(𝑡)+𝜆𝑊₂𝑢(𝑡))𝑢(𝑡),_′(𝑡)=(𝐶(𝑡)−𝜆𝑊₁(𝑡))𝑥(𝑡)−𝐴^∗(𝑡)𝑢(𝑡)(1.1) 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 [5–9, 11], Krall [18–20], 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

Δ𝑥(𝑡)=𝐴(𝑡)𝑥(𝑡+1)+(𝐵(𝑡)+𝜆𝑊₂(𝑡))𝑢(𝑡),Δ𝑢(𝑡)=(𝐶(𝑡)−𝜆𝑊₁(𝑡))𝑥(𝑡+1)−𝐴^∗(𝑡)𝑢(𝑡)(1.2) 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 𝕋 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

𝑥^Δ(𝑡)=𝐴(𝑡)𝑥^𝜎(𝑡)+(𝐵(𝑡)+𝜆𝑊₂𝑢(𝑡))𝑢(𝑡),_Δ(𝑡)=(𝐶(𝑡)−𝜆𝑊₁(𝑡))𝑥^𝜎(𝑡)−𝐴^∗(𝑡)𝑢(𝑡),(1.3) where 𝑡 takes values in a time scale 𝕋, 𝜎(𝑡)∶=inf{𝑠∈𝕋∣𝑠>𝑡} is the forward jump operator on 𝕋, 𝑥^𝜎=𝑥∘𝜎, and Δ 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 [5–9, 11, 18–20, 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) 𝑀(𝜆) 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. 𝕋 is a time scale that is unbounded above, that is, 𝕋 is a closed subset of ℝ such that sup𝕋=∞. We let 𝑎∈𝕋 and define 𝕋=𝕋∩[𝑎,∞).

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

𝒥𝑦^Δ(𝑡)∶=(𝜆𝒲(𝑡)+𝒫(𝑡))𝑦(𝑡)for𝑡∈𝕋,(2.1) where

𝑥𝑢,𝑥𝑦=𝑦=^𝜎𝑢,𝒫=−𝐶𝐴^∗𝑊𝐴𝐵,𝒲=¹00𝑊₂,𝒥=0−𝐼_𝑛𝐼_𝑛0(2.2) are subject to the following assumptions.

Assumption 2. 𝐴, 𝐵, 𝐶, 𝑊₁, 𝑊₂ are 𝑛×𝑛 complex-valued matrix functions belonging to 𝐶_rd(𝕋), 𝐴^∗(𝑡) is the complex conjugate transpose of 𝐴(𝑡), and 𝐵(𝑡), 𝐶(𝑡), 𝑊₁(𝑡), 𝑊₂(𝑡) are Hermitian,𝑊₁(𝑡), 𝑊₂(𝑡) are nonnegative definite, and 𝐴(𝑡)∶=𝐼_𝑛−𝜇(𝑡)𝐴(𝑡)isnonsingularon𝕋,(2.3)𝐼_𝑛 is the 𝑛×𝑛 identity matrix, and 𝜇 is the graininess of 𝕋 defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡.

Remark 2.1. If 𝕋=ℝ, then 𝑥^Δ=𝑥^′ and all points in 𝕋 satisfy 𝜎(𝑡)=𝑡, and (1.3) becomes (1.1). If 𝕋=ℤ, then 𝑥^Δ=Δ𝑥 and all points in 𝕋 satisfy 𝜎(𝑡)=𝑡+1, and (1.3) turns into (1.2).

Assumption 3. We always assume that the following definiteness condition holds: for any nontrivial solution 𝑦 of (2.1), we have ^𝑐_𝑎𝑦^∗(𝜏)𝒲(𝜏)𝑦(𝜏)Δ𝜏>0,∀𝑐∈𝕋⧵{𝑎}.(2.4)

Remark 2.2. If 𝕋=ℝ, then the condition (2.4) is just the same as Atkinson's definiteness condition. In the case of 𝕋=ℕ, the condition is the one used in [34, 35].

By a solution of (2.1), we mean an 𝑛×1 vector-valued function 𝑦 satisfying (2.1) on 𝕋.

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

Theorem 2.3 (Existence and Uniqueness Theorem). For arbitrary initial data 𝑡₀∈𝕋,𝑦₀∈ℂ^2𝑛, the initial value problem of (2.1) with 𝑦(𝑡₀)=𝑦₀ has a unique solution on 𝕋.

Proof. By [43, Proposition 1.1], we can rewrite (2.1) as 𝑦^Δ(𝑡)=ℋ(𝑡,𝜆)𝑦(𝑡),𝑡∈𝕋,(2.5) where ℋ(⋅,𝜆)=𝐴𝐴𝐴(𝐵+𝜆𝑊₂)(𝐶−𝜆𝑊₁)𝐴𝜇(𝐶−𝜆𝑊₁)𝐴(𝐵+𝜆𝑊₂)−𝐴^∗(2.6) and ℋ(⋅,𝜆) is symplectic with respect to 𝕋, that is, ℋ^∗(⋅,𝜆)𝒥+𝒥ℋ(⋅,𝜆)+𝜇ℋ^∗(⋅,𝜆)𝒥ℋ(⋅,𝜆)=0on𝕋,(2.7) and hence 𝐼_2𝑛+𝜇ℋ(⋅,𝜆) is symplectic. So 𝐼_2𝑛+𝜇ℋ(⋅,𝜆) 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 𝑦₁,𝑦₂ are solutions of (2.1), then any linear combination of 𝑦₁ and 𝑦₂ is also a solution of (2.1).

Proposition 2.5. There exist 2𝑛 linearly independent solutions 𝑦₁,…,𝑦_2𝑛 of the system (2.1), and every solution 𝑦 of the system (2.1) can be expressed in the form 𝑦=𝑐₁𝑦₁+⋯+𝑐_2𝑛𝑦_2𝑛, where 𝑐₁,…,𝑐_2𝑛∈ℂ are constants.

Every such set of 2𝑛 linearly independent solutions 𝑦₁,𝑦₂,…,𝑦_2𝑛 is called a fundamental solution set. The matrix-valued function 𝑌=(𝑦₁,𝑦₂,…,𝑦_2𝑛) is called a fundamental matrix for the system (2.1).

Corollary 2.6. Let 𝑍 be a fundamental matrix for (2.1). Then every solution of (2.1) can be expressed by 𝑧=𝑍𝑐 for some 𝑐∈ℂ^2𝑛.

Lemma 2.7. Let 𝑌(⋅,𝜆) be a fundamental matrix for the system (2.1). Then 𝑌^∗(𝑡,𝜆)𝒥𝑌(𝑡,𝜆)=𝑌^∗(𝑎,𝜆)𝒥𝑌(𝑎,𝜆),∀𝑡∈𝕋.(2.8)

Proof. From (2.5) and (2.7), we have (𝑌^∗(⋅,𝜆)𝒥𝑌(⋅,𝜆))^Δ=(𝑌^∗)^Δ(⋅,𝜆)𝒥𝑌^𝜎(⋅,𝜆)+𝑌^∗(⋅,𝜆)𝒥𝑌^Δ(⋅,𝜆)=(𝑌^∗)^Δ(⋅,𝜆)𝒥(𝑌(⋅,𝜆)+𝜇𝑌^Δ(⋅,𝜆))+𝑌^∗(⋅,𝜆)𝒥𝑌^Δ(⋅,𝜆)=𝑌^∗(⋅,𝜆)[ℋ^∗(⋅,𝜆)𝒥+𝒥ℋ(⋅,𝜆)+𝜇ℋ^∗(⋅,𝜆)𝒥ℋ(⋅,𝜆)]𝑌(⋅,𝜆)=0(2.9) on 𝕋, and so by [39, Corollary 1.68] there exists a constant matrix 𝐶 with 𝑌^∗(⋅,𝜆)𝒥𝑌(⋅,𝜆)=𝐶 on 𝕋. 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:

ℑ𝜆=𝜆−𝜆2𝑖,ℑ𝑀=𝑀−𝑀^∗.2𝑖(2.10)

3. Eigenvalue Problems

Let 𝑦∈ℂ^2𝑛 be defined on [𝑎,𝑏]⊂𝕋 with 𝑏∈𝕋 and let ℳ,𝒩∈ℂ^2𝑛×2𝑛. We consider the boundary condition

ℳ𝑦(𝑎)+𝒩𝑦(𝑏)=0.(3.1)

Definition 3.1. The boundary condition (3.1) is called formally self adjoint if 𝑦^∗₁𝒥𝑦₂∣^𝑏_𝑎=0,∀𝑦₁,𝑦₂∈ℂ^2𝑛satisfying(3.1).(3.2)

Lemma 3.2. Let ℳ and 𝒩 be 2𝑛×2𝑛 matrices such that rank(ℳ,𝒩)=2𝑛. Then the boundary condition (3.1) is formally self adjoint if and only if ℳ𝒥ℳ^∗=𝒩𝒥𝒩^∗.

Proof. Let 𝑃=(^𝑃1_𝑃2) be any matrix with Im𝑃=Ker(ℳ,𝒩). Then ℳ𝑃₁+𝒩𝑃₂=0, rank𝑃=2𝑛, and so ℳ𝑦(𝑎)+𝒩𝑦(𝑏)=0 if and only if 𝑦(𝑎)=𝑃₁𝑐 and 𝑦(𝑏)=𝑃₂𝑐 for some 𝑐∈ℂ^2𝑛. This yields that the boundary condition (3.1) is formally self adjoint if and only if 𝑦^∗₁𝒥𝑦₂∣^𝑏_𝑎=𝑐^∗₁(𝑃^∗₂𝒥𝑃₂−𝑃^∗₁𝒥𝑃₁)𝑐₂=0,∀𝑐₁,𝑐₂∈ℂ^2𝑛,(3.3) that is, 𝑃^∗₂𝒥𝑃₂=𝑃^∗₁𝒥𝑃₁.(3.4)
First, assume that ℳ𝒥ℳ^∗=𝒩𝒥𝒩^∗. From rank(ℳ,𝒩)=2𝑛, we can conclude that
Im𝒥ℳ^∗−𝒥𝒩^∗=Ker(ℳ,𝒩).(3.5) Hence, the matrix 𝑃 above can be taken to be (^𝒥ℳ∗_{−𝒥𝒩∗}) and then ℳ𝒥ℳ^∗=𝒩𝒥𝒩^∗ yields 𝑃^∗₂𝒥𝑃₂=𝑃^∗₁𝒥𝑃₁, which means that the boundary condition (3.1) is formally self adjoint.
Next, assume that the boundary condition is self adjoint, that is, 𝑃^∗₂𝒥𝑃₂=𝑃^∗₁𝒥𝑃₁. Then 𝑃^∗₁𝒥𝑃₁−𝑃^∗₂𝒥𝑃₂=0, 𝑃^∗₁ℳ^∗+𝑃^∗₂𝒩^∗=0 and rank𝑃=rank(ℳ,𝒩)=2𝑛 imply that
Ker(𝑃^∗₁,𝑃^∗₂ℳ)=Im^∗𝒩^∗=Im𝒥𝑃₁−𝒥𝑃₂.(3.6) Hence ℳ^∗=𝒥𝑃₁𝑆 and 𝒩^∗=−𝒥𝑃₂𝑆 for some invertible matrix 𝑆, and it follows that ℳ𝒥ℳ^∗=𝑆^∗𝑃^∗₁𝒥^∗𝒥𝒥𝑃₁𝑆=𝑆^∗𝑃^∗₂𝒥^∗𝒥𝒥𝑃₂𝑆=𝑁𝒥𝑁^∗,(3.7) which completes the proof.

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

𝛼𝑦(𝑎)=0,𝛽𝑦(𝑏)=0,(3.8) where 𝛼 and 𝛽 are 𝑛×2𝑛 matrices satisfying the self-adjoint conditions

rank𝛼=𝑛,𝛼𝛼^∗=𝐼_𝑛,𝛼𝒥𝛼^∗=0,rank𝛽=𝑛,𝛽𝛽^∗=𝐼_𝑛,𝛽𝒥𝛽^∗=0.(3.9) Since (3.9) can be written as ℳ𝑦(𝑎)=0, 𝒩𝑦(𝑏)=0, where

𝛼00𝛽ℳ=,𝒩=,(3.10) we have ℳ𝒥ℳ^∗=(^𝛼₀)𝒥(𝛼^∗0)=0 and 𝒩𝒥𝒩^∗=(⁰_𝛽)𝒥(0𝛽^∗)=0. Hence, by Lemma 3.2, the boundary condition (3.8) is self adjoint.

Let 𝜃(⋅,𝜆) and 𝜙(⋅,𝜆) be the 2𝑛×𝑛 matrix-valued solutions of (2.1) satisfying

𝜃(𝑎,𝜆)=𝛼^∗,𝜙(𝑎,𝜆)=𝒥𝛼^∗.(3.11)

It is clear that 𝛼𝜃(𝑎,𝜆)=𝐼_𝑛 and 𝛼𝜙(𝑎,𝜆)=0. Set 𝑌=(𝜃𝜙). Then 𝑌(⋅,𝜆) is the fundamental matrix for (2.1) satisfying 𝑌(𝑎,𝜆)=(𝛼^∗𝒥𝛼^∗).

Lemma 3.3. Let 𝑌(⋅,𝜆) be the fundamental matrix for (2.1) satisfying 𝑌(𝑎,𝜆)=(𝛼^∗𝒥𝛼^∗). Then 𝑌^∗(⋅,𝜆)𝒥𝑌(⋅,𝜆)=𝑌(⋅,𝜆)𝒥𝑌^∗(⋅,𝜆)=𝒥on𝕋.(3.12)

Proof. From Lemma 2.7, 𝑌^∗(𝑡,𝜆)𝒥𝑌(𝑡,𝜆)=𝑌^∗(𝑎,𝛼𝜆)𝒥𝑌(𝑎,𝜆)=𝛼𝒥^∗𝒥(𝛼^∗𝒥𝛼^∗)=𝒥,∀𝑡∈𝕋.(3.13) Furthermore, −𝒥𝑌^∗(⋅,𝜆)𝒥𝑌(⋅,𝜆)=𝐼_2𝑛 on 𝕋 implies 𝒥𝑌(⋅,𝜆)(−𝒥𝑌^∗(⋅,𝜆))=𝐼_2𝑛 on 𝕋. It follows that 𝑌(⋅,𝜆)𝒥𝑌^∗(⋅,𝜆)=𝒥 on 𝕋. This completes the proof.

Theorem 3.4. Assume (3.9). Then 𝜆 is an eigenvalue of the problem (2.1), (3.8) if and only if det(𝛽𝜙(𝑏,𝜆))=0, and 𝑦(⋅,𝜆) is a corresponding eigenfunction if and only if there exists a vector 𝜉∈ℂ^𝑛 such that 𝑦(⋅,𝜆)=𝜙(⋅,𝜆)𝜉 on 𝕋, where 𝜉 is a nonzero solution of the equation 𝛽𝜙(𝑏,𝜆)𝜉=0.

Proof. Assume (3.9). Let 𝜆 be an eigenvalue of the eigenvalue problem (2.1), (3.8) with corresponding eigenfunction 𝑦(⋅,𝜆). Then there exists a unique constant vector 𝜂∈ℂ^2𝑛⧵{0} such that 𝑦(𝑡,𝜆)=𝑌(𝑡,𝜆)𝜂,∀𝑡∈𝕋.(3.14) Then, using (3.8) and (3.9), 0=𝛼𝑦(𝑎,𝜆)=𝛼𝑌(𝑎,𝜆)𝜂=𝛼(𝛼^∗𝒥𝛼^∗)𝜂=(𝐼_𝑛𝜁𝜉0)𝜂=𝜁,where𝜂=(3.15) with 𝜁,𝜉∈ℂ^𝑛. Thus 𝑦(𝑡,𝜆)=𝜙(𝑡,𝜆)𝜉, and (3.8) implies that 𝛽𝜙(𝑏,𝜆)𝜉=0. Clearly, 𝜉≠0, since 𝑦(⋅,𝜆)≠0. Hence 𝜉 is a nonzero solution of 𝛽𝜙(𝑏,𝜆)𝜉=0. Thus det(𝛽𝜙(𝑏,𝜆))=0.
Conversely, if 𝜆 satisfies det(𝛽𝜙(𝑏,𝜆))=0, then 𝛽𝜙(𝑏,𝜆)𝜉=0 has a nonzero solution 𝜉. Let 𝑦(⋅,𝜆)=𝜙(⋅,𝜆)𝜉. Then 𝛽𝑦(𝑏,𝜆)=0. Moreover, 𝛼𝑦(𝑎,𝜆)=𝛼𝜙(𝑎,𝜆)𝜉=𝛼𝒥𝛼^∗=0 by (3.9). Taking into account rank𝜙(𝑎,𝜆)=rank(𝒥𝛼^∗)=𝑛, we get that 𝑦(⋅,𝜆) is a nontrivial solution of (2.1). This completes the proof.

Lemma 3.5. Let 𝑦(⋅,𝜆) and 𝑦(⋅,𝜈) be any solutions of (2.1) corresponding to the parameters 𝜆, 𝜈∈ℂ. Then 𝑦^∗(𝑡,𝜈)𝒥𝑦(𝑡,𝜆)∣^𝑏_𝑎=(𝜆−𝜈)^𝑏_𝑎𝑦^∗(𝑡,𝜈)𝒲(𝑡)𝑦(𝑡,𝜆)Δ𝑡.(3.16) In particular, 𝑦^∗(𝑡,𝜆)𝒥𝑦(𝑡,𝜆)∣^𝑏_𝑎=2𝑖ℑ𝜆^𝑏_𝑎𝑦^∗(𝑡,𝜆)𝒲(𝑡)𝑦(𝑡,𝜆)Δ𝑡.(3.17)

Proof. Set (𝑙𝑦)(𝑡,𝜆)=𝒥𝑦^Δ(𝑡,𝜆)−𝒫(𝑡)𝑦(𝑡,𝜆).(3.18) Then from [44, Lemma 2] we have 𝑦^∗(𝑡,𝜈)𝒥𝑦(𝑡,𝜆)∣^𝑏_𝑎=^𝑏_𝑎[𝑦^∗(𝑡,𝜈)(𝑙𝑦)(𝑡,𝜆)−(𝑙𝑦)^∗=(𝑡,𝜈)𝑦(𝑡,𝜆)]Δ𝑡_𝑏𝑎[𝑦^∗(𝑡,𝜈)𝜆𝒲(𝑡)𝑦(𝑡,𝜆)−(𝜈𝒲(𝑡)𝑦(𝑡,𝜈))^∗𝑦(𝑡,𝜆)]Δ𝑡=(𝜆−𝜈)^𝑏_𝑎𝑦^∗(𝑡,𝜈)𝒲(𝑡)𝑦(𝑡,𝜆)Δ𝑡.(3.19) 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 𝜆 be an eigenvalue of (2.1), (3.8) with corresponding eigenfunction 𝑦(⋅,𝜆). Hence 𝑦(⋅,𝜆) satisfies (2.1) and 𝑦(𝑎,𝜆)∈Ker𝛼=Im𝒥𝛼^∗,𝑦(𝑏,𝜆)∈Ker𝛽=Im𝒥𝛽^∗,(3.20) which follows from (3.9) and [10, Corollary 3.1.3]. Thus there exist 𝑐₁,𝑐₂∈ℂ^𝑛 such that 𝑦(𝑎,𝜆)=𝒥𝛼^∗𝑐₁,𝑦(𝑏,𝜆)=𝒥𝛽^∗𝑐₂.(3.21) Using Lemma 3.5 and (3.9), we have 2𝑖ℑ𝜆^𝑏_𝑎𝑦^∗(𝑡,𝜆)𝒲(𝑡)𝑦(𝑡,𝜆)Δ𝑡=𝑦^∗(𝑡,𝜆)𝒥𝑦(𝑡,𝜆)∣^𝑏_𝑎=𝑐^∗₂𝛽𝒥^∗𝒥𝒥𝛽^∗𝑐₂−𝑐^∗₁𝛼𝒥^∗𝒥𝒥𝛼^∗𝑐₁=0(3.22) so that ℑ𝜆=0 and 𝜆∈ℝ. Now let 𝑦(⋅,𝜆) and 𝑦(⋅,𝜈) be eigenfunctions corresponding to the eigenvalues 𝜆≠𝜈. Then, using Lemma 3.5 and proceeding as above, we have (𝜆−𝜈)^𝑏_𝑎𝑦^∗(𝑡,𝜈)𝒲(𝑡)𝑦(𝑡,𝜆)Δ𝑡=𝑦^∗(𝑡,𝜈)𝒥𝑦(𝑡,𝜆)∣^𝑏_𝑎=0(3.23) so that 𝑦(⋅,𝜆) and 𝑦(⋅,𝜈) 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 𝛼,𝛽 and 𝑌(⋅,𝜆) be defined as in Section 3. Suppose 𝜆∈ℂ⧵ℝ and set

𝜒_𝑏𝐼(⋅,𝜆)=𝑌(⋅,𝜆)^𝑛𝑀(𝑏,𝜆),where𝑀(𝑏,𝜆)=𝑀_𝛽(𝑏,𝜆)=−(𝛽𝜙(𝑏,𝜆))⁻¹𝛽𝜃(𝑏,𝜆)(4.1)

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

ℰ(𝑀,𝑏,𝜆)∶=−𝑖sgn(ℑ𝜆)(𝐼_𝑛𝑀^∗)𝑌^∗𝐼(𝑏,𝜆)𝒥𝑌(𝑏,𝜆)^𝑛𝑀,𝐼𝜒(⋅,𝜆)=𝑌(⋅,𝜆)^𝑛𝑀.(4.2) It is clear that

ℰ(𝑀(𝑏,𝜆),𝑏,𝜆)=−𝑖sgn(ℑ𝜆)𝜒^∗_𝑏(𝑏,𝜆)𝒥𝜒_𝑏(𝑏,𝜆).(4.3)

Definition 4.1. Let 𝜆∈ℂ⧵ℝ. The sets 𝔇(𝑏,𝜆)={𝑀∈ℂ^𝑛×𝑛∣ℰ(𝑀,𝑏,𝜆)≤0}and𝒦(𝑏,𝜆)={𝑀∈ℂ^𝑛×𝑛∣ℰ(𝑀,𝑏,𝜆)=0}(4.4) are called a Weyl disk and a Weyl circle, respectively.

Theorem 4.2. Let 𝜆∈ℂ⧵ℝ. Then 𝒦(𝑏,𝜆)={𝑀_𝛽(𝑏,𝜆)∣𝛽satisfies(3.9)}.(4.5)

Proof. Let 𝜆∈ℂ⧵ℝ. Assume that 𝛽 satisfies (3.9). Let 𝜂∈ℂ^2𝑛. Then 𝛽𝜒_𝑏(𝑏,𝜆)𝜂=0 so that (use again (3.9) and [10, Corollary 3.1.3]) 𝜒_𝑏(𝑏,𝜆)𝜂∈Ker𝛽=Im𝒥^∗𝛽,(4.6) and thus there exists 𝑐∈ℂ^𝑛 such that 𝜒_𝑏(𝑏,𝜆)𝜂=𝒥^∗𝛽𝑐. Hence 𝜂^∗𝜒^∗_𝑏(𝑏,𝜆)𝒥𝜒_𝑏(𝑏,𝜆)𝜂=𝑐^∗𝛽^∗𝒥𝒥𝒥^∗𝛽𝑐=𝑐^∗𝛽^∗𝒥𝛽𝑐=0(4.7) by (3.9). So 𝜒^∗_𝑏(𝑏,𝜆)𝒥𝜒_𝑏(𝑏,𝜆)=0, that is, ℰ(𝑀(𝑏,𝜆),𝑏,𝜆)=0.
Conversely, if ℰ(𝑀,𝑏,𝜆)=0, then
0=𝐼_𝑛𝑀^∗𝑌^∗𝐼(𝑏,𝜆)𝒥𝑌(𝑏,𝜆)^𝑛𝑀=𝛾𝒥𝛾^∗,where𝛾=(𝐼_𝑛𝑀^∗)𝑌^∗(𝑏,𝜆)𝒥.(4.8) Then rank𝛾=𝑛 and 𝛾𝜒(𝑏,𝜆)=0. Since 𝛾𝛾^∗=(𝐼_𝑛𝑀^∗)𝑌^∗𝐼(𝑏,𝜆)𝑌(𝑏,𝜆)^𝑛𝑀>0,(4.9) we can define 𝛽=(𝛾𝛾^∗)^−1/2𝛾. Then 𝛽 satisfies (3.9) and 𝛽𝑌(𝑏,𝜆)(^𝐼𝑛_𝑀)=0. It follows that 𝑀=−(𝛽𝜙(𝑏,𝜆))⁻¹𝛽𝜃(𝑏,𝜆)=𝑀(𝑏,𝜆).

Let

ℱ(𝑏,𝜆)∶=−𝑖sgn(ℑ𝜆)𝑌^∗(𝑏,𝜆)𝒥𝑌(𝑏,𝜆).(4.10) Then ℱ(𝑏,𝜆) is a 2𝑛×2𝑛 Hermitian matrix and

ℰ(𝑀,𝑏,𝜆)=(𝐼_𝑛𝑀^∗𝐼)ℱ(𝑏,𝜆)^𝑛𝑀.(4.11)

Lemma 4.3. For 𝜆∈ℂ⧵ℝ and 𝑏≥𝑎, we have ℱ(𝑏,𝜆)=sgn(ℑ𝜆)(−𝑖𝒥+2ℑ𝜆^𝑏_𝑎𝑌^∗(𝑡,𝜆)𝒲(𝑡)𝑌(𝑡,𝜆)Δ𝑡),(4.12)_𝑏𝑎𝜒^∗1(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡=2|ℑ𝜆|ℰ(𝑀,𝑏,𝜆)+ℑ𝑀.ℑ𝜆(4.13)

Proof. From Lemma 3.5, we obtain 𝑌^∗(𝑏,𝜆)𝒥𝑌(𝑏,𝜆)=𝑌^∗(𝑎,𝜆)𝒥𝑌(𝑎,𝜆)+2𝑖ℑ𝜆^𝑏_𝑎𝑌^∗(𝑡,𝜆)𝒲(𝑡)𝑌(𝑡,𝜆)Δ𝑡=𝒥+2𝑖ℑ𝜆^𝑏_𝑎𝑌^∗(𝑡,𝜆)𝒲(𝑡)𝑌(𝑡,𝜆)Δ𝑡,(4.14) and so (4.12) follows from (4.10). From (4.12), we obtain ^𝑏_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡=(𝐼_𝑛𝑀^∗)^𝑏_𝑎𝑌^∗𝐼(𝑡,𝜆)𝒲(𝑡)𝑌(𝑡,𝜆)Δ𝑡^𝑛𝑀=12|ℑ𝜆|(𝐼_𝑛𝑀^∗𝐼)[ℱ(𝑏,𝜆)+𝑖sgn(ℑ𝜆)𝒥]^𝑛𝑀=12|ℑ𝜆|(𝐼_𝑛𝑀^∗𝐼)ℱ(𝑏,𝜆)^𝑛𝑀+ℑ𝑀=1ℑ𝜆2|ℑ𝜆|ℰ(𝑀,𝑏,𝜆)+ℑ𝑀.ℑ𝜆(4.15) This completes the proof.

Theorem 4.4. Let 𝜆∈ℂ⧵ℝ. Then 𝔇(𝑏₂,𝜆)⊂𝔇(𝑏₁,𝜆)forany𝑏₁,𝑏₂∈𝕋with𝑏₁<𝑏₂.(4.16)

Proof. Let 𝜆∈ℂ⧵ℝ and 𝑏₁<𝑏₂. Assume 𝑀∈𝔇(𝑏₂,𝜆). Then ℰ(𝑀,𝑏₂,𝜆)≤0. By Lemma 3.5 and Assumption 2, ℱ(𝑏₂,𝜆)−ℱ(𝑏₁,𝜆)=2|ℑ𝜆|^𝑏2_𝑏1𝑌^∗(𝑡,𝜆)𝒲(𝑡)𝑌(𝑡,𝜆)Δ𝑡>0,(4.17) which implies that ℰ(𝑀,𝑏₂,𝜆)≥ℰ(𝑀,𝑏₁,𝜆). From this, we have ℰ(𝑀,𝑏₁,𝜆)≤0. Thus 𝑀∈𝔇(𝑏₁,𝜆).

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

𝐹ℱ(𝑏,𝜆)=¹¹(𝑏,𝜆)𝐹₁₂𝐹(𝑏,𝜆)_∗12(𝑏,𝜆)𝐹₂₂(𝑏,𝜆),(4.18) where 𝐹₁₁(𝑏,𝜆),𝐹₁₂(𝑏,𝜆), and 𝐹₂₂(𝑏,𝜆) are 𝑛×𝑛 matrices.

Lemma 4.5. For 𝜆∈ℂ⧵ℝ, 𝐹₁₁(𝑏,𝜆) and 𝐹₂₂(𝑏,𝜆) are positive definite and nondecreasing in 𝑏.

Proof. From (4.10), (4.12), and (4.13), we have 𝐹₁₁(𝑏,𝜆)=−𝑖sgn(ℑ𝜆)𝜃^∗(𝑏,𝜆)𝒥𝜃(𝑏,𝜆)=2|ℑ𝜆|^𝑏_𝑎𝜃^∗𝐹(𝑡,𝜆)𝒲(𝑡)𝜃(𝑡,𝜆)Δ𝑡,₂₂(𝑏,𝜆)=−𝑖sgn(ℑ𝜆)𝜙^∗(𝑏,𝜆)𝒥𝜙(𝑏,𝜆)=2|ℑ𝜆|^𝑏_𝑎𝜙^∗(𝑡,𝜆)𝒲(𝑡)𝜙(𝑡,𝜆)Δ𝑡.(4.19) Employing Assumption 2 completes the proof.

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

ℰ(𝑀,𝑏,𝜆)=𝑀^∗𝐹₂₂(𝑏,𝜆)𝑀+𝐹₁₂(𝑏,𝜆)𝑀+𝑀^∗𝐹^∗₁₂(𝑏,𝜆)+𝐹₁₁(𝑏,𝜆)=[𝑀+𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆)]^∗𝐹₂₂(𝑏,𝜆)[𝑀+𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆)]+𝐹₁₁(𝑏,𝜆)−𝐹₁₂(𝑏,𝜆)𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆).(4.20)

Lemma 4.6. For 𝜆∈ℂ⧵ℝ, 𝐹₁₂(𝑏,𝜆)𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆)−𝐹₁₁(𝑏,𝜆)=𝐹⁻¹₂₂(𝑏,𝜆).

Proof. By applying Lemma 3.3 twice, we find ℱ^∗(𝑏,𝜆)𝒥ℱ(𝑏,𝜆)=(−𝑖^∗sgn(ℑ𝜆))(−𝑖sgn(ℑ𝜆))𝑌^∗(𝑏,𝜆)𝒥^∗𝑌(𝑏,𝜆)𝒥𝑌^∗(𝑏,𝜆)𝒥𝑌(𝑏,𝜆)=−𝑌^∗(𝑏,𝜆)𝒥^∗𝒥𝒥𝑌(𝑏,𝜆)=−𝑌^∗(𝑏,𝜆)𝒥𝑌(𝑏,𝜆)=−𝒥.(4.21) Hence 𝐹₁₂(𝑏,𝜆)𝐹₁₂(𝑏,𝜆)−𝐹₁₁(𝑏,𝜆)𝐹₂₂(𝑏,𝜆)=𝐼_𝑛,𝐹₂₂(𝑏,𝜆)𝐹₁₂(𝑏,𝜆)−𝐹^∗₁₂(𝑏,𝜆)𝐹₂₂(𝑏,𝜆)=0.(4.22) From the second relation in (4.22), we have (observe Lemma 4.3) 𝐹₁₂(𝑏,𝜆)𝐹⁻¹₂₂(𝑏,𝜆)=𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆),(4.23) and hence, using also the first relation in (4.22), we obtain 𝐹₁₂(𝑏,𝜆)𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆)−𝐹₁₁(𝑏,𝜆)=𝐹₁₂(𝑏,𝜆)𝐹₁₂(𝑏,𝜆)𝐹⁻¹₂₂(𝑏,𝜆)−𝐹₁₁(𝑏,𝜆)=(𝐼_𝑛+𝐹₁₁(𝑏,𝜆)𝐹₂₂(𝑏,𝜆))𝐹⁻¹₂₂(𝑏,𝜆)−𝐹₁₁(𝑏,𝜆)=𝐹⁻¹₂₂(𝑏,𝜆),(4.24) which completes the proof.

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

ℰ(𝑀,𝑏,𝜆)=(𝑀−ℭ(𝑏,𝜆))^∗ℛ⁻²(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))−ℛ²(𝑏,𝜆),(4.25) where

ℭ(𝑏,𝜆)=−𝐹⁻¹₂₂(𝑏,𝜆)𝐹^∗₁₂(𝑏,𝜆),ℛ(𝑏,𝜆)=𝐹^−1/2₂₂(𝑏,𝜆).(4.26)

Definition 4.7. ℭ(𝑏,𝜆) is called the center of the Weyl disk 𝔇(𝑏,𝜆) or the Weyl circle 𝒦(𝑏,𝜆), while ℛ(𝑏,𝜆) and ℛ(𝑏,𝜆) are called the matrix radii of 𝔇(𝑏,𝜆) or 𝒦(𝑏,𝜆).

Theorem 4.8. Define the unit matrix circle and the unit matrix disk by 𝜕𝐷={𝑈∈ℂ^𝑛×𝑛∣𝑈^∗𝑈=𝐼_𝑛}and𝐷={𝑉∈ℂ^𝑛×𝑛∣𝑉^∗𝑉≤𝐼_𝑛},(4.27) respectively. Then 𝒦(𝑏,𝜆)={ℭ(𝑏,𝜆)+ℛ(𝑏,𝜆)𝑈ℛ(𝑏,𝜆)∣𝑈∈𝜕𝐷},𝔇(𝑏,𝜆)={ℭ(𝑏,𝜆)+ℛ(𝑏,𝜆)𝑉ℛ(𝑏,𝜆)∣𝑉∈𝐷}.(4.28)

Proof. We only prove the first statement as the second one can be shown similarly. From (4.25), ℰ(𝑀,𝑏,𝜆)=0ifandonlyif[ℛ⁻¹(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))ℛ⁻¹(𝑏,𝜆)]^∗[ℛ⁻¹(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))ℛ⁻¹(𝑏,𝜆)]=𝐼_𝑛.(4.29) First, let 𝑀∈𝒦(𝑏,𝜆) and put 𝑈=ℛ⁻¹(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))ℛ⁻¹(𝑏,𝜆). Then 𝑀=ℭ(𝑏,𝜆)+ℛ(𝑏,𝜆)𝑈ℛ(𝑏,𝜆) and (4.29) yields 𝑈^∗𝑈=𝐼_𝑛. Conversely, let 𝑈 be unitary and define 𝑀=ℭ(𝑏,𝜆)+ℛ(𝑏,𝜆)𝑈ℛ(𝑏,𝜆). Then 𝑈=ℛ⁻¹(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))ℛ⁻¹(𝑏,𝜆), so that [ℛ⁻¹(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))ℛ⁻¹(𝑏,𝜆)]^∗[ℛ⁻¹(𝑏,𝜆)(𝑀−ℭ(𝑏,𝜆))ℛ⁻¹(𝑏,𝜆)]=𝐼_𝑛,(4.30) and hence (4.29) yields 𝑀∈𝒦(𝑏,𝜆).

Theorem 4.9. For all 𝜆∈ℂ⧵ℝ, lim_𝑏→∞ℛ(𝑏,𝜆) exists and lim_𝑏→∞ℛ(𝑏,𝜆)≥0.

Proof. From Lemma 4.5, 𝐹₂₂(𝑏,𝜆)>0 is Hermitian and nondecreasing in 𝑏. Thus ℛ(𝑏,𝜆)=𝐹^−1/2₂₂(𝑏,𝜆)>0 is Hermitian and nonincreasing in 𝑏. Hence lim_𝑏→∞ℛ(𝑏,𝜆) exists and is nonnegative definite.

Theorem 4.10. For all 𝜆∈ℂ⧵ℝ, lim_𝑏→∞ℭ(𝑏,𝜆) exists.

Proof. Let 𝑏₁,𝑏₂∈𝕋 with 𝑏₁<𝑏₂. Let 𝑉∈𝐷 and define 𝑀=ℭ(𝑏₂,𝜆)+ℛ(𝑏₂,𝜆)𝑉ℛ(𝑏₂,𝜆).(4.31) By Theorem 4.8, 𝑀∈𝔇(𝑏₂,𝜆). Hence, by Theorem 4.4, 𝑀∈𝔇(𝑏₁,𝜆). Again by Theorem 4.8, there exists Φ(𝑉)∈𝐷 with 𝑀=ℭ(𝑏₁,𝜆)+ℛ(𝑏₁,𝜆)Φ(𝑉)ℛ(𝑏₁,𝜆).(4.32) Thus Φ∶𝐷→𝐷 satisfies Φ(𝑉)=ℛ⁻¹(𝑏₁,𝜆)[ℭ(𝑏₂,𝜆)−ℭ(𝑏₁,𝜆)+ℛ(𝑏₂,𝜆)𝑉ℛ(𝑏₂,𝜆)]ℛ⁻¹(𝑏₁,𝜆)(4.33) for all 𝑉∈𝐷. This implies Φ(𝑉_𝐼)−Φ(𝑉_𝐼𝐼)=ℛ⁻¹(𝑏₁,𝜆)ℛ(𝑏₁,𝜆)[𝑉_𝐼−𝑉_𝐼𝐼]ℛ(𝑏₂,𝜆)ℛ⁻¹(𝑏₂,𝜆)(4.34) for all 𝑉_𝐼,𝑉_𝐼𝐼∈𝐷. Thus Φ∶𝐷→𝐷 is continuous and hence has a fixed point 𝑉∈𝐷 by Brouwer's fixed point theorem. Letting Φ(𝑉)=𝑉 in (4.33), we have ‖ℭ(𝑏₂,𝜆)−ℭ(𝑏₁,𝜆)‖=‖ℛ(𝑏₁,𝜆)𝑉ℛ(𝑏₁,𝜆)−ℛ(𝑏₂,𝜆)𝑉ℛ(𝑏₂,𝜆)‖≤‖ℛ(𝑏₁,𝜆)𝑉ℛ(𝑏₁,𝜆)−ℛ(𝑏₁,𝜆)𝑉ℛ(𝑏₂,𝜆)‖+‖ℛ(𝑏₁,𝜆)𝑉ℛ(𝑏₂,𝜆)−ℛ(𝑏₂,𝜆)𝑉ℛ(𝑏₂,𝜆)‖≤‖ℛ(𝑏₁,𝜆)‖‖ℛ(𝑏₂,𝜆)−ℛ(𝑏₁,𝜆)‖+‖ℛ(𝑏₂,𝜆)−ℛ(𝑏₁,𝜆)‖‖ℛ(𝑏₂,𝜆)‖,(4.35) where ‖⋅‖ is a matrix norm. Using Theorem 4.9 completes the proof.

Definition 4.11. Let 𝜆∈ℂ⧵ℝ and define ℭ₀(𝜆)∶=lim_𝑏→∞ℭ(𝑏,𝜆),ℛ₀(𝜆)∶=lim_𝑏→∞ℛ(𝑏,𝜆).(4.36) Then ℭ₀(𝜆) is called the center, and ℛ₀(𝜆) and ℛ₀(𝜆) are called the matrix radii of the limiting set 𝔇₀(𝜆)∶={ℭ₀(𝜆)+ℛ₀(𝜆)𝑉ℛ₀(𝜆)∣𝑉∈𝐷}.(4.37)

The following result gives another expression for 𝔇₀(𝜆).

Theorem 4.12. The set 𝔇₀(𝜆) is given by 𝔇₀(𝜆)=⋂_𝑏≥𝑎𝔇(𝑏,𝜆).

Proof. If 𝑀∈𝔇₀(𝜆), then there exists 𝑉∈𝐷 such that 𝑀=ℭ₀(𝜆)+ℛ₀(𝜆)𝑉ℛ₀(𝜆). Hence 𝑀=lim_𝑏→∞𝑀(𝑏), where 𝑀(𝑏)=ℭ(𝑏,𝜆)+ℛ(𝑏,𝜆)𝑉ℛ(𝑏,𝜆). Let 𝑏̃≥𝑎. Then 𝑀(𝑏)∈𝔇(𝑏,𝜆)⊂𝔇(𝑏̃,𝜆) for all 𝑏≥𝑏̃ by Theorem 4.4 and thus 𝑀=lim_𝑏→∞𝑀(𝑏)∈𝔇(𝑏̃,𝜆). Therefore 𝑀∈⋂_𝑏≥𝑎𝔇(𝑏,𝜆).
Conversely, if 𝑀∈⋂_𝑏≥𝑎𝔇(𝑏,𝜆), then for all 𝑏≥𝑎, there exists 𝑉_𝑏∈𝐷 such that 𝑀=ℭ(𝑏,𝜆)+ℛ(𝑏,𝜆)𝑉_𝑏𝑅(𝑏,𝜆). Since 𝐷 is compact, there exist a sequence {𝑏_𝑘} and 𝑉∈𝐷 such that 𝑉_𝑏𝑘→𝑉 as 𝑘→∞. Thus 𝑀=ℭ₀(𝜆)+ℛ₀(𝜆)𝑉ℛ₀(𝜆)∈𝔇₀(𝜆).

Theorem 4.13. For all 𝜆∈ℂ⧵ℝ and for 𝑀∈𝔇₀(𝜆), we have ℑ𝜆⋅ℑ𝑀>0.

Proof. Assume that 𝜆∈ℂ⧵ℝ and let 𝑀∈𝔇₀(𝜆). Fix an arbitrary 𝑏>𝑎. From Theorem 4.12, 𝔇₀(𝜆)⊂𝔇(𝑏,𝜆). Hence 𝑀∈𝔇(𝑏,𝜆), and thus ℰ(𝑀,𝑏,𝜆)≤0. Therefore, (4.13) and Assumption 2 yield ℑ𝑀≥ℑ𝜆^𝑏_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡>0.(4.38) The proof is complete.

Definition 4.14. Let 𝑀 be an 𝑛×𝑛 matrix and 𝜆∈ℂ⧵ℝ. We say that (1)𝑀 lies in the limit circle if 𝑀∈𝔇₀(𝜆);(2)𝑀 lies on the boundary of the limit circle if 𝑀∈𝔇₀(𝜆) and there exists a sequence 𝑏_𝑘→∞ as 𝑘→∞ such that lim_𝑘→∞ℰ(𝑀,𝑏_𝑘,𝜆)=0.

Theorem 4.15. Let 𝑀∈ℂ^𝑛×𝑛 and 𝜆∈ℂ⧵ℝ. Then (1)𝑀 lies in the limit circle if and only if ^∞_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡≤ℑ𝑀;ℑ𝜆(4.39)(2)𝑀 lies on the boundary of the limit circle if and only if ^∞_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡=ℑ𝑀.ℑ𝜆(4.40)

Proof. Assume 𝑀∈𝔇₀(𝜆). Let 𝑏>𝑎. Then 𝑀∈𝔇(𝑏,𝜆) by Theorem 4.12. Hence ℰ(𝑀,𝑏,𝜆)≤0. From (4.13), we have that ^𝑏_𝑎𝜒^∗1(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡=2|ℑ𝜆|ℰ(𝑀,𝑏,𝜆)+ℑ𝑀≤ℑ𝜆ℑ𝑀.ℑ𝜆(4.41) Letting 𝑏→∞, we arrive at (4.39). Conversely, assume that (4.39) holds. Let 𝑏≥𝑎. By Assumption 2, ^𝑏_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡≤^∞_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡≤ℑ𝑀.ℑ𝜆(4.42) So ℰ(𝑀,𝑏,𝜆)≤0 by (4.13). This shows that 𝑀∈𝔇(𝑏,𝜆). Using Theorem 4.12 yields 𝑀∈𝔇₀(𝜆). This proves (1), and (2) can be concluded immediately by result (1) and (4.13).

Theorem 4.16. Let 𝜆∈ℂ⧵ℝ. Then 𝑀 lies on the boundary of the limit circle if and only if lim_𝑡→∞𝜒^∗(𝑡,𝜆)𝒥𝜒(𝑡,𝜆)=0.

Proof. From Lemma 3.5, for any 𝑡>𝑎, we have that 𝜒^∗(⋅,𝜆)𝒥𝜒(⋅,𝜆)∣^𝑡_𝑎=2𝑖ℑ𝜆^𝑡_𝑎𝜒^∗(𝑠,𝜆)𝒲(𝑠)𝜒(𝑠,𝜆)Δ𝑠.(4.43) Since 𝜒^∗(𝑎,𝜆)𝒥𝜒(𝑎,𝜆)=𝑀^∗−𝑀=−2𝑖ℑ𝑀,(4.44) we get 𝜒^∗(𝑡,𝜆)𝒥𝜒(𝑡,𝜆)=2𝑖ℑ𝜆^𝑡_𝑎𝜒^∗(𝑠,𝜆)𝒲(𝑠)𝜒(𝑠,𝜆)Δ𝑠−2𝑖ℑ𝑀.(4.45) From Theorem 4.15, 𝑀 is on the boundary of the limit circle if and only if ℑ𝜆⋅^∞_𝑎𝜒^∗(𝑡,𝜆)𝒲(𝑡)𝜒(𝑡,𝜆)Δ𝑡−ℑ𝑀=0.(4.46) So by (4.45), we have that 𝑀 is on the boundary of the limit circle if and only if lim_𝑡→∞𝜒^∗(𝑡,𝜆)𝒥𝜒(𝑡,𝜆)=0.(4.47) 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).