# Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

Authors: Roman Šimon Hilscher; Petr Zemánek
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce theM(λ)-function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.


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## Body

## 1. Introduction

In this paper we develop systematically the Weyl-Titchmarsh theory for time scale symplectic systems. Such systems unify and extend the classical linear Hamiltonian differential systems and discrete symplectic and Hamiltonian systems, including the Sturm-Liouville differential and difference equations of arbitrary even order. As the research in the Weyl-Titchmarsh theory has been very active in the last years, we contribute to this development by presenting a theory which directly generalizes and unifies the results in several recent papers, such as [1–4] and partly in [5–14].Historically, the theory nowadays called by Weyl and Titchmarsh started in [15] by the investigation of the second-order linear differential equation
(1.1)(r(t)z′(t))+q(t)z(t)=λz(t),t∈[0,∞),
where r,q:[0,∞)→ℝ are continuous, r(t)>0, and λ∈ℂ, is a spectral parameter. By using a geometrical approach it was showed that (1.1) can be divided into two classes called the limit circle and limit point meaning that either all solutions of (1.1) are square integrable for all λ∈ℂ∖ℝ or there is a unique (up to a multiplicative constant) square-integrable solution of (1.1) on [0,∞). Analytic methods for the investigation of (1.1) have been introduced in a series of papers starting with [16]; see also [17]. We refer to [18–20] for an overview of the original contributions to the Weyl-Titchmarsh theory for (1.1); see also [21]. Extensions of the Weyl-Titchmarsh theory to more general equations, namely, to the linear Hamiltonian differential systems (1.2)z′(t)=[λA(t)+B(t)]z(t),t∈[0,∞),
was initiated in [22] and developed further in [6, 8, 10, 11, 23–38].According to [19], the first paper dealing with the parallel discrete time Weyl theory for second-order difference equations appears to be the work mentioned in [39]. Since then a long time elapsed until the theory of difference equations attracted more attention. The Weyl-Titchmarsh theory for the second-order Sturm-Liouville difference equations was developed in [22, 40, 41]; see also the references in [19]. For higher-order Sturm-Liouville difference equations and linear Hamiltonian difference systems, such as
(1.3)Δxk=Akxk+1+(Bk+λWk[2])uk,Δuk=(Ck-λWk[1])xk+1-Ak*uk,k∈[0,∞)Z,
where Ak, Bk, Ck, Wk[1], Wk[2] are complex n×n matrices such that Bk and Ck are Hermitian and Wk[1] and Wk[2] are Hermitian and nonnegative definite, the Weyl-Titchmarsh theory was studied in [9, 14, 42]. Recently, the results for linear Hamiltonian difference systems were generalized in [1, 2] to discrete symplectic systems (1.4)xk+1=Akxk+Bkuk,uk+1=Ckxk+Dkuk+λWkxk+1,k∈[0,∞)Z,
where 𝒜k, ℬk, 𝒞k, 𝒟k, 𝒲k are complex n×n matrices such that 𝒲k is Hermitian and nonnegative definite and the 2n×2n transition matrix in (1.4) is symplectic, that is, (1.5)Sk:=(AkBkCkDk),Sk*JSk=J,J:=(0I-I0).In the unifying theory for differential and difference equations—the theory of time scales—the classification of second-order Sturm-Liouville dynamic equations(1.6)yΔΔ(t)+q(t)yσ(t)=λyσ(t),t∈[a,∞)T,
to be of the limit point or limit circle type is given in [4, 43]. These two papers seem to be the only ones on time scales which are devoted to the Weyl-Titchmarsh theory for the second order dynamic equations. Another way of generalizing the Weyl-Titchmarsh theory for continuous and discrete Hamiltonian systems was presented in [3, 5]. In these references the authors consider the linear Hamiltonian system (1.7)xΔ(t)=A(t)xσ(t)+[B(t)+λW2(t)]u(t),uΔ(t)=[C(t)-λW1(t)]xσ(t)-A*(t)u(t),t∈[a,∞)T,
on the so-called Sturmian or general time scales, respectively. Here fΔ(t) is the time scale Δ-derivative and fσ(t):=f(σ(t)), where σ(t) is the forward jump at t; see the time scale notation in Section 2.In the present paper we develop the Weyl-Titchmarsh theory for more general linear dynamic systems, namely, the time scale symplectic systems(Sλ)xΔ(t)=A(t)x(t)+B(t)u(t),uΔ(t)=C(t)x(t)+D(t)u(t)-λW(t)xσ(t),t∈[a,∞)T,
where 𝒜, ℬ, 𝒞, 𝒟, 𝒲 are complex n×n matrix functions on [a,∞)𝕋, 𝒲(t) is Hermitian and nonnegative definite, λ∈ℂ, and the 2n×2n coefficient matrix in system (𝒮λ) satisfies
(1.8)S(t):=(A(t)B(t)C(t)D(t)),S*(t)J+JS(t)+μ(t)S*(t)JS(t)=0,t∈[a,∞)T,
where μ(t):=σ(t)-t is the graininess of the time scale. The spectral parameter λ is only in the second equation of system (𝒮λ). This system was introduced in [44], and it naturally unifies the previously mentioned continuous, discrete, and time scale linear Hamiltonian systems (having the spectral parameter in the second equation only) and discrete symplectic systems into one framework. Our main results are the properties of the M(λ) function, the geometric description of the Weyl disks, and characterizations of the limit point and limit circle cases for the time scale symplectic system (𝒮λ). In addition, we give a formula for the L𝒲2 solutions of a nonhomogeneous time scale symplectic system in terms of its Green function. These results generalize and unify in particular all the results in [1–4] and some results from [5–14]. The theory of time scale symplectic systems or Hamiltonian systems is a topic with active research in recent years; see, for example, [44–51]. This paper can be regarded not only as a completion of these papers by establishing the Weyl-Titchmarsh theory for time scale symplectic systems but also as a comparison of the corresponding continuous and discrete time results. The references to particular statements in the literature are displayed throughout the text. Many results of this paper are new even for (1.6), being a special case of system (𝒮λ). An overview of these new results for (1.6) will be presented in our subsequent work.This paper is organized as follows. In the next section we recall some basic notions from the theory of time scales and linear algebra. In Section3 we present fundamental properties of time scale symplectic systems with complex coefficients, including the important Lagrange identity (Theorem 3.5) and other formulas involving their solutions. In Section 4 we define the time scale M(λ)-function for system (𝒮λ) and establish its basic properties in the case of the regular spectral problem. In Section 5 we introduce the Weyl disks and circles for system (𝒮λ) and describe their geometric structure in terms of contractive matrices in ℂn×n. The properties of the limiting Weyl disk and Weyl circle are then studied in Section 6, where we also prove that system (𝒮λ) has at least n linearly independent solutions in the space L𝒲2 (see Theorem 6.7). In Section 7 we define the system (𝒮λ) to be in the limit point and limit circle case and prove several characterizations of these properties. In the final section we consider the system (𝒮λ) with a nonhomogeneous term. We construct its Green function, discuss its properties, and characterize the L𝒲2 solutions of this nonhomogeneous system in terms of the Green function (Theorem 8.5). A certain uniqueness result is also proven for the limit point case.


## 2. Time Scales

Following [52, 53], a time scale 𝕋 is any nonempty and closed subset of ℝ. A bounded time scale can be therefore identified as [a,b]𝕋:=[a,b]∩𝕋 which we call the time scale interval, where a:=min𝕋 and b:=max𝕋. Similarly, a time scale which is unbounded above has the form [a,∞)𝕋:=[a,∞)∩𝕋. The forward and backward jump operators on a time scale are denoted by σ(t) and ρ(t) and the graininess function by μ(t):=σ(t)-t. If not otherwise stated, all functions in this paper are considered to be complex valued. A function f on [a,b]𝕋 is called piecewise  rd-continuous; we write f∈Cprd on [a,b]𝕋 if the right-hand limit f(t+) exists finite at all right-dense points t∈[a,b)𝕋, and the left-hand limit f(t-) exists finite at all left-dense points t∈(a,b]𝕋 and f is continuous in the topology of the given time scale at all but possibly finitely many right-dense points t∈[a,b)𝕋. A function f on [a,∞)𝕋 is piecewise  rd-continuous; we write f∈Cprd on [a,∞)𝕋 if f∈Cprd on [a,b]𝕋 for every b∈(a,∞)𝕋. An n×n matrix-valued function f is called regressive on a given time scale interval if I+μ(t)f(t) is invertible for all t in this interval.The time scaleΔ-derivative of a function f at a point t is denoted by fΔ(t); see [52, Definition  1.10]. Whenever fΔ(t) exists, the formula fσ(t)=f(t)+μ(t)fΔ(t) holds true. The product rule for the Δ-differentiation of the product of two functions has the form (2.1)(fg)Δ(t)=fΔ(t)g(t)+fσ(t)gΔ(t)=fΔ(t)gσ(t)+f(t)gΔ(t).
A function f on [a,b]𝕋 is called piecewise  rd-continuouslyΔ-differentiable; we write f∈Cprd1 on [a,b]𝕋; if it is continuous on [a,b]𝕋, then fΔ(t) exists at all except for possibly finitely many points t∈[a,ρ(b)]𝕋, and fΔ∈Cprd on [a,ρ(b)]𝕋. As a consequence we have that the finitely many points ti at which fΔ(ti) does not exist belong to (a,b)𝕋 and these points ti are necessarily right-dense and left-dense at the same time. Also, since at those points we know that fΔ(ti+) and fΔ(ti-) exist finite, we replace the quantity fΔ(ti) by fΔ(ti±) in any formula involving fΔ(t) for all t∈[a,ρ(b)]𝕋. Similarly as above we define f∈Cprd1 on [a,∞)𝕋. The time scale integral of a piecewise rd-continuous function f over [a,b]𝕋 is denoted by ∫abf(t)Δt and over [a,∞)𝕋 by ∫a∞f(t)Δt provided this integral is convergent in the usual sense; see [52, Definitions  1.71 and  1.82].Remark 2.1.
As it is known in [52, Theorem  5.8] and discussed in [54, Remark  3.8], for a fixed t0∈[a,b]𝕋 and a piecewise rd-continuous n×n matrix function A(·) on [a,b]𝕋 which is regressive on [a,t0)𝕋, the initial value problem yΔ(t)=A(t)y(t) for t∈[a,ρ(b)]𝕋 with y(t0)=y0 has a unique solution y(·)∈Cprd1 on [a,b]𝕋 for any y0∈ℂn. Similarly, this result holds on [a,∞)𝕋.Let us recall some matrix notations from linear algebra used in this paper. Given a complex square matrixM, by M*, M>0, M≥0, M<0, M≤0, rankM, KerM, defM, we denote, respectively, the conjugate transpose, positive definiteness, positive semidefiniteness, negative definiteness, negative semidefiniteness, rank, kernel, and the defect (i.e., the dimension of the kernel) of the matrix M. Moreover, we will use the notation Im(M):=(M-M*)/(2i) and Re(M):=(M+M*)/2 for the Hermitian components of the matrix M; see [55, pages  268-269] or [56, Fact  3.5.24]. This notation will be also used with λ∈ℂ, and in this case Im(λ) and Re(λ) represent the imaginary and real parts of λ.Remark 2.2.
If the matrixIm(M) is positive or negative definite, then the matrix M is necessarily invertible. The proof of this fact can be found, for example, in [2, Remark  2.6].In order to simplify the notation we abbreviate[fσ(t)]* and [f*(t)]σ by fσ*(t). Similarly, instead of [fΔ(t)]* and [f*(t)]Δ we will use fΔ*(t).


## 3. Time Scale Symplectic Systems

Let𝒜(·), ℬ(·), 𝒞(·), 𝒟(·), 𝒲(·) be n×n piecewise rd-continuous functions on [a,∞)𝕋 such that 𝒲(t)≥0 for all t∈[a,∞)𝕋; that is, 𝒲(t) is Hermitian and nonnegative definite, satisfying identity (1.8). In this paper we consider the linear system (𝒮λ) introduced in the previous section. This system can be written as (Sλ)zΔ(t,λ)=S(t)z(t,λ)+λJW̃(t)zσ(t,λ),t∈[a,∞)T,
where the 2n×2n matrix 𝒲̃(t) is defined and has the property (3.1)W̃(t):=(W(t)000),JW̃(t)=(00-W(t)0).
The system (𝒮λ) can be written in the equivalent form
(3.2)zΔ(t,λ)=S(t,λ)z(t,λ),t∈[a,∞)T,
where the matrix 𝒮(t,λ) is defined through the matrices 𝒮(t) and 𝒲̃(t) from (1.8) and (3.1) by (3.3)S(t,λ):=S(t)+λJW̃(t)[I+μ(t)S(t)]=(A(t)B(t)C(t)-λW(t)[I+μ(t)A(t)]D(t)-λμ(t)W(t)B(t)).
By using the identity in (1.8), a direct calculation shows that the matrix function 𝒮(·,·) satisfies (3.4)S*(t,λ)J+JS(t,λ̅)+μ(t)S*(t,λ)JS(t,λ̅)=0,t∈[a,∞)T,λ∈C.
Here 𝒮*(t,λ)=[𝒮(t,λ)]*, and λ̅ is the usual conjugate number to λ.Remark 3.1.
The name time scalesymplectic  system or Hamiltonian system has been reserved in the literature for the system of the form
(3.5)zΔ(t)=S(t)z(t),t∈[a,∞)T,
in which the matrix function 𝕊(·) satisfies the identity in (1.8); see [44–47, 57], and compare also, for example, with [58–61]. Since for a fixed λ,ν∈ℂ the matrix 𝒮(t,λ) from (3.3) satisfies
(3.6)S*(t,λ)J+JS(t,ν)+μ(t)S*(t,λ)JS(t,ν)=(λ̅-ν)[I+μ(t)S*(t)]W̃(t)[I+μ(t)S(t)],
it follows that the system (𝒮λ) is a true time scale symplectic system according to the above terminology only for λ∈ℝ, while strictly speaking (𝒮λ) is  not a time scale symplectic system for λ∈ℂ∖ℝ. However, since (𝒮λ) is a perturbation of the time scale symplectic system (𝒮0) and since the important properties of time scale symplectic systems needed in the presented Weyl-Titchmarsh theory, such as (3.4) or (3.8), are satisfied in an appropriate modification, we accept with the above understanding the same terminology for the system (𝒮λ) for any λ∈ℂ.Equation (3.4) represents a fundamental identity for the theory of time scale symplectic systems (𝒮λ). Some important properties of the matrix 𝒮(t,λ) are displayed below. Note that formula (3.7) is a generalization of [46, equation  (10.4)] to complex values of λ.Lemma 3.2.
Identity (3.4) is equivalent to the identity
(3.7)S(t,λ̅)J+JS*(t,λ)+μ(t)S(t,λ̅)JS*(t,λ)=0,t∈[a,∞)T,λ∈C.
In this case for any λ∈ℂ we have
(3.8)[I+μ(t)S*(t,λ)]J[I+μ(t)S(t,λ̅)]=J,t∈[a,∞)T,(3.9)[I+μ(t)S(t,λ̅)]J[I+μ(t)S*(t,λ)]=J,t∈[a,∞)T,
and the matrices I+μ(t)𝒮(t,λ) and I+μ(t)𝒮(t,λ̅) are invertible with
(3.10)[I+μ(t)S(t,λ)]-1=-J[I+μ(t)S*(t,λ̅)]J,t∈[a,∞)T.Proof.
Lett∈[a,∞)𝕋 and λ∈ℂ be fixed. If t is right-dense, that is, μ(t)=0, then identity (3.4) reduces to 𝒮*(t,λ)𝒥+𝒥𝒮(t,λ̅)=0. Upon multiplying this equation by 𝒥 from the left and right side, we get identity (3.7) with μ(t)=0. If t is right scattered, that is, μ(t)>0, then (3.4) is equivalent to (3.8). It follows that the determinants of I+μ(t)𝒮(t,λ) and I+μ(t)𝒮(t,λ̅) are nonzero proving that these matrices are invertible with the inverse given by (3.10). Upon multiplying (3.8) by the invertible matrices [I+μ(t)𝒮(t,λ̅)]𝒥 from the left and -[I+μ(t)𝒮(t,λ̅)]-1𝒥 from the right and by using 𝒥2=-I, we get formula (3.9), which is equivalent to (3.7) due to μ(t)>0.Remark 3.3.
Equation (3.10) allows writing the system (𝒮λ) in the equivalent adjoint form
(3.11)zΔ(t,λ)=JS*(t,λ̅)Jzσ(t,λ),t∈[a,∞)T.
System (3.11) can be found, for example, in [47, Remark  3.1(iii)] or [50, equation  (3.2)] in the connection with optimality conditions for variational problems over time scales.In the following result we show that (3.4) guarantees, among other properties, the existence and uniqueness of solutions of the initial value problems associated with (𝒮λ).Theorem 3.4 (existence and uniqueness theorem).
Letλ∈ℂ, t0∈[a,∞)𝕋, and z0∈ℂ2n be given. Then the initial value problem (𝒮λ) with z(t0)=z0 has a unique solution z(·,λ)∈Cprd1 on the interval [a,∞)𝕋.Proof.
The coefficient matrix of system (𝒮λ), or equivalently of system (3.2), is piecewise rd-continuous on [a,∞)𝕋. By Lemma 3.2, the matrix I+μ(t)𝒮(t,λ) is invertible for all t∈[a,∞)𝕋, which proves that the function 𝒮(·,λ) is regressive on [a,∞)𝕋. Hence, the result follows from Remark 2.1.If not specified otherwise, we use a common agreement that2n-vector solutions of system (𝒮λ) and 2n×n-matrix solutions of system (𝒮λ) are denoted by small letters and capital letters, respectively, typically by z(·,λ) or z̃(·,λ) and Z(·,λ) or Z̃(·,λ).Next we establish several identities involving solutions of system (𝒮λ) or solutions of two such systems with different spectral parameters. The first result is the Lagrange identity known in the special cases of continuous time linear Hamiltonian systems in [11, Theorem  4.1] or [8, equation  (2.23)], discrete linear Hamiltonian systems in [9, equation  (2.55)] or [14, Lemma  2.2], discrete symplectic systems in [1, Lemma  2.6] or [2, Lemma  2.3], and time scale linear Hamiltonian systems in [3, Lemma  3.5] and [5, Theorem  2.2].Theorem 3.5 (Lagrange identity).
Letλ,ν∈ℂ and m∈ℕ be given. If z(·,λ) and z(·,ν) are 2n×m solutions of systems (𝒮λ) and (𝒮ν), respectively, then
(3.12)[z*(t,λ)Jz(t,ν)]Δ=(λ̅-ν)zσ*(t,λ)W̃(t)zσ(t,ν),t∈[a,∞)T.Proof.
Formula (3.12) follows from the time scales product rule (2.1) by using the relation zσ(t,λ)=[I+μ(t)𝒮(t,λ)]z(t,λ) and identity (3.6).As consequences of Theorem3.5, we obtain the following.Corollary 3.6.
Letλ,ν∈ℂ and m∈ℕ be given. If z(·,λ) and z(·,ν) are 2n×m solutions of systems (𝒮λ) and (𝒮ν), respectively, then for all t∈[a,∞)𝕋 we have
(3.13)z*(t,λ)Jz(t,ν)=z*(a,λ)Jz(a,ν)+(λ̅-ν)∫atzσ*(s,λ)W̃(s)zσ(s,ν)Δs.One can easily see that ifz(·,λ) is a solution of system (𝒮λ), then z(·,λ̅) is a solution of system (𝒮λ̅). Therefore, Theorem 3.5 with ν=λ̅ yields a Wronskian-type property of solutions of system (𝒮λ).Corollary 3.7.
Letλ∈ℂ and m∈ℕ be given. For any 2n×m solution z(·,λ) of systems (𝒮λ)
(3.14)z*(t,λ)Jz(t,λ̅)≡z*(a,λ)Jz(a,λ̅),isconstanton[a,∞)T.The following result gives another interesting property of solutions of system (𝒮λ) and (𝒮λ̅).Lemma 3.8.
Letλ∈ℂ and m∈ℕ be given. For any 2n×m solutions z(·,λ) and z̃(·,λ) of system (𝒮λ), the 2n×2n matrix function K(·,λ) defined by
(3.15)K(t,λ):=z(t,λ)z̃*(t,λ̅)-z̃(t,λ)z*(t,λ̅),t∈[a,∞)T,
satisfies the dynamic equation
(3.16)KΔ(t,λ)=S(t,λ)K(t,λ)+[I+μ(t)S(t,λ)]K(t,λ)S*(t,λ̅),t∈[a,∞)T,
and the identities K*(t,λ)=-K(t,λ̅) and
(3.17)Kσ(t,λ)=[I+μ(t)S(t,λ)]K(t,λ)[I+μ(t)S*(t,λ̅)],t∈[a,∞)T.Proof.
Having thatz(·,λ) and z̃(·,λ) are solutions of system (𝒮λ), it follows that z(·,λ̅) and z̃(·,λ̅) are solutions of system (𝒮λ̅). The results then follow by direct calculations.Remark 3.9.
The content of Lemma3.8 appears to be new both in the continuous and discrete time cases. Moreover, when the matrix function K(·,λ)≡K(λ) is constant, identity (3.17) yields for any right-scattered t∈[a,∞)𝕋 that
(3.18)S(t,λ)K(λ)+K(λ)S*(t,λ̅)+μ(t)S(t,λ)K(λ)S*(t,λ̅)=0.
It is interesting to note that this formula is very much like (3.7). More precisely, identity (3.7) is a consequence of (3.18) for the case of K(λ)≡𝒥.Next we present properties of certain fundamental matricesΨ(·,λ) of system (𝒮λ), which are generalizations of the corresponding results in [46, Section  10.2] to complex λ. Some of these results can be proven under the weaker condition that the initial value of Ψ(a,λ) does depend on λ and satisfies Ψ*(a,λ)𝒥Ψ(a,λ̅)=𝒥. However, these more general results will not be needed in this paper.Lemma 3.10.
Letλ∈ℂ be fixed. If  Ψ(·,λ) is a fundamental matrix of system (𝒮λ) such that Ψ(a,λ) is symplectic and independent of λ, then for any t∈[a,∞)𝕋 we have
(3.19)Ψ*(t,λ)JΨ(t,λ̅)=J,Ψ-1(t,λ)=-JΨ*(t,λ̅)J,Ψ(t,λ)JΨ*(t,λ̅)=J.Proof.
Identity (3.19)(i) is a consequence of Corollary 3.7, in which we use the fact that Ψ(a,λ) is symplectic and independent of λ. The second identity in (3.19) follows from the first one, while the third identity is obtained from the equation Ψ(t,λ)Ψ-1(t,λ)=I.Remark 3.11.
If the fundamental matrixΨ(·,λ)=(Z(·,λ)Z̃(·,λ)) in Lemma 3.10 is partitioned into two 2n×n blocks, then (3.19)(i) and (3.19)(iii) have, respectively, the form
(3.20)Z*(t,λ)JZ(t,λ̅)=0,Z*(t,λ)JZ̃(t,λ̅)=I,Z̃*(t,λ)JZ̃(t,λ̅)=0,(3.21)Z(t,λ)Z̃*(t,λ̅)-Z̃(t,λ)Z*(t,λ̅)=J.
Observe that the matrix on the left-hand side of (3.21) represents a constant matrix K(t,λ) from Lemma 3.8 and Remark 3.9.Corollary 3.12.
Under the conditions of Lemma3.10, for any t∈[a,∞)𝕋, we have
(3.22)Ψσ(t,λ)JΨ*(t,λ̅)=[I+μ(t)S(t,λ)]J,
which in the notation of Remark 3.11 has the form
(3.23)Zσ(t,λ)Z̃*(t,λ̅)-Z̃σ(t,λ)Z*(t,λ̅)=[I+μ(t)S(t,λ)]J.Proof.
Identity (3.22) follows from the equation Ψσ(t,λ)=[I+μ(t)𝒮(t,λ)]Ψ(t,λ) by applying formula (3.19)(ii).


## 4.M(λ)-Function for Regular Spectral Problem

In this section we consider the regular spectral problem on the time scale interval[a,b]𝕋 with some fixed b∈(a,∞)𝕋. We will specify the corresponding boundary conditions in terms of complex n×2n matrices from the set (4.1)Γ:={α∈Cn×2n,αα*=I,αJα*=0}.
The two defining conditions for α∈ℂn×2n in (4.1) imply that the 2n×2n matrix (α*-𝒥α*) is unitary and symplectic. This yields the identity (4.2)(α*-Jα*)(ααJ)=I∈C2n×2n,thatis,α*α-Jα*αJ=I.
The last equation also implies, compare with [60, Remark  2.1.2], that (4.3)Ker⁡α=Im⁡Jα*.Letα,β∈Γ be fixed and consider the boundary value problem (4.4)(Sλ),αz(a,λ)=0,βz(b,λ)=0.
Our first result shows that the boundary conditions in (4.4) are equivalent with the boundary conditions phrased in terms of the images of the 2n×2n matrices (4.5)Ra:=(Jα*0),Rb:=(0-Jβ*),
which satisfy Ra*𝒥Ra=0, Rb*𝒥Rb=0, and rank(Ra*Rb*)=2n.Lemma 4.1.
Letα,β∈Γ and λ∈ℂ be fixed. A solution z(·,λ) of system (𝒮λ) satisfies the boundary conditions in (4.4) if and only if there exists a unique vector ξ∈ℂ2n such that
(4.6)z(a,λ)=Raξ,z(b,λ)=Rbξ.Proof.
Assume that (4.4) holds. Identity (4.3) implies the existence of vectors ξa,ξb∈ℂn such that z(a,λ)=-𝒥α*ξa and z(b,λ)=-𝒥β*ξb. It follows that z(·,λ) satisfies (4.6) with ξ:=(-ξa*ξb*)*. It remains to prove that ξ is unique such a vector. If z(·,λ) satisfies (4.6) and also z(a,λ)=Raζ and z(b,λ)=Rbζ for some ξ,ζ∈ℂ2n, then Ra(ξ-ζ)=0 and Rb(ξ-ζ)=0. Hence, 𝒥α*(I0)(ξ-ζ)=0 and -𝒥β*(0I)(ξ-ζ)=0. If we multiply the latter two equalities by α𝒥 and β𝒥, respectively, and use αα*=I=ββ*, then we obtain (I0)(ξ-ζ)=0 and (0I)(ξ-ζ)=0. This yields ξ-ζ=0, which shows that the vector ξ in (4.6) is unique. The opposite direction, that is, that (4.6) implies (4.4), is trivial.Following the standard terminology, see, for example, [62, 63], a number λ∈ℂ is an eigenvalue of (4.4) if this boundary value problem has a solution z(·,λ)≢0. In this case the function z(·,λ) is called the eigenfunction corresponding to the eigenvalue λ, and the dimension of the space of all eigenfunctions corresponding to λ (together with the zero function) is called the geometric  multiplicity of λ.Givenα∈Γ, we will utilize from now on the fundamental matrix Ψ(·,λ,α) of system (𝒮λ) satisfying the initial condition from (4.4), that is, (4.7)ΨΔ(t,λ,α)=S(t,λ)Ψ(t,λ,α),t∈[a,ρ(b)]T,Ψ(a,λ,α)=(α*-Jα*).
Then Ψ(a,λ,α) does not depend on λ, and it is symplectic and unitary with the inverse Ψ-1(a,λ,α)=Ψ*(a,λ,α). Hence, the properties of fundamental matrices derived earlier in Lemma 3.10, Remark 3.11, and Corollary 3.12 apply for the matrix function Ψ(·,λ,α).The following assumption will be imposed in this section when studying the regular spectral problem.Hypothesis 4.2.
For everyλ∈ℂ, we have
(4.8)∫abΨσ*(t,λ,α)W̃(t)Ψσ(t,λ,α)Δt>0.Condition (4.8) can be written in the equivalent form as (4.9)∫abzσ*(t,λ)W̃(t)zσ(t,λ)Δt>0,
for every nontrivial solution z(·,λ) of system (𝒮λ). Assumptions (4.8) and (4.9) are equivalent by a simple argument using the uniqueness of solutions of system (𝒮λ). The latter form (4.9) has been widely used in the literature, such as in the continuous time case in [8, Hypothesis  2.2], [30, equation  (1.3)], [26, equation  (2.3)], in the discrete time case in [9, Condition  (2.16)], [14, equation  (1.7)], [1, Assumption  2.2], [2, Hypothesis  2.4], and in the time scale Hamiltonian case in [3, Assumption  3] and [5, Condition  (3.9)].Following Remark3.11, we partition the fundamental matrix Ψ(·,λ,α) as (4.10)Ψ(⋅,λ,α)=(Z(⋅,λ,α)Z̃(⋅,λ,α)),
where Z(·,λ,α) and Z̃(·,λ,α) are the 2n×n solutions of system (𝒮λ) satisfying Z(a,λ,α)=α* and Z̃(a,λ,α)=-𝒥α*. With the notation (4.11)Λ(λ,α,β):=Ψ(b,λ,α)Ψ*(a,λ,α)Ra-Rb=(-Z̃(b,λ,α)Jβ*),
we have the classical characterization of the eigenvalues of (4.4); see, for example, the continuous time in [64, Chapter  4], the discrete time in [14, Theorem  2.3, Lemma  2.4], [2, Lemma  2.9, Theorem  2.11], and the time scale case in [62, Lemma  3], [63, Corollary  1].Proposition 4.3.
Forα,β∈Γ and λ∈ℂ, we have with notation (4.11) the following. (i)
The numberλ is an eigenvalue of (4.4) if and only if detΛ(λ,α,β)=0.(ii)
The algebraic multiplicity of the eigenvalueλ, that is, the number defΛ(λ,α,β), is equal to the geometric multiplicity of λ.(iii)
Under Hypothesis4.2, the eigenvalues of (4.4) are real, and the eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the semi-inner product
(4.12)〈z(⋅,λ),z(⋅,ν)〉W,b:=∫abzσ*(t,λ)W̃(t)zσ(t,ν)Δt.Proof.
The arguments are here standard, and we refer to [44, Section  5], [63, Corollary  1], [3, Theorem  3.6].The next algebraic characterization of the eigenvalues of (4.4) is more appropriate for the development of the Weyl-Titchmarsh theory for (4.4), since it uses the matrix βZ̃(b,λ,α) which has dimension n instead of using the matrix Λ(λ,α,β) which has dimension 2n. Results of this type can be found in special cases of system (𝒮λ) in [8, Lemma  2.5], [11, Theorem  4.1], [9, Lemma  2.8], [14, Lemma  3.1], [1, Lemma  2.5], [3, Theorem  3.4], and [2, Lemma  3.1].Lemma 4.4.
Letα,β∈Γ and λ∈ℂ be fixed. Then λ is an eigenvalue of (4.4) if and only if detβZ̃(b,λ,α)=0. In this case the algebraic and geometric multiplicities of λ are equal to defβZ̃(b,λ,α).Proof.
One can follow the same arguments as in the proof of the corresponding discrete symplectic case in [2, Lemma  3.1]. However, having the result of Proposition 4.3, we can proceed directly by the methods of linear algebra. In this proof we abbreviate Λ:=Λ(λ,α,β) and Z̃:=Z̃(b,λ,α). Assume that Λ is singular, that is, -Z̃c+𝒥β*d=0 for some vectors c,d∈ℂn, not both zero. Then Z̃c=𝒥β*d, which yields that βZ̃c=0. If c=0, then 𝒥β*d=0, which implies upon the multiplication by β𝒥 from the left that d=0. Since not both c and d can be zero, it follows that c≠0 and the matrix βZ̃ is singular. Conversely, if βZ̃c=0 for some nonzero vector c∈ℂn, then -Z̃c+𝒥β*d=0; that is, Λ is singular, with the vector d:=-β𝒥Z̃c. Indeed, by using identity (4.2) we have 𝒥β*d=-𝒥β*β𝒥Z̃c=(I-β*β)Z̃c=Z̃c. From the above we can also see that the number of linearly independent vectors in KerβZ̃ is the same as the number of linearly independent vectors in KerΛ. Therefore, by Proposition 4.3(ii), the algebraic and geometric multiplicities of λ as an eigenvalue of (4.4) are equal to defβZ̃.Since the eigenvalues of (4.4) are real, it follows that the matrix βZ̃(b,λ,α) is invertible for every λ∈ℂ except for at most n real numbers. This motivates the definition of the M(λ)-function for the regular spectral problem.Definition 4.5 (M(λ)-function).
Letα,β∈Γ. Whenever the matrix βZ̃(b,λ,α) is invertible for some value λ∈ℂ, we define the Weyl-TitchmarshM(λ)-function as the n×n matrix
(4.13)M(λ)=M(λ,b)=M(λ,b,α,β):=-[βZ̃(b,λ,α)]-1βZ(b,λ,α).The above definition of theM(λ)-function is a generalization of the corresponding definitions for the continuous and discrete linear Hamiltonian and symplectic systems in [8, Definition  2.6], [9, Definition  2.9], [14, equation  (3.10)], [1, page  2859], [2, Definition  3.2] and time scale linear Hamiltonian systems in [3, equation  (4.1)]. The dependence of the M(λ)-function on b, α, and β will be suppressed in the notation, and M(λ,b) or M(λ,b,α,β) will be used only in few situations when we emphasize the dependence on b (such as at the end of Section 5) or on α and β (as in Lemma 4.14). By [65, Corollary  4.5], see also [44, Remark  2.2], the M(·)-function is an entire function in λ. Another important property of the M(λ)-function is established in the following.Lemma 4.6.
Letα,β∈Γ and λ∈ℂ∖ℝ. Then
(4.14)M*(λ)=M(λ̅).Proof.
We abbreviateZ(λ):=Z(b,λ,α) and Z̃(λ):=Z̃(b,λ,α). By using the definition of M(λ) in (4.13) and identity (3.21), we have
(4.15)M*(λ)-M(λ̅)=[βZ̃(λ̅)]-1β[Z(λ̅)Z̃*(λ)-Z̃(λ̅)Z*(λ)]β*[βZ̃(λ)]*-1=(3.21)[βZ̃(λ̅)]-1βJβ*[βZ̃(λ)]*-1=0,
because β∈Γ. Hence, equality (4.14) holds true.The following solution plays an important role in particular in the results concerning the square integrable solutions of system (𝒮λ).Definition 4.7 (Weyl solution).
For any matrixM∈ℂn×n, we define the so-called Weyl  solution of system (𝒮λ) by
(4.16)X(⋅,λ,α,M):=Ψ(⋅,λ,α)(IM*)*=Z(⋅,λ,α)+Z̃(⋅,λ,α)M,
where Z(·,λ,α) and Z̃(·,λ,α) are defined in (4.10).The function𝒳(·,λ,α,M), being a linear combination of two solutions of system (𝒮λ), is also a solution of this system. Moreover, α𝒳(a,λ,α,M)=I, and, if βZ̃(b,λ,α) is invertible, then β𝒳̃(b,λ,α,M)=βZ̃(b,λ,α)[M-M(λ)]. Consequently, if we take M:=M(λ) in Definition 4.7, then β𝒳(b,λ,α,M(λ))=0; that is, the Weyl solution 𝒳(·,λ,α,M(λ)) satisfies the right endpoint boundary condition in (4.4).Following the corresponding notions in [8, equation  (2.18)], [9, equation  (2.51)], [14, page  471], [1, page  2859], [2, equation  (3.13)], [3, equation  (4.2)], we define the Hermitian n×n matrix function ℰ(M) for system (𝒮λ).Definition 4.8.
For a fixedα∈Γ and λ∈ℂ∖ℝ, we define the matrix function
(4.17)E:Cn×n⟶Cn×n,E(M)=E(M,b):=iδ(λ)X*(b,λ,α,M)JX(b,λ,α,M),
where δ(λ):=sgn Im(λ).For brevity we suppress the dependence of the functionℰ(·) on b and λ. In few cases we will need ℰ(M) depending on b (as in Theorem 5.1 and Definition 6.2) and in such situations we will use the notation ℰ(M,b). Since (i𝒥)*=i𝒥, it follows that ℰ(M) is a Hermitian matrix for any M∈ℂn×n. Moreover, from Corollary 3.6, we obtain the identity (4.18)E(M)=-2δ(λ)Im⁡(M)+2|Im⁡(λ)|∫abXσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt,
where we used the fact that (4.19)X*(a,λ,α,M)JX(a,λ,α,M)=(4.7)M-M*=2iIm⁡(M).Next we define the Weyl disk and Weyl circle for the regular spectral problem. The geometric characterizations of the Weyl disk and Weyl circle in terms of the contractive or unitary matrices which justify the terminology “disk” or “circle” will be presented in Section5.Definition 4.9 (Weyl disk and Weyl circle).
For a fixedα∈Γ and λ∈ℂ∖ℝ, the set
(4.20)D(λ)=D(λ,b):={M∈Cn×n,E(M)≤0},
is called the Weyl  disk, and the set
(4.21)C(λ)=C(λ,b):=∂D(λ)={M∈Cn×n,E(M)=0},
is called the Weyl  circle.The dependence of the Weyl disk and Weyl circle onb will be again suppressed. In the following result we show that the Weyl circle consists of precisely those matrices M(λ) with β∈Γ. This result generalizes the corresponding statements in [8, Lemma  2.8], [9, Lemma  2.13], [14, Lemma  3.3], [1, Theorem  3.1], [2, Theorem  3.6], and [3, Theorem  4.2].Theorem 4.10.
Letα∈Γ, λ∈ℂ∖ℝ, and M∈ℂn×n. The matrix M belongs to the Weyl circle C(λ) if and only if there exists β∈Γ such that β𝒳(b,λ,α,M)=0. In this case and under Hypothesis 4.2, we have with such a matrix β that M=M(λ) as defined in (4.13).Proof.
Assume thatM∈C(λ), that is, ℰ(M)=0. Then, with the vector
(4.22)β:=X*(b)J=(IM*)Ψ*(b,λ,α)J∈Cn×2n,
where 𝒳(b) denotes 𝒳(b,λ,α,M), we have
(4.23)βX(b)=X*(b)JX(b)=[1(iδ(λ))]E(M)=0.
Moreover, rankβ=n, because the matrices Ψ(b,λ,α) and 𝒥 are invertible and rank(IM*)=n. In addition, the identity 𝒥*=𝒥-1 yields
(4.24)βJβ*=X*(b)JX(b)=(4.23)0.
Now, if the condition ββ*=I is not satisfied, then we replace β by β̃:=(ββ*)-1/2β (note that ββ*>0, so that (ββ*)-1/2 is well defined), and in this case
(4.25)β̃X(b)=(ββ*)-1/2βX(b)=(4.23)0,β̃Jβ̃*=(ββ*)-1/2βJβ*(ββ*)-1/2=(4.24)0,β̃β̃*=(ββ*)-1/2ββ*(ββ*)-1/2=I.
Conversely, suppose that for a given M∈ℂn×n there exists β∈Γ such that β𝒳(b)=0. Then from (4.3) it follows that 𝒳(b)=𝒥β*P for the matrix P:=-β𝒥𝒳(b)∈ℂn×n. Hence,
(4.26)E(M)=iδ(λ)P*βJ*JJβ*P=iδ(λ)P*βJβ*P=0,
that is, M∈C(λ). Finally, since λ∈ℂ∖ℝ, then by Proposition 4.3(iii) the number λ is not an eigenvalue of (4.4), which by Lemma 4.4 shows that the matrix βZ̃(b,λ,α) is invertible. The definition of the Weyl solution in (4.16) then yields
(4.27)βZ(b,λ,α)+βZ̃(b,λ,α)M=βX(b,λ,α,M)=0,
which implies that M=-[βZ̃(b,λ,α)]-1βZ(b,λ,α)=M(λ).Remark 4.11.
The matrixP:=-β𝒥𝒳(b,λ,α,M)∈ℂn×n from the proof of Theorem 4.10 is invertible. This fact was not needed in that proof. However, we show that P is invertible because this argument will be used in the proof of Lemma 4.14. First we prove that KerP=Ker𝒳(b,λ,α,M). For if Pd=0 for some d∈ℂn, then from identity (4.2) we get 𝒳(b,λ,α,M)d=(I-β*β)𝒳(b,λ,α,M)d=𝒥β*Pd=0. Therefore, KerP⊆Ker𝒳(b,λ,α,M). The opposite inclusion follows by the definition of P. And since, by (4.16), rank𝒳(b,λ,α,M)=rank(IM*)*=n, it follows that Ker𝒳(b,λ,α,M)={0}. Hence, KerP={0} as well; that is, the matrix P is invertible.The next result contains a characterization of the matricesM∈ℂn×n which lie “inside” the Weyl disk D(λ). In the previous result (Theorem 4.10) we have characterized the elements of the boundary of the Weyl disk D(λ), that is, the elements of the Weyl circle C(λ), in terms of the matrices β∈Γ. For such β we have β𝒥β*=0, which yields iδ(λ)β𝒥β*=0. Comparing with that statement we now utilize the matrices β∈ℂn×2n which satisfy iδ(λ)β𝒥β*>0. In the special cases of the continuous and discrete time, this result can be found in [8, Lemma  2.13], [9, Lemma  2.18], and [2, Theorem  3.13].Theorem 4.12.
Letα∈Γ, λ∈ℂ∖ℝ, and M∈ℂn×n. The matrix M satisfies ℰ(M)<0 if and only if there exists β∈ℂn×2n such that iδ(λ)β𝒥β*>0 and β𝒳(b,λ,α,M)=0. In this case and under Hypothesis 4.2, we have with such a matrix β that M=M(λ) as defined in (4.13) and β may be chosen so that ββ*=I.Proof.
ForM∈ℂn×n consider on [a,b]𝕋 the Weyl solution
(4.28)X(⋅):=X(⋅,λ,α,M)=(X1(⋅)X2(⋅)),withn×nblocksX1(⋅)andX2(⋅).
Suppose first that ℰ(M)<0. Then the matrices 𝒳j(b), j∈{1,2}, are invertible. Indeed, if one of them is singular, then there exists a nonzero vector v∈ℂn such that 𝒳1(b)v=0 or 𝒳2(b)v=0. Then
(4.29)v*E(M)v=iδ(λ)v*X*(b)JX(b)v=iδ(λ)v*[X1*(b)X2(b)-X2*(b)X1(b)]v=0,
which contradicts ℰ(M)<0. Now we set β1:=I, β2:=-𝒳1(b)𝒳2-1(b), and β:=(β1β2). Then for this 2n×n matrix β we have β𝒳(b)=0 and, by a similar calculation as in (4.29),
(4.30)E(M)=iδ(λ)X*(b)JX(b)=iδ(λ)X2*(b)(β2β1*-β1β2*)X2(b)=2δ(λ)X2*(b)Im⁡(β1β2*)X2(b)=-iδ(λ)X2*(b)βJβ*X2(b),
where we used the equality β𝒥β*=2iIm(β1β2*). Since ℰ(M)<0 and 𝒳2(b) is invertible, it follows that iδ(λ)β𝒥β*>0. Conversely, assume that for a given matrix M∈ℂn×n there is β=(β1β2)∈ℂn×2n satisfying iδ(λ)β𝒥β*>0 and β𝒳(b)=0. Condition iδ(λ)β𝒥β*>0 is equivalent to Im(β1β2*)<0 when Im(λ)>0 and to Im(β1β2*)>0 when Im(λ)<0. The positive or negative definiteness of Im(β1β2*) implies the invertibility of β1 and β2; see Remark 2.2. Therefore, from the equality β1𝒳1(b)+β2𝒳2(b)=β𝒳(b)=0, we obtain 𝒳1(b)=-β1-1β2𝒳2(b), and so
(4.31)E(M)=iδ(λ)[X1*(b)X2(b)-X2*(b)X1(b)]=iδ(λ)X2*(b)β1-1(β2β1*-β1β2*)β1*-1X2(b)=-iδ(λ)X2*(b)β1-1βJβ*β1*-1X2(b).
The matrix 𝒳2(b) is invertible, because if 𝒳2(b)d=0 for some nonzero vector d∈ℂn, then 𝒳1(b)d=-β1-1β2𝒳2(b)d=0, showing that rank𝒳(b)<n. This however contradicts rank𝒳(b)=n which we have from the definition of the Weyl solution 𝒳(·) in (4.16). Consequently, (4.31) yields through iδ(λ)β𝒥β*>0 that ℰ(M)<0.
If the matrixβ does not satisfy ββ*=I, then we modify it according to the procedure described in the proof of Theorem 4.10. Finally, since λ∈ℂ∖ℝ, we get from Proposition 4.3(iii) and Lemma 4.4 that the matrix βZ̃(b,λ,α) is invertible which in turn implies through the calculation in (4.27) that M=-[βZ̃(b,λ,α)]-1βZ(b,λ,α)=M(λ).In the following lemma we derive some additional properties of the Weyl disk and theM(λ)-function. Special cases of this statement can be found in [8, Lemma  2.9], [33, Theorem  3.1], [9, Lemma  2.14], [14, Lemma  3.2(ii)], [1, Theorem  3.7], [2, Lemma  3.7], and [3, Theorem  4.13].Theorem 4.13.
Letα∈Γ and λ∈ℂ∖ℝ. For any matrix M∈D(λ) we have
(4.32)δ(λ)Im⁡(M)≥|Im⁡(λ)|∫abXσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt≥0.
In addition, under Hypothesis 4.2, we have δ(λ)Im(M)>0.Proof.
By identity (4.18), for any matrix M∈D(λ), we have
(4.33)2δ(λ)Im⁡(M)=-E(M)+2|Im⁡(λ)|∫abXσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt≥2|Im⁡(λ)|∫abXσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt,
which yields together with 𝒲̃(t)≥0 on [a,ρ(b)]𝕋 the inequalities in (4.32). The last assertion in Theorem 4.13 is a simple consequence of Hypothesis 4.2.In the last part of this section we wish to study the effect of changingα, which is one of the parameters of the M(λ)-function and the Weyl solution 𝒳(·,λ,α,M), when α varies within the set Γ. For this purpose we will use the M(λ)-function with all its arguments in the following two statements.Lemma 4.14.
Letα,β,γ∈Γ and λ∈ℂ∖ℝ. Then
(4.34)M(λ,b,α,β)=[αJγ*+αγ*M(λ,b,γ,β)][αγ*-αJγ*M(λ,b,γ,β)]-1.Proof.
LetM(b,λ,α,β) and M(b,λ,γ,β) be given via (4.13), and consider the Weyl solutions 𝒳α(·):=𝒳(·,λ,α,M(b,λ,α,β)) and 𝒳γ(·):=𝒳(·,λ,γ,M(b,λ,γ,β)) defined by (4.16) with M=M(b,λ,α,β) and M=M(b,λ,γ,β), respectively. First we prove that the two Weyl solutions 𝒳α(·) and 𝒳γ(·) differ by a constant nonsingular multiple. By definition, β𝒳α(b)=0 and β𝒳γ(b)=0, which implies through (4.3) that 𝒳α(b)=𝒥β*Pα and 𝒳γ(b)=𝒥β*Pγ for some matrices Pα,Pγ∈ℂn×n, which are invertible by Remark 4.11. This implies that 𝒳α(b)Pα-1=𝒥β*=𝒳γ(b)Pγ-1. Consequently, 𝒳α(b)=𝒳γ(b)P, where P:=Pγ-1Pα. By the uniqueness of solutions of system (𝒮λ), see Theorem 3.4, we obtain that 𝒳α(·)=𝒳γ(·)P on [a,b]𝕋. Upon the evaluation at t=a we get
(4.35)Ψ(a,λ,α)(IM(λ,b,α,β))=Ψ(a,λ,γ)(IM(λ,b,γ,β))P.
Since the matrices Ψ(a,λ,α)=(α*-𝒥α*) and Ψ(a,λ,γ)=(γ*-𝒥γ*) are unitary, it follows from (4.35) that
(4.36)(IM(λ,b,α,β))=(ααJ)(γ*-Jγ*)(IM(λ,b,γ,β))P=(αγ*-αJγ*M(λ,b,γ,β)αJγ*+αγ*M(λ,b,γ,β))P.
The first row above yields that P=[αγ*-α𝒥γ*M(λ,b,γ,β)]-1, while the second row is then written as identity (4.34).Corollary 4.15.
Letα,β,γ∈Γ and λ∈ℂ∖ℝ. With notation (4.16) and (4.13) we have
(4.37)X(⋅,λ,α,M(λ,b,α,β))=X(⋅,λ,γ,M(λ,b,γ,β))[αγ*-αJγ*M(λ,b,γ,β)]-1.Proof.
The above identity follows from (4.35) and the formula for the matrix P from the end of the proof of Lemma 4.14.


## 5. Geometric Properties of Weyl Disks

In this section we study the geometric properties of the Weyl disks as the pointb moves through the interval [a,∞)𝕋. Our first result shows that the Weyl disks D(λ,b) are nested. This statement generalizes the results in [11, Theorem  4.5], [66, Section  3.2.1], [9, equation  (2.70)], [14, Theorem  3.1], [3, Theorem  4.4], and [5, Theorem  3.3(i)].Theorem 5.1 (nesting property of Weyl disks).
Letα∈Γ and λ∈ℂ∖ℝ. Then
(5.1)D(λ,b2)⊆D(λ,b1),foreveryb1,b2∈[a,∞)T,b1<b2.Proof.
Letb1,b2∈[a,∞)𝕋 with b1<b2, and take M∈D(λ,b2), that is, ℰ(M,b2)≤0. From identity (4.18) with b=b1 and later with b=b2 and by using 𝒲̃(·)≥0, we have
(5.2)E(M,b1)=(4.18)-2δ(λ)Im⁡(M)+2|Im⁡(λ)|∫ab1Xσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt≤-2δ(λ)Im⁡(M)+2|Im⁡(λ)|∫ab2Xσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt=(4.18)E(M,b2)≤0.
Therefore, by Definition 4.9, the matrix M belongs to D(λ,b1), which shows the result.Similarly for the regular case (Hypothesis4.2) we now introduce the following assumption.Hypothesis 5.2.
There existsb0∈(a,∞)𝕋 such that Hypothesis 4.2 is satisfied with b=b0; that is, inequality (4.8) holds with b=b0 for every λ∈ℂ.From Hypothesis5.2 it follows by 𝒲̃(·)≥0 that inequality (4.8) holds for every b∈[b0,∞)𝕋.For the study of the geometric properties of Weyl disks we will use the following representation:(5.3)E(M,b)=iδ(λ)X*(b,λ,α,M)JX(b,λ,α,M)=(IM*)(F(b,λ,α)G*(b,λ,α)G(b,λ,α)H(b,λ,α))(IM),
of the matrix ℰ(M,b), where we define on [a,∞)𝕋 the n×n matrices (5.4)F(⋅,λ,α):=iδ(λ)Z*(⋅,λ,α)JZ(⋅,λ,α),G(⋅,λ,α):=iδ(λ)Z̃*(⋅,λ,α)JZ(⋅,λ,α),H(⋅,λ,α):=iδ(λ)Z̃*(⋅,λ,α)JZ̃(⋅,λ,α).
Since ℰ(M,b) is Hermitian, it follows that ℱ(·,λ,α) and ℋ(·,λ,α) are also Hermitian. Moreover, by (4.7), we have ℋ(a,λ,α)=0. In addition, if b∈[b0,∞)𝕋, then Corollary 3.7 and Hypothesis 5.2 yield for any λ∈ℂ∖ℝ(5.5)H(b,λ,α)=2|Im⁡(λ)|∫abZ̃σ*(t,λ,α)W̃(t)Z̃σ(t,λ,α)Δt>0.
Therefore, ℋ(b,λ,α) is invertible (positive definite) for all b∈[b0,∞)𝕋 and monotone nondecreasing as b→∞, with a consequence that ℋ-1(b,λ,α) is monotone nonincreasing as b→∞. The following factorization of ℰ(M,b) holds true; see also [2, equation  (4.11)].Lemma 5.3.
Letα∈Γ and λ∈ℂ∖ℝ. With the notation (5.4), for any M∈ℂn×n and b∈[a,∞)𝕋 we have
(5.6)E(M,b)=F(b,λ,α)-G*(b,λ,α)H-1(b,λ,α)G(b,λ,α)+[G*(b,λ,α)H-1(b,λ,α)+M*]H(b,λ,α)[H-1(b,λ,α)G(b,λ,α)+M],
whenever the matrix ℋ(b,λ,α) is invertible.Proof.
The result is shown by a direct calculation.The following identity is a generalization of its corresponding versions in [11, Lemma  4.3], [1, Lemma  3.3], [14, Proposition  3.2], [2, Lemma  4.2], [3, Lemma  4.6], and [5, Theorem  5.6].Lemma 5.4.
Letα∈Γ and λ∈ℂ∖ℝ. With the notation (5.4), for any b∈[a,∞)𝕋, we have
(5.7)G*(b,λ,α)H-1(b,λ,α)G(b,λ,α)-F(b,λ,α)=H-1(b,λ̅,α),
whenever the matrices ℋ(b,λ,α) and ℋ(b,λ̅,α) are invertible.Proof.
In order to simplify and abbreviate the notation we introduce the matrices(5.8)F:=F(b,λ,α),G:=G(b,λ,α),H:=H(b,λ,α),F̃:=F(b,λ̅,α),G̃:=G(b,λ̅,α),H̃:=H(b,λ̅,α),
and use the notation Z(λ) and Z̃(λ) for Z(b,λ,α) and Z̃(b,λ,α), respectively. Then, since ℱ*=ℱ and δ(λ)δ(λ̅)=-1, we get the identities
(5.9)G*F̃-F*G̃=Z*(λ)J[Z̃(λ)Z*(λ̅)-Z(λ)Z̃*(λ̅)]JZ(λ̅)=(3.21)Z*(λ)JZ(λ̅)=(3.20)0,(5.10)HG̃*-GH*=Z̃*(λ)J[Z̃(λ)Z*(λ̅)-Z(λ)Z̃*(λ̅)]JZ̃(λ̅)=(3.21)Z̃*(λ)JZ̃(λ̅)=(3.20)0,(5.11)GG̃-HF̃=Z̃*(λ)J[Z(λ)Z̃*(λ̅)-Z̃(λ)Z*(λ̅)]JZ(λ̅)=(3.21)-Z̃*(λ)JZ(λ̅)=(3.20)I,(5.12)G*G̃*-FH̃=Z*(λ)J[Z̃(λ)Z*(λ̅)-Z(λ)Z̃*(λ̅)]JZ̃(λ̅)=(3.21)Z*(λ)JZ̃(λ̅)=(3.20)I.
Hence, by using that ℋ̃ is Hermitian, we see that
(5.13)H̃-1=(5.12)G*G̃*H̃-1-F=G*G̃*H̃*-1-F=(5.10)G*H-1G-F.
Identity (5.7) is now proven.Corollary 5.5.
Letα∈Γ and λ∈ℂ∖ℝ. Under Hypothesis 5.2, the matrix ℋ(b,λ,α) is invertible for every b∈[b0,∞)𝕋, and for these values of b we have
(5.14)G*(b,λ,α)H-1(b,λ,α)G(b,λ,α)-F(b,λ,α)>0.Proof.
Sinceb∈[b0,∞)𝕋, then identity (5.5) yields that ℋ(b,λ,α)>0 and ℋ(b,λ̅,α)>0. Consequently, inequality (5.14) follows from (5.7) of Lemma 5.4.In the next result we justify the terminology for the setsD(λ,b) and C(λ,b) in Definition 4.9 to be called a “disk” and a “circle.” It is a generalization of [14, Theorem  3.1], [2, Theorem  5.4], [5, Theorem  3.3(iii)]; see also [66, Theorem  3.5], [26, pages  70-71], [8, page  3485], [14, Proposition  3.3], [1, Theorem  3.3], [3, Theorem  4.8]. Consider the sets 𝒱 and 𝒰 of contractive and unitary matrices in ℂn×n, respectively, that is, (5.15)V:={V∈Cn×n,V*V≤I},U:=∂V={U∈Cn×n,U*U=I}.
The set 𝒱 is known to be closed (in fact compact, since 𝒱 is bounded) and convex.Theorem 5.6.
Letα∈Γ and λ∈ℂ∖ℝ. Under Hypothesis 5.2, for every b∈[b0,∞)𝕋, the Weyl disk and Weyl circle have the representations
(5.16)D(λ,b)={P(λ,b)+R(λ,b)VR(λ̅,b),V∈V},(5.17)C(λ,b)={P(λ,b)+R(λ,b)UR(λ̅,b),U∈U},
where, with the notation (5.4),
(5.18)P(λ,b):=-H-1(λ,b,α)G(λ,b,α),R(λ,b):=H-1/2(λ,b,α).
Consequently, for every b∈[b0,∞)𝕋, the sets D(λ,b) are closed and convex.The representations ofD(λ,b) and C(λ,b) in (5.16) and (5.17) can be written as D(λ,b)=P(λ,b)+R(λ,b)𝒱R(λ̅,b) and C(λ,b)=P(λ,b)+R(λ,b)𝒰R(λ̅,b). The importance of the matrices P(λ,b) and R(λ,b) is justified in the following.Definition 5.7.
Forα∈Γ, λ∈ℂ∖ℝ, and b∈[a,∞)𝕋 such that ℋ(λ,b,α) and ℋ(λ̅,b,α) are positive definite, the matrix P(λ,b) is called the center of the Weyl disk or the Weyl circle. The matrices R(λ,b) and R(λ̅,b) are called the matrix  radii of the Weyl disk or the Weyl circle.Proof of Theorem5.6.
By (5.5) and for any b∈[b0,∞)𝕋, the matrices ℋ:=ℋ(λ,b,α) and ℋ̃:=ℋ(λ̅,b,α) are positive definite, so that the matrices P:=P(λ,b), R(λ):=R(λ,b), and R(λ̅):=R(λ̅,b) are well defined. By Definition 4.9, for M∈D(λ,b), we have ℰ(M,b)≤0, which in turn with notation (5.8) implies by Lemmas 5.3 and 5.4 that
(5.19)-R2(λ̅)+(M*-P*)R-2(λ)(M-P)=(5.7)F-G*H-1G+(H-1G+M)*H(H-1G+M)=E(M,b)≤0.
Therefore, the matrix
(5.20)V:=R-1(λ)(M-P)R-1(λ̅),
satisfies V*V≤I. This relation between the matrices M∈D(λ,b) and V∈𝒱 is bijective (more precisely, it is a homeomorphism), and the inverse to (5.20) is given by M=P+R(λ)VR(λ̅). The latter formula proves that the Weyl disk D(λ,b) has the representation in (5.16). Moreover, since by the definition M∈C(λ,b) means that ℰ(M,b)=0, it follows that the elements of the Weyl circle C(λ,b) are in one-to-one correspondence with the matrices V defined in (5.20) which, similarly as in (5.19), now satisfy V*V=I. Hence, the representation of C(λ,b) in (5.17) follows. The fact that for b∈[b0,∞)𝕋 the sets D(λ,b) are closed and convex follows from the same properties of the set 𝒱, being homeomorphic to D(λ,b).


## 6. Limiting Weyl Disk and Weyl Circle

In this section we study the limiting properties of the Weyl disk and Weyl circle and their center and matrix radii. Since under Hypothesis5.2 the matrix function ℋ(·,λ,α) is monotone nondecreasing as b→∞, it follows from the definition of R(λ,b) and R(λ̅,b) in (5.18) that the two matrix functions R(λ,·) and R(λ̅,·) are monotone nonincreasing for b→∞. Furthermore, since R(λ,b) and R(λ̅,b) are Hermitian and positive definite for b∈[b0,∞)𝕋, the limits (6.1)R+(λ):=lim⁡b→∞R(λ,b),R+(λ̅):=lim⁡b→∞R(λ̅,b),
exist and satisfy R+(λ)≥0 and R+(λ̅)≥0. The index “+” in the above notation as well as in Definition 6.2 refers to the limiting disk at +∞. In the following result we will see that the center P(λ,b) also converges to a limiting matrix when b→∞. This is a generalization of [11, Theorem  4.7], [1, Theorem  3.5], [14, Proposition  3.5], [2, Theorem  4.5], and [3, Theorem  4.10].Theorem 6.1.
Letα∈Γ and λ∈ℂ∖ℝ. Under Hypothesis 5.2, the center P(λ,b) converges as b→∞ to a limiting matrix P+(λ)∈ℂn×n, that is,
(6.2)P+(λ):=lim⁡b→∞P(λ,b).Proof.
We prove that the matrix functionP(λ,·) satisfies the Cauchy convergence criterion. Let b1,b2∈[b0,∞)𝕋 be given with b1<b2. By Theorem 5.1, we have that D(λ,b2)⊆D(λ,b1). Therefore, by (5.16) of Theorem 5.6, for a matrix M∈D(λ,b2), there are (unique) matrices V1,V2∈𝒱 such that
(6.3)M=P(λ,bj)+R(λ,bj)VjR(λ̅,bj),j∈{1,2}.
Upon subtracting the two equations in (6.3), we get
(6.4)P(λ,b2)-P(λ,b1)+R(λ,b2)V2R(λ̅,b2)=R(λ,b1)V1R(λ̅,b1).
This equation, when solved for V1 in terms of V2, has the form
(6.5)V1=R-1(λ,b1)[P(λ,b2)-P(λ,b1)+R(λ,b2)V2R(λ̅,b2)]R-1(λ̅,b1)=:T(V2),
which defines a continuous mapping T:𝒱→𝒱, T(V2)=V1. Since 𝒱 is compact, it follows that the mapping T has a fixed point in 𝒱, that is, T(V)=V for some matrix V∈𝒱. Equation T(V)=V implies that
(6.6)P(λ,b2)-P(λ,b1)=R(λ,b1)VR(λ̅,b1)-R(λ,b2)VR(λ̅,b2)=[R(λ,b1)-R(λ,b2)]VR(λ̅,b1)-R(λ,b2)V[R(λ̅,b1)-R(λ̅,b2)].
Hence, by ∥V∥≤1, we have
(6.7)‖P(λ,b2)-P(λ,b1)‖≤‖R(λ,b1)-R(λ,b2)‖‖R(λ̅,b1)‖+‖R(λ,b2)‖‖R(λ̅,b1)-R(λ̅,b2)‖.
Since the functions R(λ,·) and R(λ̅,·) are monotone nonincreasing, they are bounded; that is, for some K>0, we have ∥R(λ,b)∥≤K and ∥R(λ̅,b)∥≤K for all b∈[b0,∞)𝕋.
Letε>0 be arbitrary. The convergence of R(λ,b) and R(λ̅,b) as b→∞ yields the existence of b3∈[b0,∞)𝕋 such that for every b1,b2∈[b3,∞)𝕋 with b1<b2 we have
(6.8)‖R(ν,b1)-R(ν,b2)‖≤ε(2K),ν∈{λ,λ̅}.
Using estimate (6.8) in inequality (6.7) we obtain for b2>b1≥b3(6.9)‖P(λ,b2)-P(λ,b1)‖<ε(2K)⋅K+ε(2K)⋅K=ε.
This means that the limit P+(λ)∈ℂn×n in (6.2) exists, which completes the proof.By Theorems5.1 and 5.6 we know that the Weyl disks D(λ,b) are closed, convex, and nested as b→∞. Thereore the limit of D(λ,b) as b→∞ is a closed, convex, and nonempty set. This motivates the following definition, which can be found in the special cases of system (𝒮λ) in [26, Theorem  3.3], [1, Theorem  3.6], [2, Definition  4.7], and [3, Theorem  4.12].Definition 6.2 (limiting Weyl disk).
Letα∈Γ and λ∈ℂ∖ℝ. Then the set
(6.10)D+(λ):=⋂b∈[a,∞)TD(λ,b),
is called the limiting  Weyl  disk. The matrix P+(λ) from Theorem 6.1 is called the center of D+(λ) and the matrices R+(λ) and R+(λ̅) from (6.1) its matrix  radii.As a consequence of Theorem5.6, we obtain the following characterization of the limiting Weyl disk.Corollary 6.3.
Letα∈Γ and λ∈ℂ∖ℝ. Under Hypothesis 5.2, we have
(6.11)D+(λ)=P+(λ)+R+(λ)VR+(λ̅),
where 𝒱 is the set of all contractive matrices defined in (5.15).From now on we assume that Hypothesis5.2 holds, so that the limiting center P+(λ) and the limiting matrix radii R+(λ) and R+(λ̅) of D+(λ) are well defined.Remark 6.4.
By means of the nesting property of the disks (Theorem5.1) and Theorems 4.10 and 4.12, it follows that the elements of the limiting Weyl disk D+(λ) are of the form
(6.12)M+(λ)∈D+(λ),M+(λ)=lim⁡b→∞M(λ,b,α,β(b)),
where β(b)∈ℂn×2n satisfies β(b)β*(b)=I and iδ(λ)β(b)𝒥β*(b)≥0 for all b∈[a,∞). Moreover, from Lemma 4.6, we conclude that
(6.13)M+*(λ)=M+(λ̅).
A matrix M+(λ) from (6.12) is called a half-line  Weyl-TitchmarshM(λ)-function. Also, as noted in [2, Section  4], see also [8, Theorem  2.18], the function M+(λ) is a Herglotz function with rank n and has a certain integral representation (which will not be needed in this paper).Our next result shows another characterization of the elements ofD+(λ) in terms of the Weyl solution 𝒳(·,α,λ,M) defined in (4.16). This is a generalization of [11, page  671], [26, equation  (3.2)], [1, Theorem  3.8(i)], [2, Theorem  4.8], and [3, Theorem  4.15].Theorem 6.5.
Letα∈Γ, λ∈ℂ∖ℝ, and M∈ℂn×n. The matrix M belongs to the limiting Weyl disk D+(λ) if and only if
(6.14)∫a∞Xσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt≤Im⁡(M)Im⁡(λ).Proof.
By Definition6.2, we have M∈D+(λ) if and only if M∈D(λ,b), that is, ℰ(M,b)≤0, for all b∈[a,∞)𝕋. Therefore, by formula (4.18), we get
(6.15)∫abXσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt=E(M,b)2|Im⁡(λ)|+δ(λ)Im⁡(M)|Im⁡(λ)|≤Im⁡(M)Im⁡(λ),
for every b∈[a,∞)𝕋, which is equivalent to inequality (6.14).Remark 6.6.
In [1, Definition  3.4], the notion of a boundary of the limiting Weyl disk D+(λ) is discussed. This would be a “limiting Weyl circle” according to Definitions 4.9 and 6.2. The description of matrices M∈ℂn×n laying on this boundary follows from Theorems 6.5 and 4.10, giving for such matrices M the equality
(6.16)∫a∞Xσ*(t,λ,α,M)W̃(t)Xσ(t,λ,α,M)Δt=Im⁡(M)Im⁡(λ).
Condition (6.16) is also equivalent to
(6.17)lim⁡t→∞X*(t,λ,α,M)JX(t,λ,α,M)=0.
This is because, by (4.19) and the Lagrange identity (Corollary 3.6),
(6.18)X*(t,λ,α,M)JX(t,λ,α,M)=2iIm⁡(λ)[Im⁡(M)Im⁡(λ)-∫atXσ*(s,λ,α,M)W̃(s)Xσ(s,λ,α,M)Δs],
for every t∈[a,∞)𝕋. From this we can see that the integral on the right-hand side above converges for t→∞ and (6.16) holds if and only if condition (6.17) is satisfied. Characterizations (6.16) and (6.17) of the matrices M on the boundary of the limiting Weyl disk D+(λ) generalize the corresponding results in [1, Theorems  3.8(ii) and  3.9]; see also [14, Theorem  6.3].Consider the linear space of square integrableCprd1 functions (6.19)LW2=LW2[a,∞)T:={z:[a,∞)T⟶C2n,z∈Cprd1,‖z(⋅)‖W<∞},
where we define (6.20)‖z(⋅)‖W:=〈z(⋅),z(⋅)〉W,〈z(⋅),z̃(⋅)〉W:=∫a∞zσ*(t)W̃(t)z̃σ(t)Δt.
In the following result we prove that the space L𝒲2 contains the columns of the Weyl solution 𝒳(·,λ,α,M) when M belongs to the limiting Weyl disk D+(λ). This implies that there are at least n linearly independent solutions of system (𝒮λ) in L𝒲2. This is a generalization of [11, Theorem  5.1], [14, Theorem  4.1], [2, Theorem  4.10], and [5, page  716].Theorem 6.7.
Letα∈Γ, λ∈ℂ∖ℝ, and M∈D+(λ). The columns of 𝒳(·,λ,α,M) form a linearly independent system of solutions of system (𝒮λ), each of which belongs to L𝒲2.Proof.
Letzj(·):=𝒳(·,λ,α,M)ej for j∈{1,…,n} be the columns of the Weyl solution 𝒳(·,λ,α,M), where ej is the jth unit vector. We prove that the functions z1(·),…,zn(·) are linearly independent. Assume that ∑j=1ncjzj(·)=0 on [a,∞)𝕋 for some c1,…,cn∈ℂ. Then 𝒳(·,λ,α,M)c=0, where c:=(c1*,…,cn*)*∈ℂn. It follows by (4.19) that
(6.21)2ic*Im⁡(M)c=c*X*(a,λ,α,M)JX(a,λ,α,M)c=0,
which implies the equality c*δ(λ)Im(M)c=0. Using that M∈D+(λ)⊆D(λ,b) for some b∈[b0,∞)𝕋, we obtain from Theorem 4.13 that the matrix δ(λ)Im(M) is positive definite. Hence, c=0 so that the functions z1(·),…,zn(·) are linearly independent. Finally, for every j∈{1,…,n} we get from Theorem 6.5 the inequality
(6.22)‖zj(⋅)‖W2=∫a∞zjσ*(t)W̃(t)zjσ(t)Δt≤(6.14)ej*Im⁡(M)Im⁡(λ)ej≤‖δ(λ)Im⁡(M)‖|Im⁡(λ)|<∞.
Thus, zj(·)∈L𝒲2 for every j∈{1,…,n}, and the proof is complete.Denote by𝒩(λ) the linear space of all square integrable solutions of system (𝒮λ), that is, (6.23)N(λ):={z(⋅)∈LW2,z(⋅)solves(Sλ)}.
Then as a consequence of Theorem 6.7 we obtain the estimate (6.24)dim⁡⁡N(λ)≥n,foreachλ∈C∖R.
Next we discuss the situation when dim𝒩(λ)=n for some λ∈ℂ∖ℝ.Lemma 6.8.
Letα∈Γ, λ∈ℂ∖ℝ, and dim𝒩(λ)=n. Then the matrix radii of the limiting Weyl disk D+(λ) satisfy R+(λ)=0=R+(λ̅). Consequently, the set D+(λ) consists of the single matrix M=P+(λ), that is, the center of D+(λ), which is given by formula (6.2) of Theorem 6.1.Proof.
With the matrix radiiR+(λ) and R+(λ̅) of D+(λ) defined in (6.1) and with the Weyl solution 𝒳(·,λ,α,M) given by a matrix M∈D+(λ), we observe that the columns of 𝒳(·,λ,α,M) form a basis of the space 𝒩(λ). Since the columns of the fundamental matrix Ψ(·,λ,α)=(Z(·,λ,α)Z̃(·,λ,α)) span all solutions of system (𝒮λ), the definition of 𝒳(·,λ,α,M)=Z(·,λ,α)+Z̃(·,λ,α)M yields that the columns of Z̃(·,λ,α) together with the columns of 𝒳(·,λ,α,M) form a basis of all solutions of system (𝒮λ). Hence, from dim𝒩(λ)=n and Theorem 6.7, we get that the columns of Z̃(·,λ,α) do not belong to L𝒲2. Consequently, by formula (5.5), the Hermitian matrix functions ℋ(·,λ,α) and ℋ(·,λ̅,α) defined in (5.4) are monotone nondecreasing on [a,∞)𝕋 without any upper bound; that is, their eigenvalues—being real—tend to ∞. Therefore, the functions R(λ,·) and R(λ̅,·) as defined in (5.18) have limits at ∞ equal to zero; that is, R+(λ)=0 and R+(λ̅)=0. The fact that the set D+(λ)={P+(λ)} then follows from the characterization of D+(λ) in Corollary 6.3.In the final result of this section, we establish another characterization of the matricesM from the limiting Weyl disk D+(λ). In comparison with Theorem 6.5, we now use a similar condition to the one in Theorem 4.12 for the regular spectral problem. However, a stronger assumption than Hypothesis 5.2 is now required for this result to hold; compare with [9, Lemma  2.21] and [2, Theorem  4.16].Hypothesis 6.9.
For everya0,b0∈(a,∞)𝕋 with a0<b0 and for every λ∈ℂ, we have
(6.25)∫a0b0Ψσ*(t,λ,α)W̃(t)Ψσ(t,λ,α)Δt>0.Under Hypothesis6.9, the Weyl disks D(λ,b) converge to the limiting disk “monotonically” as b→∞; that is, the limiting Weyl disk D+(λ) is “open” in the sense that all of its elements lie inside D+(λ). This can be interpreted in view of Theorem 4.12 as ℰ(M,t)<0 for all t∈[a,∞)𝕋.Theorem 6.10.
Letα∈Γ, λ∈ℂ∖ℝ, and M∈ℂn×n. Under Hypothesis 6.9, the matrix M∈D+(λ) if and only if
(6.26)E(M,t)<0,∀t∈[a,∞)T.Proof.
If condition (6.26) holds, then M∈D+(λ) follows from the definition of D+(λ). Conversely, suppose that M∈D+(λ), and let t∈[a,∞)𝕋 be given. Then for any b∈(t,∞)𝕋 we have by formula (4.18) that
(6.27)E(M,t)=-2δ(λ)Im⁡(M)+2|Im⁡(λ)|∫atXσ*(s,λ,α,M)W̃(s)Xσ(s,λ,α,M)Δs=E(M,b)-2|Im⁡(λ)|∫tbXσ*(s,λ,α,M)W̃(s)Xσ(s,λ,α,M)Δs,
where we used the property ∫atf(s)Δs=∫abf(s)Δs-∫tbf(s)Δs. Since M∈D+(λ) is assumed, we have M∈D(λ,b), that is, ℰ(M,b)≤0, while Hypothesis 6.9 implies the positivity of the integral over [t,b]𝕋 in (6.27). Consequently, (6.27) yields that ℰ(M,t)<0.Remark 6.11.
If we partition the Weyl solution𝒳(·,λ):=𝒳(·,λ,α,M) into two n×n blocks 𝒳1(·,λ) and 𝒳2(·,λ) as in (4.28), then condition (6.26) can be written as
(6.28)δ(λ)Im⁡(X1*(t,λ)X2(t,λ))>0,∀t∈[a,∞)T.
Therefore, by Remark 2.2, the matrices 𝒳1(t,λ) and 𝒳2(t,λ) are invertible for all t∈[a,∞)𝕋. A standard argument then yields that the quotient Q(·,λ):=𝒳2(·,λ)𝒳1-1(·,λ) satisfies the Riccati  matrix  equation (suppressing the argument t in the coefficients)
(6.29)QΔ-(C+DQ)+Qσ(A+BQ)+λW[I+μ(A+BQ)]=0,t∈[a,∞)T,
see [57, Theorem  3], [48, Section  6], and [49].


## 7. Limit Point and Limit Circle Criteria

Throughout this section we assume that Hypothesis5.2 is satisfied. The results from Theorem 6.7 and Lemma 6.8 motivate the following terminology; compare with [4, page  75], [43, Definition  1.2] in the time scales scalar case n=1, with [8, page  3486], [36, page  1668], [30, page  274], [38, Definition  3.1], [37, Definition  1], [67, page  2826] in the continuous case, and with [14, Definition  5.1], [2, Definition  4.12] in the discrete case.Definition 7.1 (limit point and limit circle case for system(𝒮λ)).
The system (𝒮λ) is said to be in the limit  point  case at ∞ (or of the limit  point  type) if
(7.1)dim⁡⁡N(λ)=n,∀λ∈C∖R.
The system (𝒮λ) is said to be in the limit  circle  case at ∞ (or of the limit  circle  type) if
(7.2)dim⁡⁡N(λ)=2n,∀λ∈C∖R.Remark 7.2.
According to Remark6.4 (in which β(b)≡β), the center P+(λ) of the limiting Weyl disk D+(λ) can be expressed in the limit point case as
(7.3)P+(λ)=M+(λ)=lim⁡b→∞M(λ,b,α,β),
where β∈Γ is arbitrary but fixed.Next we establish the first main result of this section. Its continuous time version can be found in [30, Theorem  2.1], [11, Theorem  8.5] and the discrete time version in [9, Lemma  3.2], [2, Theorem  4.13].Theorem 7.3.
Let the system (𝒮λ) be in the limit point or limit circle case, fix α∈Γ, and let λ,ν∈ℂ∖ℝ. Then
(7.4)lim⁡t→∞X+*(t,λ,α,M+(λ))JX+(t,ν,α,M+(ν))=0,
where 𝒳+(·,λ,α,M+(λ)) and 𝒳+(·,ν,α,M+(ν)) are the Weyl solutions of (𝒮λ) and (𝒮ν), respectively, defined by (4.16) through the matrices M+(λ) and M+(ν), which are determined by the limit in (6.12).Proof.
For everyt∈[a,∞)𝕋 and matrices β(t)∈ℂn×2n such that β(t)β*(t)=I and iδ(λ)β(t)𝒥β*(t)≥0 and for κ∈{λ,ν}, we define the matrix (compare with Definition 4.5)
(7.5)M(κ,t,α,β(t)):=-[β(t)Z̃(t,κ,α)]-1β(t)Z(t,κ,α).
Then, by Theorems 4.10 and 4.12, we have M(κ,t,α,β(t))∈D(κ,t). Following the notation in (4.16), we consider the Weyl solutions 𝒳(·,κ):=𝒳(·,κ,α,M(κ,t,α,β(·))). Similarly, let 𝒳+(·,κ):=𝒳(·,κ,α,M+(κ)) be the Weyl solutions corresponding to the matrices M+(κ)∈D+(κ) from the statement of this theorem.
First assume that the system (𝒮λ) is of the limit point type. In this case, by Remark 7.2, we may take β(t)∈Γ for all t∈[a,∞)𝕋. Hence, from Theorem 4.10, we get that β(·)𝒳(·,κ)=0 on [a,∞)𝕋. By (4.3), for each t∈[a,∞)𝕋 and κ∈{λ,ν}, there is a matrix Qκ(t)∈ℂn×n such that 𝒳(·,κ)=𝒥β*(·)Qκ(·) on [a,∞)𝕋. Hence, we have on [a,∞)𝕋(7.6)X+*(t,λ)JX+(t,ν)+F(t,λ,ν,β(t))+G(t,λ,ν,β(t))=X*(t,λ)JX(t,ν)=Qλ*(t)β(t)Jβ*(t)Qν(t)=0,
where we define
(7.7)F(t,λ,ν,β(t)):=X+*(t,λ)JZ̃(t,ν,α)[M(ν,t,α,β(t))-M+(ν)],G(t,λ,ν,β(t)):=[M*(λ,t,α,β(t))-M+*(λ)]Z̃*(t,λ,α)JX(t,ν).
If we show that
(7.8)lim⁡t→∞F(t,λ,ν,β(t))=0,lim⁡t→∞G(t,λ,ν,β(t))=0,
then (7.6) implies the result claimed in (7.4). First we prove the second limit in (7.8). Pick any t∈[b0,∞)𝕋. By Theorem 5.6, Corollary 6.3, and D+(λ)⊆D(λ,t), we have
(7.9)M(λ,t,α,β(t))=P(λ,t)+R(λ,t)U(t)R(λ̅,t),M+(λ)=P(λ,t)+R(λ,t)V(t)R(λ̅,t),
where U(t)∈𝒰 and V(t)∈𝒱. Therefore,
(7.10)M(λ,t,α,β(t))-M+(λ)=R(λ,t)[U(t)-V(t)]R(λ̅,t).
Since Z̃(·,λ,α) and 𝒳(·,ν) are, respectively, solutions of systems (𝒮λ) and (𝒮ν) which satisfy Z̃*(a,λ,α)𝒥𝒳(a,ν)=-I, it follows from Corollary 3.6 that
(7.11)Z̃*(t,λ,α)JX(t,ν)=-I+(λ̅-ν)∫atZ̃σ*(s,λ,α)W̃(s)Xσ(s,ν)Δs.
Hence, we can write
(7.12)G(t,λ,ν,β(t))=R(λ̅,t)[U*(t)-V*(t)]R(λ,t)[(λ̅-ν)∫atZ̃σ*(s,λ,α)W̃(s)Xσ(s,ν)Δs-I],
where we used the Hermitian property of R(λ,t) and R(λ̅,t). Since we now assume that system (𝒮λ) is in the limit point case, we know from Lemma 6.8 that limt→∞R(λ,t)=0 and limt→∞R(λ̅,t)=0. Therefore, in order to establish (7.8)(ii), it is sufficient to show that
(7.13)R(λ,t)∫atZ̃σ*(s,λ,α)W̃(s)Xσ(s,ν)Δs,
is bounded for t∈[b0,∞)𝕋. Let η∈ℂn be a unit vector, and denote by 𝒳j(·,ν):=𝒳(·,ν)ej the jth column of 𝒳(·,ν) for j∈{1,…,n}. With the definition of R(λ,·) in (5.18) we have
(7.14)|∫atη*R(λ,s)Z̃σ*(s,λ,α)W̃(s)Xjσ(s,ν)Δs|≤∫at|W̃1/2(s)Z̃σ*(s,λ,α)R(λ,s)η||W̃1/2(s)Xjσ(s,ν)|Δs≤C-S(∫atη*R(λ,s)Z̃σ*(s,λ,α)W̃(s)Z̃σ(s,λ,α)R(λ,s)ηΔs)1/2×(∫atXjσ*(s,ν)W̃(s)Xjσ(s,ν)Δs)1/2,
where the last step follows from the Cauchy-Schwarz inequality (C-S) on time scales. From (5.5) we obtain
(7.15)H-1/2(t,λ,α)∫atZ̃σ*(s,λ,α)W̃(s)Z̃σ(s,λ,α)ΔsH-1/2(t,λ,α)=12|Im⁡(λ)|I,
so that the first term in the product in (7.14) is bounded by 1/2|Im(λ)|. Moreover, from formula (4.18) we get that the second term in the product in (7.14) is bounded by the number [ej*Im(M(ν,t,α,β(t)))ej]/Im(ν). Hence, upon recalling the limit in (6.12), we conclude that the product in (7.14) is bounded by
(7.16)12|Im⁡(λ)|⋅ej*Im⁡(M+(ν))ejIm⁡(ν),
which is independent of t. Consequently, the second limit in (7.8) is established. The first limit in (7.8) is then proven in a similar manner. The proof for the limit point case is finished.
If the system (𝒮λ) is in the limit circle case, then for κ∈{λ,ν} the columns of Z̃(·,κ,α) and 𝒳+(·,κ) belong to L𝒲2; hence, they are bounded in the L𝒲2 norm. In this case the limits in (7.8) easily follow from the limit (6.12) for M+(κ), κ∈{λ,ν}.In the next result we provide a characterization of the system (𝒮λ) being of the limit point type. Special cases of this statement can be found, for example, in [14, Theorem  6.12] and [2, Theorem  4.14].Theorem 7.4.
Letα∈Γ. The system (𝒮λ) is in the limit point case if and only if, for every λ∈ℂ∖ℝ and every square integrable solutions z1(·,λ) and z2(·,λ̅) of (𝒮λ) and (𝒮λ̅), respectively, we have
(7.17)z1*(t,λ)Jz2(t,λ̅)=0,∀t∈[b0,∞)T.Proof.
Let (𝒮λ) be in the limit point case. Fix any λ∈ℂ∖ℝ, and suppose that z1(·,λ) and z2(·,λ̅) are solutions of (𝒮λ) and (𝒮λ̅), respectively. Then, by Theorem 6.7 and Remark 6.4, there are vectors ξ1,ξ2∈ℂn such that z1(·,λ)=𝒳+(·,λ)ξ1 and z2(·,λ̅)=𝒳+(·,λ̅)ξ2 on [a,∞)𝕋, where 𝒳+(·,κ):=𝒳+(·,κ,α,M+(κ)) are the Weyl solutions corresponding to some matrices M+(κ)∈D+(κ) for κ∈{λ,λ̅}. In fact, by Lemma 6.8, the matrix M+(κ) is equal to the center of the disk D+(κ). It follows that for any t∈[b0,∞)𝕋 equality
(7.18)X+*(t,λ)JX+(t,λ̅)=(4.16)(IM+*(λ))Ψ*(t,λ,α)JΨ(t,λ̅,α)(IM+*(λ̅))*=(3.19)(i)M+*(λ̅)-M+*(λ)=(6.13)0,
holds, so that (7.17) is established. Conversely, let ν∈ℂ∖ℝ be arbitrary but fixed, set λ:=ν̅, and suppose that, for every square integrable solutions z1(·,λ) and z2(·,ν) of (𝒮λ) and (𝒮ν), condition (7.17) is satisfied. From Theorem 6.7 we know that for M+(κ)∈D+(κ) the columns 𝒳+[j](·,κ), j∈{1,…,n}, of the Weyl solution 𝒳+(·,κ) are linearly independent square integrable solutions of (𝒮κ), κ∈{λ,ν}. Therefore, dim𝒩(λ)≥n, and dim𝒩(ν)≥n. Moreover, by identity (3.19)(i), we have
(7.19)X+*(t,λ)JX+[j](t,ν)=0,∀t∈[b0,∞)T,j∈{1,…,n}.
Let z(·,ν) be any square integrable solution of system (𝒮ν). Then, by our assumption (7.17),
(7.20)X+*(t,λ)Jz(t,ν)=0,∀t∈[b0,∞)T.
From (7.19) and (7.20) it follows that the vectors 𝒳+[j](a,ν), j∈{1,…,n}, and z(a,ν) are solutions of the linear homogeneous system
(7.21)X+*(a,λ)Jη=0.
Since, by Theorem 6.7, the vectors 𝒳+[j](a,ν) for j∈{1,…,n} represent a basis of the solution space of system (7.21), there exists a vector ξ∈ℂn such that z(a,ν)=𝒳+(a,ν)ξ. By the uniqueness of solutions of system (𝒮ν) we then get z(·,ν)=𝒳+(·,ν)ξ on [a,∞)𝕋. Hence, the solution z(·,ν) is square integrable and dim𝒩(ν)=n. Since ν∈ℂ∖ℝ was arbitrary, it follows that the system (𝒮λ) is in the limit point case.As a consequence of the above result, we obtain a characterization of the limit point case in terms of a condition similar to (7.17), but using a limit. This statement is a generalization of [30, Corollary  2.3], [9, Corollary  3.3], [14, Theorem  6.14], [2, Corollary  4.15], [1, Theorem  3.9], [3, Theorem  4.16].Corollary 7.5.
Letα∈Γ. The system (𝒮λ) is in the limit point case if and only if, for every λ,ν∈ℂ∖ℝ and every square integrable solutions z1(·,λ) and z2(·,ν) of (𝒮λ) and (𝒮ν), respectively, we have
(7.22)lim⁡t→∞z1*(t,λ)Jz2(t,ν)=0.Proof.
The necessity follows directly from Theorem7.3. Conversely, assume that condition (7.22) holds for every λ,ν∈ℂ∖ℝ and every square integrable solutions z1(·,λ) and z2(·,ν) of (𝒮λ) and (𝒮ν). Fix λ∈ℂ∖ℝ, and set ν:=λ̅. By Corollary 3.7 we know that z1*(·,λ)𝒥z2(·,ν) is constant on [a,∞)𝕋. Therefore, by using condition (7.22), we can see that identity (7.17) must be satisfied, which yields by Theorem 7.4 that the system (𝒮λ) is of the limit point type.


## 8. Nonhomogeneous Time Scale Symplectic Systems

In this section we consider the nonhomogeneous time scale symplectic system(8.1)zΔ(t,λ)=S(t,λ)z(t,λ)-JW̃(t)fσ(t),t∈[a,∞)T,
where the matrix function 𝒮(·,λ) and 𝒲̃(·) are defined in (3.3) and (3.1), f∈L𝒲2, and where the associated homogeneous system (𝒮λ) is either of the limit point or limit circle type at ∞. Together with system (8.1) we consider a second system of the same form but with a different spectral parameter and a different nonhomogeneous term (8.2)yΔ(t,ν)=S(t,ν)y(t,ν)-JW̃(t)gσ(t),t∈[a,∞)T,
with g∈L𝒲2. The following is a generalization of Theorem 3.5 to nonhomogeneous systems.Theorem 8.1 (Lagrange identity).
Letλ,ν∈ℂ and m∈ℕ be given. If z(·,λ) and y(·,ν) are 2n×m solutions of systems (8.1) and (8.2), respectively, then
(8.3)[z*(t,λ)Jy(t,ν)]Δ=(λ̅-ν)zσ*(t,λ)W̃(t)yσ(t,ν)-fσ*(t)W̃(t)yσ(t,ν)+zσ*(t,λ)W̃(t)gσ(t),t∈[a,∞)T.Proof.
Formula (8.3) follows by the product rule (2.1) with the aid of the relation
(8.4)zσ(t,λ)=[I+μ(t)S(t,λ)]z(t,λ)+μ(t)W̃(t)fσ(t),
and identity (3.6).Forα∈Γ, λ∈ℂ∖ℝ, and t,s∈[a,∞)𝕋, we define the function (8.5)G(t,s,λ,α):={Z̃(t,λ,α)X+*(s,λ̅,α),fort∈[a,s)T,X+(t,λ,α)Z̃*(s,λ̅,α),fort∈[s,∞)T,
where Z̃(·,λ,α) is the solution of system (𝒮λ) given in (4.10), that is, Z̃(a,λ,α)=-𝒥α*, and 𝒳+(·,λ,α):=𝒳(·,λ,α,M+(λ)) is the Weyl solution of (𝒮λ) as in (4.16) determined by a matrix M+(λ)∈D+(λ). This matrix M+(λ)∈D+(λ) is arbitrary but fixed throughout this section. By interchanging the order of the arguments t and s, we have (8.6)G(t,s,λ,α)={X+(t,λ,α)Z̃*(s,λ̅,α),fors∈[a,t]T,Z̃(t,λ,α)X+*(s,λ̅,α),fors∈(t,∞)T.
In the literature the function G(·,·,λ,α) is called a resolvent kernel, compare with [30, page  283], [32, page  15], [2, equation  (5.4)], and in this section it will play a role of the Green function.Lemma 8.2.
Letα∈Γ and λ∈ℂ∖ℝ. Then
(8.7)X+(t,λ,α)Z̃*(t,λ̅,α)-Z̃(t,λ,α)X+*(t,λ̅,α)=J,∀t∈[a,∞)T.Proof.
Identity (8.7) follows by a direct calculation from the definition of 𝒳+(·,λ,α) via (4.16) with a matrix M+(λ)∈D+(λ) by using formulas (3.21) and (6.13).In the next lemma we summarize the properties of the functionG(·,·,λ,α), which together with Proposition 8.4 and Theorem 8.5 justifies the terminology “Green function” of the system (8.1); compare with [68, Section  4]. A discrete version of the following result can be found in [2, Lemma  5.1].Lemma 8.3.
Letα∈Γ and λ∈ℂ∖ℝ. The function G(·,·,λ,α) has the following properties: (i)
G*(t,s,λ,α)=G(s,t,λ̅,α) for every t,s∈[a,∞)𝕋, t≠s,(ii)
G*(t,t,λ,α)=G(t,t,λ̅,α)-𝒥 for every t∈[a,∞)𝕋,(iii)
G(σ(t),σ(t),λ,α)=[I+μ(t)𝒮(t,λ)]G(t,σ(t),λ,α)+𝒥 for every right-scattered point t∈[a,∞)𝕋,(iv)
for everyt,s∈[a,∞)𝕋 such that t∉𝒯(s), the function G(·,s,λ,α) solves the homogeneous system (𝒮λ) on the set 𝒯(s), where
(8.8)T(s):={τ∈[a,∞)T,τ≠ρ(s)ifsisleft-scattered},(v)
the columns ofG(·,s,λ,α) belong to L𝒲2 for every s∈[a,∞)𝕋, and the columns of G(t,·,λ,α) belong to L𝒲2 for every t∈[a,∞)𝕋.Proof.
Condition (i) follows from the definition ofG(·,s,λ,α) in (8.5). Condition (ii) is a consequence of Lemma 8.2. Condition (iii) is proven from the definition of G(σ(t),σ(t),λ,α) in (8.5) by using Lemma 8.2 and Z̃(t,λ,α)=Z̃σ(t,λ,α)-μ(t)𝒮(t,λ)Z̃(t,λ,α). Concerning condition (iv), the function G(·,s,λ,α) solves the system (𝒮λ) on [s,∞)𝕋 because 𝒳+(·,λ,α) solves this system on [s,∞)𝕋. If s∈(a,∞)𝕋 is left-dense, then G(·,s,λ,α) solves (𝒮λ) on [a,s)𝕋, since Z̃(·,λ,α) solves this system on [a,s)𝕋. For the same reason G(·,s,λ,α) solves (𝒮λ) on [a,ρ(s))𝕋 if s∈(a,∞)𝕋 is left-scattered. Condition (v) follows from the definition of G(·,s,λ,α) in (8.5) used with t≥s and from the fact that the columns of 𝒳+(·,λ,α) belong to L𝒲2, by Theorem 6.7. The columns of G(t,·,λ,α) then belong to L𝒲2 by part (i) of this lemma.Since by Lemma8.3(v) the columns of G(t,·,λ,α) belong to L𝒲2, the function (8.9)ẑ(t,λ,α):=-∫a∞G(t,σ(s),λ,α)W̃(s)fσ(s)Δs,t∈[a,∞)T,
is well defined whenever f∈L𝒲2. Moreover, by using (8.6), we can write ẑ(t,λ,α) as (8.10)ẑ(t,λ,α)=-X+(t,λ,α)∫atZ̃σ*(s,λ̅,α)W̃(s)fσ(s)Δs-Z̃(t,λ,α)∫t∞X+σ*(s,λ̅,α)W̃(s)fσ(s)Δs,t∈[a,∞)T.Proposition 8.4.
Forα∈Γ, λ∈ℂ∖ℝ, and f∈L𝒲2, the function ẑ(·,λ,α) defined in (8.9) solves the nonhomogeneous system (8.1) with the initial condition αẑ(a,λ,α)=0.Proof.
By the time scales product rule (2.1) when we Δ-differentiate expression (8.10), we have for every t∈[a,∞)𝕋 (suppressing the dependence on α in the the following calculation)
(8.11)ẑΔ(t,λ)=-X+Δ(t,λ)∫atZ̃σ*(s,λ̅)W̃(s)fσ(s)Δs-X+σ(t,λ)Z̃σ*(t,λ̅)W̃(t)fσ(t)-Z̃Δ(t,λ)∫t∞X+σ*(s,λ̅)W̃(s)fσ(s)Δs+Z̃σ(t,λ)X+σ*(t,λ̅)W̃(t)fσ(t)=S(t,λ)ẑ(t,λ)-[X+σ(t,λ)Z̃σ*(t,λ̅)-Z̃σ(t,λ)X+σ*(t,λ̅)]W̃(t)fσ(t)=(8.7)S(t,λ)ẑ(t,λ)-JW̃(t)fσ(t).
This shows that ẑ(·,λ,α) is a solution of system (8.1). From (8.10) with t=a, we get
(8.12)αẑ(a,λ,α)=-αZ̃(a,λ,α)∫a∞X+σ*(s,λ̅,α)W̃(s)fσ(s)Δs=0,
where we used the initial condition Z̃(a,λ,α)=-𝒥α* and α𝒥α*=0 coming from α∈Γ.The following theorem provides further properties of the solutionẑ(·,λ,α) of system (8.1). It is a generalization of [10, Lemma  4.2], [11, Theorem  7.5], [2, Theorem  5.2] to time scales.Theorem 8.5.
Letα∈Γ, λ∈ℂ∖ℝ, and f∈L𝒲2. Suppose that system (𝒮λ) is in the limit point or limit circle case. Then the solution ẑ(·,λ,α) of system (8.1) defined in (8.9) belongs to L𝒲2 and satisfies
(8.13)‖ẑ(⋅,λ,α)‖W≤1|Im⁡(λ)|‖f‖W,(8.14)lim⁡t→∞X+*(t,ν,α)Jẑ(t,λ,α)=0,foreveryν∈C∖R.Proof.
To shorten the notation we suppress the dependence onα in all quantities appearing in this proof. Assume first that system (𝒮λ) is in the limit point case. For every r∈[a,∞)𝕋 we define the function fr(·):=f(·) on [a,r]𝕋 and fr(·):=0 on (r,∞)𝕋 and the function
(8.15)ẑr(t,λ):=-∫a∞G(t,σ(s),λ)W̃(s)frσ(s)Δs=-∫arG(t,σ(s),λ)W̃(s)fσ(s)Δs.
For every t∈[r,∞)𝕋 we have as in (8.10) that
(8.16)ẑr(t,λ)=-X+(t,λ)g(r,λ),g(r,λ):=∫arZ̃σ*(s,λ̅)W̃(s)fσ(s)Δs.
Since by Theorem 6.7 the solution 𝒳+(·,λ)∈L𝒲2, (8.16) shows that ẑr(·,λ), being a multiple of 𝒳+(·,λ), also belongs to L𝒲2. Moreover, by Theorem 7.3,
(8.17)lim⁡t→∞ẑr*(t,λ)Jẑr(t,λ)=(8.16)g*(r,λ)lim⁡t→∞X+*(t,λ)JX+(t,λ)g(r,λ)=(7.4)0.
On the other hand, ẑr*(a,λ)𝒥ẑr(a,λ)=0, and for any t∈[a,∞)𝕋 identity (8.3) implies
(8.18)ẑr*(t,λ)Jẑr(t,λ)=-2iIm⁡(λ)∫atẑrσ*(s,λ)W̃(s)ẑrσ(s,λ)Δs+2iIm⁡(∫atẑrσ*(s,λ)W̃(s)frσ(s)Δs).
Combining (8.18), where t→∞, formula (8.17), and the definition on fr(·) yields
(8.19)‖ẑr(⋅,λ)‖W2=∫a∞ẑrσ*(s,λ)W̃(s)ẑrσ(s,λ)Δs=1Im⁡(λ)Im⁡(∫arẑrσ*(s,λ)W̃(s)fσ(s)Δs).
By using the Cauchy-Schwarz inequality (C-S) on time scales and 𝒲̃(·)≥0, we then have
(8.20)‖ẑr(⋅,λ)‖W2=12iIm⁡(λ)[∫arẑrσ*(s,λ)W̃(s)fσ(s)Δs-∫arfσ*(s)W̃(s)ẑrσ(s,λ)Δs]≤1|Im⁡(λ)||∫arẑrσ*(s,λ)W̃(s)fσ(s)Δs|≤C-S1|Im⁡(λ)|(∫arẑrσ*(s,λ)W̃(s)ẑrσ(s,λ)Δs)1/2(∫arfσ*(s)W̃(s)fσ(s)Δs)1/2≤1|Im⁡(λ)|‖ẑr(⋅,λ)‖W‖f‖W.
Since ∥ẑr(·,λ)∥𝒲 is finite by ẑr(·,λ)∈L𝒲2, we get from the above calculation that
(8.21)‖ẑr(⋅,λ)‖W≤1|Im⁡(λ)|‖f‖W.
We will prove that (8.21) implies estimate (8.13) by the convergence argument. For any t,r∈[a,∞)𝕋 we observe that
(8.22)ẑ(t,λ)-ẑr(t,λ)=-∫r∞G(t,σ(s),λ)W̃(s)fσ(s)Δs.
Now we fix q∈[a,r)𝕋. By the definition of G(·,·,λ) in (8.5) we have for every t∈[a,q]𝕋(8.23)ẑ(t,λ)-ẑr(t,λ)=-Z̃(t,λ)∫r∞X+*(σ(s),λ̅)W̃(s)fσ(s)Δs.
Since the functions 𝒳+(·,λ̅) and f(·) belong to L𝒲2, it follows that the right-hand side of (8.23) converges to zero as r→∞ for every t∈[a,q]𝕋. Hence, ẑr(·,λ) converges to the function ẑ(·,λ) uniformly on [a,q]𝕋. Since ẑ(·,λ)=ẑr(·,λ) on [a,q]𝕋, we have by 𝒲̃(·)≥0 and (8.21) that
(8.24)∫aqẑσ*(s,λ)W̃(s)ẑσ(s,λ)Δs≤‖ẑr(⋅,λ)‖W2≤(8.21)1|Im⁡(λ)|2‖f‖W2.
Since q∈[a,∞)𝕋 was arbitrary, inequality (8.24) implies the result in (8.13). In the limit circle case inequality (8.13) follows by the same argument by using the fact that all solutions of system (𝒮λ) belong to L𝒲2.
Now we prove the existence of the limit (8.14). Assume that the system (𝒮λ) is in the limit point case, and let ν∈ℂ∖ℝ be arbitrary. Following the argument in the proof of [30, Lemma  4.1] and [2, Theorem  5.2], we have from identity (8.3) that for any r,t∈[a,∞)𝕋(8.25)X+*(t,ν)Jẑr(t,λ)=X+*(a,ν)Jẑr(a,λ)+(ν̅-λ)∫atX+σ*(s,ν)W̃(s)ẑrσ(s,λ)Δs+∫atX+σ*(s,ν)W̃(s)frσ(s)Δs.
Since for t∈[r,∞)𝕋 equality (8.16) holds, it follows that
(8.26)lim⁡t→∞X+*(t,ν)Jẑr(t,λ)=-lim⁡t→∞X+*(t,ν)JX+(t,λ)g(r,λ)=(7.4)0.
Hence, by (8.25),
(8.27)X+*(a,ν)Jẑr(a,λ)=(λ-ν̅)∫a∞X+σ*(s,ν)W̃(s)ẑrσ(s,λ)Δs-∫arX+σ*(s,ν)W̃(s)fσ(s)Δs.
By the uniform convergence of ẑr(·,λ) to ẑ(·,λ) on compact intervals, we get from (8.27) with r→∞ the equality
(8.28)X+*(a,ν)Jẑ(a,λ)=(λ-ν̅)∫a∞X+σ*(s,ν)W̃(s)ẑσ(s,λ)Δs-∫a∞X+σ*(s,ν)W̃(s)fσ(s)Δs.
On the other hand, by (8.3), we obtain for every t∈[a,∞)𝕋(8.29)X+*(t,ν)Jẑ(t,λ)=X+*(a,ν)Jẑ(a,λ)+(ν̅-λ)∫atX+σ*(s,ν)W̃(s)ẑσ(s,λ)Δs+∫atX+σ*(s,ν)W̃(s)fσ(s)Δs.
Upon taking the limit in (8.29) as t→∞ and using equality (8.28), we conclude that the limit in (8.14) holds true.
In the limit circle case, the limit in (8.14) can be proved similarly as above, because all the solutions of system (𝒮λ) now belong to L𝒲2. However, in this case, we can apply a direct argument to show that (8.14) holds. By formula (8.10) we get for every t∈[a,∞)𝕋(8.30)X+*(t,ν)Jẑ(t,λ)=-X+*(t,ν)JX+(t,λ)∫atZ̃σ*(s,λ̅)W̃(s)fσ(s)Δs-X+*(t,ν)JZ̃(t,λ)∫t∞X+σ*(s,λ̅)W̃(s)fσ(s)Δs.
The limit of the first term in (8.30) is zero because 𝒳+*(t,ν)𝒥𝒳+(t,λ) tends to zero for t→∞ by (7.4), and it is multiplied by a convergent integral as t→∞. Since the columns of Z̃(·,λ) belong to L𝒲2, the function 𝒳+*(·,ν)𝒥Z̃(·,λ) is bounded on [a,∞)𝕋, and it is multiplied by an integral converging to zero as t→∞. Therefore, formula (8.14) follows.In the last result of this paper we construct another solution of the nonhomogeneous system (8.1) satisfying condition (8.14) and such that it starts with a possibly nonzero initial condition at t=a. It can be considered as an extension of Theorem 8.5.Corollary 8.6.
Letα∈Γ and λ∈ℂ∖ℝ. Assume that (𝒮λ) is in the limit point or limit circle case. For f∈L𝒲2 and v∈ℂn we define
(8.31)z̃(t,λ,α):=X+(t,λ,α)v+ẑ(t,λ,α),∀t∈[a,∞)T,
where ẑ(·,λ,α) is given in (8.9). Then z̃(·,λ,α) solves the system (8.1) with αz̃(a,λ,α)=v,
(8.32)‖z̃(⋅,λ,α)‖W≤1|Im⁡(λ)|‖f‖W+‖X+(⋅,λ,α)v‖W,(8.33)lim⁡t→∞X+*(t,ν,α)Jz̃(t,λ,α)=0,foreveryν∈C∖R.
In addition, if the system (𝒮λ) is in the limit point case, then z̃(·,λ,α) is the only L𝒲2 solution of (8.1) satisfying αz̃(a,λ,α)=v.Proof.
As in the previous proof we suppress the dependence onα. Since the function 𝒳+(·,λ)v solves (𝒮λ), it follows from Proposition 8.4 that z̃(·,λ,α) solves the system (8.1) and αz̃(a,λ)=α𝒳+(a,λ)v=v. Next, z̃(·,λ)∈L𝒲2 as a sum of two L𝒲2 functions. The limit in (8.33) follows from the limit (8.14) of Theorem 8.5 and from identity (7.4), because
(8.34)lim⁡t→∞X+*(t,ν)Jz̃(t,λ)=lim⁡t→∞{X+*(t,ν)JX+(t,λ)v+X+*(t,ν)Jẑ(t,λ)}=0.
Inequality (8.32) is obtained from estimate (8.13) by the triangle inequality.
Now we prove the uniqueness ofz̃(·,λ) in the case of (𝒮λ) being of the limit point type. If z1(·,λ) and z2(·,λ) are two L𝒲2 solutions of (8.1) satisfying αz1(a,λ)=v=αz2(a,λ), then their difference z(·,λ):=z1(·,λ)-z2(·,λ) also belongs to L𝒲2 and solves system (𝒮λ) with αz(·,λ)=0. Since z(·,λ)=Ψ(·,λ)c for some c∈ℂ2n, the initial condition αz(·,λ)=0 implies through (4.7) that z(·,λ)=Z̃(·,λ)d for some d∈ℂn. If d≠0, then z(·,λ)∉L𝒲2, because in the limit point case the columns of Z̃(·,λ) do not belong to L𝒲2, which is a contradiction. Therefore, d=0 and the uniqueness of z̃(·,λ) is established.



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*Source: 738520-2011-03-09.xml*