# Unifying discrete and continuous Weyl -Titchmarsh theory via a class of
linear Hamiltonian systems on Sturmian time scales

Douglas R. Anderson  Department of Mathematics and Computer Science, Concordia
College, Moorhead, MN 56562 USA
visiting the School of Mathematics, the University of New South Wales, Sydney
NSW 2052, Australia<EMAIL_ADDRESS>
###### Abstract.

In this study, we are concerned with introducing Weyl-Titchmarsh theory for a
class of dynamic linear Hamiltonian nabla systems over a half-line on Sturmian
time scales. After developing fundamental properties of solutions and regular
spectral problems, we introduce the corresponding maximal and minimal
operators for the system. Matrix disks are constructed and proved to be nested
and converge to a limiting set. Some precise relationships among the rank of
the matrix radius of the limiting set, the number of linearly independent
square summable solutions, and the defect indices of the minimal operator are
established. Using the above results, a classification of singular dynamic
linear Hamiltonian nabla systems is given in terms of the defect indices of
the minimal operator, and several equivalent conditions on the cases of limit
point and limit circle are obtained, respectively. These results unify and
extend certain classic and recent results on the subject in the continuous and
discrete cases, respectively, to Sturmian time scales.

###### Key words and phrases:

limit point; limit circle; spectral problem; Sturmian time scales; Sturm-
Liouville theory; linear Hamiltonian systems; nabla derivative.

###### 2000 Mathematics Subject Classification:

34B20; 34N05; 47A10; 47B25

##  1\. Introduction

Ahlbrandt [2] introduced the following class of linear discrete Hamiltonian
equations

$\nabla x(t)=H_{y}(t,x(t),y(t-1)),\quad\nabla y(t)=-H_{x}(t,x(t),y(t-1)),$

where $\nabla x(t)=x(t)-x(t-1)$, which yields a linear discrete Hamiltonian
system [5] of the form

$\nabla x(t)=A(t)x(t)+B(t)u(t-1),\quad\nabla u(t)=C(t)x(t)-A^{*}(t)u(t-1),$

where $A$, $B$, and $C$ are $d\times d$ matrices, $A^{*}$ is the complex
conjugate transpose of $A$, and $B$ and $C$ are Hermitian. Shi [41] shifted
the points one unit right and extended the analysis to develop a Weyl-
Titchmarsh theory for linear discrete Hamiltonian systems

(1.1) $J\Delta y(t)=(\lambda W(t)+P(t))R(y)(t),\quad
t\in[0,\infty)\cap\mathbb{Z},$

where $W$ and $P$ are $2d\times 2d$ complex Hermitian matrices with weight
function $W(t)\geq 0$, which is one possible discrete version (see also Clark
and Gesztesy [16]) of the classic form studied by Atkinson [11]

(1.2) $Jy^{\prime}(t)=(\lambda W(t)+P(t))y(t),\quad t\in[0,\infty).$

Shi used the partial right uniform shift operator
$R(y)(t)=(y_{1}^{\operatorname{T}}(t+1),y_{2}^{\operatorname{T}}(t))^{\operatorname{T}}$
with the vector
$y(t)=(y_{1}^{\operatorname{T}}(t),y_{2}^{\operatorname{T}}(t))^{\operatorname{T}}$
for $y_{1},y_{2}\in\mathbb{C}^{d}$ and the canonical symplectic matrix
$J=\left(\begin{smallmatrix}0&-I_{d}\\\ I_{d}&0\end{smallmatrix}\right)$. With
a view toward extending (1.1) to Sturmian time scales, we introduce the linear
Hamiltonian nabla system on Sturmian time scales

(1.3) $Jy^{\nabla}(t)=(\lambda W(t)+P(t))\Upsilon y(t),\quad
t\in[t_{0},\infty)_{\mathbb{T}}:=[t_{0},\infty)\cap\mathbb{T},\quad
J=\left(\begin{smallmatrix}0&-I_{d}\\\ I_{d}&0\end{smallmatrix}\right),$

where $W$ and $P$ are $2d\times 2d$ complex Hermitian left-dense continuous
matrix functions, $y:\mathbb{T}\rightarrow\mathbb{C}^{2d}$ is a nabla
differentiable vector function, and we use the partial left-shift operator
$\Upsilon$ defined by

(1.4) $\Upsilon y(t):=\begin{pmatrix}y_{1}(t)\\\
y_{2}^{\rho}(t)\end{pmatrix}\quad\text{for}\quad
y(t)=\begin{pmatrix}y_{1}(t)\\\ y_{2}(t)\end{pmatrix},\quad
y_{1},y_{2}:[\rho(t_{0}),\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{d}.$

Throughout this work we have the following key assumptions: following Atkinson
[11, Chapter 9], the weight function $W$ satisfies the definiteness conditions

(1.5) $\begin{cases}W(t)=\operatorname{diag}\\{W_{1}(t),W_{2}(t)\\}\geq
0,&W_{j}(t)\geq 0\;\text{is}\;d\times d\;\text{Hermitian},\quad j=1,2,\\\
\displaystyle\int_{t_{0}}^{t}(\Upsilon y)^{*}(s)W(s)\Upsilon y(s)\nabla
s>0,&\forall\;t\in[t_{1},\infty)_{\mathbb{T}}\;\text{for
some}\;t_{1}\in\mathbb{T},\end{cases}$

for every nontrivial solution $y$ of (1.3), and $P$ satisfies the block form

(1.6) $P(t)=\begin{pmatrix}-C(t)&A^{*}(t)\\\ A(t)&B(t)\end{pmatrix},\quad
I_{d}-\nu(t)A(t)\;\text{invertible},\quad t\in[t_{0},\infty)_{\mathbb{T}},$

where $A$, $B$, and $C$ are left-dense continuous $d\times d$ complex matrices
with $B$ and $C$ Hermitian. Using standard notation, $\mathbb{T}$ is an
unbounded Sturmian time scale; the left jump operator $\rho$ is given by
$\rho(t)=\sup\\{s\in\mathbb{T}:s<t\\}$ with the composition $u\circ\rho$
denoted $u^{\rho}$; the graininess function is defined by $\nu(t)=t-\rho(t)$;
and the nabla derivative of $x$ at $t\in\mathbb{T}$, denoted $x^{\nabla}(t)$,
is the vector (provided it exists) given by

$x^{\nabla}(t):=\lim_{s\rightarrow t}\frac{x^{\rho}(t)-x(s)}{\rho(t)-s}.$

Sturmian time scales, introduced in [4], are a specialized class of time
scales (closed, nonempty sets of real numbers) with the property that

(1.7) $\sigma(\rho(t))=\rho(\sigma(t))\quad\text{for all}\quad
t\in[t_{0},\infty)_{\mathbb{T}}.$

This crucial assumption allows us to identify a partial right-shift operator
$\Upsilon^{-1}$ in terms of (1.4), namely

(1.8) $\Upsilon^{-1}y(t):=\begin{pmatrix}y_{1}(t)\\\
y_{2}^{\sigma}(t)\end{pmatrix}\quad\text{for}\quad
y(t)=\begin{pmatrix}y_{1}(t)\\\ y_{2}(t)\end{pmatrix},\quad
y_{1},y_{2}:[\rho(t_{0}),\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{d};$

note that on our Sturmian time scale $\mathbb{T}$, we have
$\Upsilon(\Upsilon^{-1}y)(t)=\Upsilon^{-1}(\Upsilon y)(t)=y(t)$ for all
$t\in[t_{0},\infty)_{\mathbb{T}}$. For more on general time scales using the
nabla derivative, see [10] and [15, Chapter 3].

######  Remark 1.1.

We employ the nabla version in (1.3) for two reasons. One, it reflects the
original form introduced by Ahlbrandt [2], and two, it contains the following
two important dynamic models [7, 9]. The first is the linear Hamiltonian nabla
system on general time scales

(1.9) $x^{\nabla}(t)=A(t)x(t)+B(t)u^{\rho}(t),\quad
u^{\nabla}(t)=\left[C(t)-\lambda\omega(t)\right]x(t)-A^{*}(t)u^{\rho}(t)$

for $t\in[a,b]_{\mathbb{T}}$, where $A$, $B$, $C$ and $\omega$ are $d\times d$
matrices, $B$ and $C$ are Hermitian, $\omega>0$ is positive definite. It is
straightforward to write (1.9) in the form (1.3), by taking

$J=\left(\begin{smallmatrix}0&-I_{d}\\\ I_{d}&0\end{smallmatrix}\right),\quad
y(t)=\left(\begin{smallmatrix}x(t)\\\ u(t)\end{smallmatrix}\right),\quad
P(t)=\left(\begin{smallmatrix}-C(t)&A^{*}(t)\\\
A(t)&B(t)\end{smallmatrix}\right),\quad
W(t)=\left(\begin{smallmatrix}\omega(t)&0\\\ 0&0\end{smallmatrix}\right).$

This model includes the second-order self-adjoint matrix equation [6, 8]

$-(P_{0}X^{\Delta})^{\nabla}(t)+Q(t)X(t)=0$

for Hermitian $P_{0}$ and $Q$ with $P_{0}$ invertible, by taking $A=0$,
$B=(P_{0}^{\rho})^{-1}$, $C=Q$, and $\lambda=0$ in (1.9).

The second important dynamic model is the even-order self-adjoint Sturm-
Liouville dynamic equation

$\displaystyle My(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{n}(-1)^{n-k}\left(p_{n-k}y^{\nabla^{n-k-1}\Delta}\right)^{\Delta^{n-k-1}\nabla}(t)$
$\displaystyle=$
$\displaystyle(-1)^{n}\left(p_{n}y^{\nabla^{n-1}\Delta}\right)^{\Delta^{n-1}\nabla}(t)+\dots-\left(p_{3}y^{\Delta^{2}\nabla}\right)^{\nabla^{2}\Delta}(t)$
$\displaystyle+\left(p_{2}y^{\nabla\Delta}\right)^{\Delta\nabla}(t)-\left(p_{1}y^{\Delta}\right)^{\nabla}(t)+p_{0}(t)y(t),$

which is formally self-adjoint [9], where $p_{n}\neq 0$. We will show (1.1)
can be written in the form of (1.9), where

(1.11) $\displaystyle A=(a_{ij})_{1\leq i,j\leq n}\quad\text{with}\quad
a_{ij}=\begin{cases}1:&\text{if }j=i+1,\;1\leq i\leq n-1,\\\
0:&\text{otherwise,}\end{cases}$ $\displaystyle
B=\operatorname{diag}\left\\{0,\dots,0,\frac{1}{p_{n}^{\rho}}\right\\},\quad
C=\operatorname{diag}\left\\{p_{0},p_{1}^{\rho},p_{2}^{\rho},\ldots,p_{n-1}^{\rho}\right\\}.$

To do this, we introduce the pseudo-derivatives of the function $y$ given by

$\displaystyle y^{[k]}$ $\displaystyle=$ $\displaystyle y^{\nabla^{k}},\quad
0\leq k\leq n-1,\quad y^{[0]}=y^{\nabla^{0}}=y,$ $\displaystyle y^{[n]}$
$\displaystyle=$ $\displaystyle p_{n}y^{\nabla^{n-1}\Delta},$ $\displaystyle
y^{[n+k]}$ $\displaystyle=$ $\displaystyle
p_{n-k}y^{\nabla^{n-k-1}\Delta}-\left(y^{[n+k-1]}\right)^{\Delta}$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{k}(-1)^{k-i}\left(p_{k-i}y^{\nabla^{n-i-1}\Delta}\right)^{\Delta^{k-i}},\quad
1\leq k\leq n-1,$ $\displaystyle y^{[2n]}$ $\displaystyle=$ $\displaystyle
p_{0}y-\left(y^{[2n-1]}\right)^{\nabla}=My.$

Then using the substitution

$x=\left(\begin{smallmatrix}y^{[0]}\\\ y^{[1]}\\\ \vdots\\\
y^{[n-1]}\end{smallmatrix}\right),\quad
u=\left(\begin{smallmatrix}y^{[2n-1]}\\\ y^{[2n-2]}\\\ \vdots\\\
y^{[n]}\end{smallmatrix}\right),$

and the matrices $A$, $B$, and $C$ above in (1.11), we have that

$x^{\nabla}=A(t)x+B(t)u^{\rho},\quad u^{\nabla}=C(t)x-A^{*}(t)u^{\rho},$

and the example is complete.

There is a vast literature on continuous linear Hamiltonian systems, and of
late a growing collection of results on corresponding discrete Hamiltonian
systems. For a few of the many relevant papers in the continuous case, see
[12],[17]$-$[21],[38],[53], and see [2, 5, 13, 16, 33, 34],[40]$-$[49] for
recent work on discrete second-order difference equations and linear
Hamiltonian systems. Some of the fundamental continuous and disrete results
alluded to above have been unified and extended via dynamic equations on time
scales, introduced by Hilger [26]. For scalar Sturm-Liouville results on time
scales, see [1, 4, 35, 37], and for systems see [3, 14, 27]. Turning to Weyl
[52] and Titchmarsh [48] specifically, much has been published on the
continuous Weyl-Titchmarsh theory, for example [23, 24, 25],[28, 29],[30, 31,
32], and three substantial works on the corresponding discrete theory,
Atkinson [11], Clark and Gesztesy [16] and Shi [41]. Recently [51] made a
first start on the scalar theory on time scales. There is yet to be any
published work on a unified continuous and discrete Weyl-Titchmarsh theory,
however, for linear Hamiltonian systems; via this paper we hope to initiate
such an investigation. With Sturmian time scales there is a much broader scope
of discretization options other than just the uniform step size offered by
difference equations. In the analysis that follows, we will largely follow a
development of the theory along the lines of Shi [41].

We will proceed as follows. In Section 2, we introduce fundamental properties
for system (1.3) and introduce a Lagrange identity. Regular spectral problems
are discussed in Section 3 via separated boundary conditions, and a result on
eigenpairs is given. In Section 4 we introduce a weighted Hilbert space to
facilitate a study of maximal and minimal operators. Weyl disks and their
limiting set are the focus of Section 5, while in Section 6 we introduce the
concept of square summable solutions. In Section 7, we give a classification
of singular linear Hamiltonian nabla systems. In the final section we discuss
an alternative form to (1.3) that may also serve as a generalization of (1.1)
and (1.2) on Sturmian time scales.

##  2\. Fundamental Properties

For any given $\lambda\in\mathbb{C}$, using the assumptions on the block forms
of $W$ and $P$ in (1.5) and (1.6), respectively, we can rewrite (1.3) as the
pair of $d$-vector equations

(2.1) $\begin{cases}y_{1}^{\nabla}(t)=A(t)y_{1}(t)+\big{(}B(t)+\lambda
W_{2}(t)\big{)}y_{2}^{\rho}(t),\\\ y_{2}^{\nabla}(t)=\big{(}C(t)-\lambda
W_{1}(t)\big{)}y_{1}(t)-A^{*}(t)y_{2}^{\rho}(t).\end{cases}$

By (1.6), we have that

(2.2) $E(t):=\big{(}I_{d}-\nu(t)A(t)\big{)}^{-1}$

exists. Then we may also view solutions $y=(y_{1},y_{2})^{\operatorname{T}}$
of (1.3) and (2.1) as solutions of

(2.3)
$y^{\nabla}(t)=\mathcal{S}(t,\lambda)y(t),\quad\mathcal{S}(\cdot,\lambda):=\begin{pmatrix}A-\nu\big{(}B+\lambda
W_{2}\big{)}E^{*}\big{(}C-\lambda W_{1}\big{)}&\big{(}B+\lambda
W_{2}\big{)}E^{*}\\\ E^{*}\big{(}C-\lambda
W_{1}\big{)}&-E^{*}A^{*}\end{pmatrix}$

for $E$ in (2.2), where it is straightforward to check that
$\mathcal{S}(\cdot,\lambda)$ satisfies

(2.4)
$\mathcal{S}^{*}\left(\cdot,\overline{\lambda}\right)J+J\mathcal{S}(\cdot,\lambda)=\nu\mathcal{S}^{*}\left(\cdot,\overline{\lambda}\right)J\mathcal{S}(\cdot,\lambda)$

for $t\in\mathbb{T}$. Directly from (2.4) we have that

(2.5)
$\left(I_{2d}-\nu(t)\mathcal{S}(t,\lambda)\right)^{*}J\left(I_{2d}-\nu(t)\mathcal{S}(t,\lambda)\right)=J,$

so that $I_{2d}-\nu(t)\mathcal{S}(t,\lambda)$ is invertible and thus
$\mathcal{S}(\cdot,\lambda)$ is $\nu-$regressive. Given the results above, we
now have the following lemma.

######  Lemma 2.1.

Assume $I_{d}-\nu A$ is invertible on $\mathbb{T}$, and $\lambda\in\mathbb{C}$
is arbitrary. Then for any vector solution $y(\cdot,\lambda)$ of (1.3)λ and
for any vector solution $z\left(\cdot,\overline{\lambda}\right)$ of
(1.3)${}_{\overline{\lambda}}$ we have

(2.6)
$z^{*}\left(t,\overline{\lambda}\right)Jy(t,\lambda)=\operatorname{const}.$

###### Proof.

Let $t\in\mathbb{T}$. By using the simple useful formula $y^{\rho}=y-\nu
y^{\nabla}$ and (2.3) we have that

$y^{\rho}(t,\lambda)=y(t,\lambda)-\nu(t)y^{\nabla}(t,\lambda)=\left(I_{d}-\nu(t)\mathcal{S}(t,\lambda)\right)y(t,\lambda).$

From the nabla product rule we subsequently obtain

$\displaystyle\left(z^{*}\left(t,\overline{\lambda}\right)Jy(t,\lambda)\right)^{\nabla}$
$\displaystyle=$ $\displaystyle
z^{*\nabla}\left(t,\overline{\lambda}\right)Jy^{\rho}(t,\lambda)+z^{*}\left(t,\overline{\lambda}\right)Jy^{\nabla}(t,\lambda)$
$\displaystyle=$ $\displaystyle
z^{*}\left(t,\overline{\lambda}\right)\mathcal{S}^{*}\left(t,\overline{\lambda}\right)J\left(I_{d}-\nu(t)\mathcal{S}(t,\lambda)\right)y(t,\lambda)+z^{*}\left(t,\overline{\lambda}\right)J\mathcal{S}(t,\lambda)y(t,\lambda)$
$\displaystyle=$ $\displaystyle
z^{*}\left(t,\overline{\lambda}\right)\left[\mathcal{S}^{*}\left(t,\overline{\lambda}\right)J+J\mathcal{S}(t,\lambda)-\nu(t)\mathcal{S}^{*}\left(t,\overline{\lambda}\right)J\mathcal{S}(t,\lambda)\right]y(t,\lambda)$
$\displaystyle=$ $\displaystyle 0,$

where the last line follows from (2.4). ∎

We now define a natural dynamic nabla differential operator for (1.3) via

(2.7) $\mathscr{L}y(t):=Jy^{\nabla}(t)-P(t)\Upsilon y(t),\quad y\in{\rm
C}_{\rm ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right),$

where ${\rm C}_{\rm ld}^{1}$ is the space of $2d$-vector functions with left-
dense continuous nabla derivatives on the given time scale interval. Then we
have the following key result.

######  Theorem 2.2 (Lagrange Identity).

For all $x,y\in{\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right)$, where
$x=\left(\begin{smallmatrix}x_{1}\\\ x_{2}\end{smallmatrix}\right)$ and
$y=\left(\begin{smallmatrix}y_{1}\\\ y_{2}\end{smallmatrix}\right)$, we have

$\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon
x)^{*}\mathscr{L}y-(\mathscr{L}x)^{*}\Upsilon y\big{\\}}(t)\nabla
t=x^{*}(t)Jy(t)\Big{|}_{\rho(t_{0})}^{b}.$

###### Proof.

Suppressing the variable $t$, we have

$(\Upsilon
x)^{*}\mathscr{L}y=-x_{1}^{*}y_{2}^{\nabla}+x_{1}^{*}Cy_{1}-x_{1}^{*}A^{*}y_{2}^{\rho}+x_{2}^{\rho*}y_{1}^{\nabla}-x_{2}^{\rho*}Ay_{1}-x_{2}^{\rho*}By_{2}^{\rho},$
$(\mathscr{L}x)^{*}\Upsilon
y=-x_{2}^{\nabla*}y_{1}+x_{1}^{*}Cy_{1}-x_{2}^{\rho*}Ay_{1}+x_{1}^{\nabla*}y_{2}^{\rho}-x_{1}^{*}A^{*}y_{2}^{\rho}-x_{2}^{\rho*}By_{2}^{\rho},$

so that when we subtract the second from the first, we obtain

$\displaystyle(\Upsilon x)^{*}\mathscr{L}y-(\mathscr{L}x)^{*}\Upsilon y$
$\displaystyle=$ $\displaystyle-
x_{1}^{*}y_{2}^{\nabla}+x_{2}^{\rho*}y_{1}^{\nabla}+x_{2}^{\nabla*}y_{1}-x_{1}^{\nabla*}y_{2}^{\rho}$
$\displaystyle=$
$\displaystyle-(x_{1}^{*}y_{2})^{\nabla}+(x_{2}^{*}y_{1})^{\nabla}=\left(x^{*}Jy\right)^{\nabla}(t).$

The result follows from the fundamental theorem of calculus. ∎

######  Lemma 2.3.

Assume $I_{d}-\nu A$ is invertible on $\mathbb{T}$. For all
$\lambda,\eta\in\mathbb{C}$, let $y(\cdot,\lambda)$ and $z(\cdot,\eta)$ be any
solutions of (1.3)λ and (1.3)η , respectively. Then for any
$t\in(t_{0},\infty)_{\mathbb{T}}$, we have

$\left(\eta-\overline{\lambda}\right)\int_{\rho(t_{0})}^{t}(\Upsilon
y)^{*}(s,\lambda)W(s)\Upsilon z(s,\eta)\nabla
s=y^{*}(t,\lambda)Jz(t,\eta)\Big{|}_{\rho(t_{0})}^{t}.$

###### Proof.

By (2.7), $\mathscr{L}y(\cdot,\lambda)=\lambda W\Upsilon y(\cdot,\lambda)$ and
$\mathscr{L}z(\cdot,\eta)=\eta W\Upsilon z(\cdot,\eta)$. Use of Theorem 2.2
yields the result. ∎

##  3\. Regular Spectral Problems

In this section we analyze (1.3) over the time scale interval
$[\rho(t_{0}),b]_{\mathbb{T}}$ of finite measure, obtaining some fundamental
spectral results. To this end, consider the regular spectral problem for
system (1.3) on $[\rho(t_{0}),b]_{\mathbb{T}}$ with the separated (homogeneous
Dirichlet) boundary conditions

(3.1) $\alpha y^{\rho}(t_{0})=0,\quad\beta y(b)=0,\qquad y\in{\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right),$

where $\alpha$ and $\beta$ are (normalized) $d\times 2d$ matrices that satisfy
the following self-adjoint boundary conditions

(3.2) $\displaystyle\operatorname{rank}\alpha=d,$
$\displaystyle\alpha\alpha^{*}=I_{d},$ $\displaystyle\alpha J\alpha^{*}=0,$
(3.3) $\displaystyle\operatorname{rank}\beta=d,$
$\displaystyle\beta\beta^{*}=I_{d},$ $\displaystyle\beta J\beta^{*}=0.$

We call these boundary conditions self-adjoint as they cause the Lagrange
identity in Theorem 2.2 to equal zero for $y\in{\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right)$.

######  Lemma 3.1.

Let $\alpha$ and $\beta$ satisfy (3.2) and (3.3), respectively. Then $y\in{\rm
C}_{\rm ld}^{1}\left([\rho(t_{0}),b]_{\mathbb{T}},\mathbb{C}^{2d}\right)$
satisfies (3.1) if and only if there exists a unique vector
$\xi\in\mathbb{C}^{2d}$ such that

(3.4) $y^{\rho}(t_{0})=M\xi,\quad y(b)=N\xi,$

where $M=(-J\alpha^{*},0)$ and $N=(0,J\beta^{*})$. Additionally,

(3.5) $M^{*}JM=0=N^{*}JN,\quad\operatorname{rank}\left(\begin{smallmatrix}M\\\
N\end{smallmatrix}\right)=2d.$

###### Proof.

For (3.4), see the continuous case in Kratz [36, Proposition 2.1.1] or Zettl
[53, Theorem 10.4.3], or in the discrete case in Shi [40, Lemma 2.1]; the
time-scales case is unchanged [6, Lemma 2.4]. Equation (3.5) follows
immediately from (3.2) and (3.3), respectively. ∎

Assume that (1.6) holds on $[\rho(t_{0}),b]_{\mathbb{T}}$. For each
$\lambda\in\mathbb{C}$, let $\Phi(\cdot,\lambda)$ be a fundamental matrix
solution for (1.3)λ. It follows from (2.3) that we can view
$\Phi(\cdot,\lambda)$ as the solution of the initial value problem

$\Phi^{\nabla}=\mathcal{S}(\cdot,\lambda)\Phi,\qquad\Phi^{\rho}(t_{0})=I_{2d},$

and that a general solution of (1.3)λ can be written as

(3.6) $y(t,\lambda)=\Phi(t,\lambda)y^{\rho}(t_{0},\lambda).$

######  Theorem 3.2.

Assume (1.5) and (1.6), and let $\alpha$ and $\beta$ satisfy (3.2) and (3.3),
respectively. Then for each $b\in[t_{1},\infty)_{\mathbb{T}}$ $($where $t_{1}$
is specified in (1.5)$)$, $\lambda$ is an eigenvalue of the boundary value
problem (1.3), (3.1) if and only if

(3.7) $\det\big{(}\Phi(b,\lambda)M-N\big{)}=0$

for the fundamental matrix solution $\Phi$ in (3.6). Moreover, all the
eigenvalues of (1.3) and (3.1) are real and can be numbered serially as in

(3.8) $|\lambda_{0}(b)|\leq|\lambda_{1}(b)|\leq|\lambda_{2}(b)|\leq\cdots,$

such that the corresponding eigenfunctions $y(\cdot,\lambda_{j}(b))$ satisfy
the orthonormality relation

(3.9)
$\Big{\langle}y(\cdot,\lambda_{j}(b)),y(\cdot,\lambda_{k}(b))\Big{\rangle}_{b}:=\int_{\rho(t_{0})}^{b}(\Upsilon
y)^{*}(t,\lambda_{k}(b))W(t)\Upsilon y(t,\lambda_{j}(b))\nabla t=\delta_{jk}.$

###### Proof.

To show (3.7), recall that $\lambda$ is an eigenvalue for the boundary value
problem (1.3), (3.1) with nontrivial eigenfunction $y(\cdot,\lambda)$ if and
only if there exists a vector $\xi\in\mathbb{C}^{2d}$, $\xi\neq 0$, such that
$y^{\rho}(t_{0},\lambda)=M\xi$ and $y(b,\lambda)=N\xi$ by Lemma 3.1 and (3.4),
if and only if $\left(\Phi(b,\lambda)M-N\right)\xi=0$ by (3.6). That the
eigenvalues are real follows from Theorem 2.2 and Lemma 3.1. As in Atkinson
[11, Theorem 9.2.1], we note that $\Phi(b,\lambda)$ consists of entire
functions of $\lambda$, so that the left-hand side of (3.7) is also an entire
function. Thus it has no complex zeros, and its zeros have no finite limit
point, whence we have (3.8), which may be a finite or infinite list and
includes possible multiplicities.

If $\lambda_{j}(b)\neq\lambda_{k}(b)$, then the corresponding eigenfunctions
$y(\cdot,\lambda_{j}(b))$ and $y(\cdot,\lambda_{k}(b))$ are orthogonal by
Theorem 2.2 and Lemma 3.1. These functions can be normalized by taking
$y(t,\lambda_{j}(b))/\|y(\cdot,\lambda_{j}(b))\|_{b}$, where

$\|x\|_{b}:=\big{(}\langle x,x\rangle_{b}\big{)}^{1/2}$

in terms of (3.9). Following Atkinson [11, Section 9.3] and Shi [41, Theorem
2.3], suppose $\lambda_{j}(b)$ is an eigenvalue with multiplicity $d_{j}$. Set

(3.10)
$V_{j}=\left\\{\xi\in\mathbb{C}^{2d}:\left(\Phi(b,\lambda_{j}(b))M-N\right)\xi=0\right\\}.$

It follows that $V_{j}$ is a subspace of $\mathbb{C}^{2d}$ with $\dim
V_{j}=d_{j}$, and $\lambda_{j}(b)$ appears exactly $d_{j}$ times in (3.8), say

$\lambda_{j}(b),\quad j=j^{\prime}+1,\cdots,j^{\prime}+d_{j}.$

We will choose a basis
$\left\\{\xi_{j}\right\\}_{j=j^{\prime}+1}^{j^{\prime}+d_{j}}$ of the set
$V_{j}$ in (3.10) such that the corresponding eigenfunctions

$y(t,\lambda_{j}(b))=\Phi(t,\lambda_{j}(b))M\xi_{j},\quad
j=j^{\prime}+1,\cdots,j^{\prime}+d_{j},$

are mutually orthonormal. We apply a process of orthogonalization ala Atkinson
[11, 9.3.13]. If we write

$u_{j}(b)=y(\rho(t_{0}),\lambda_{j}(b))=M\xi_{j},\quad
y(t,\lambda_{j}(b))=\Phi(t,\lambda_{j}(b))u_{j}(b),$

then (3.9) is equivalent to

(3.11) $u^{*}_{r}(b)K(b,\lambda_{j}(b))u_{s}(b)=\delta_{rs},\quad
j^{\prime}+1\leq r,s\leq j^{\prime}+d_{j},$

where

$K(t,\lambda):=\int_{\rho(t_{0})}^{t}(\Upsilon\Phi)^{*}(\tau,\lambda)W(\tau)\Upsilon\Phi(\tau,\lambda)\nabla\tau,$

and $\Upsilon\Phi(t,\lambda)$ denotes the partial left-shift operator
$\Upsilon$ from (1.4) acting on the last $d$ rows of the fundamental matrix
$\Phi(t,\lambda)$ with respect to the variable $t$. Using the definiteness
condition in (1.5) and the invertibility of $\Phi(\cdot,\lambda_{j}(b))$, we
see that $K(b,\lambda_{j}(b))>0$. On the other hand, the space
$\widetilde{V}_{j}=\\{u:u=M\xi\\}$ has the same dimension $d_{j}$ as $V_{j}$,
as $M\xi=N\xi=0$ always implies $\xi=0$ by (3.5). By the invertibility of
$K(b,\lambda_{j}(b))$, the space

$\widehat{V}_{j}=\\{v:v=(K(b,\lambda_{j}(b)))^{1/2}u,\;u\in\widetilde{V}_{j}\\}$

has dimension $d_{j}$ as well, and an orthonormal basis
$\\{v_{r}\\}_{r=j^{\prime}+1}^{j^{\prime}+d_{j}}$, that is

$v_{r}^{*}v_{s}=\delta_{rs},\quad j^{\prime}+1\leq r,s\leq j^{\prime}+d_{j}.$

From this we recover a basis for $\widetilde{V}_{j}$, namely

$u_{r}=(K(b,\lambda_{j}(b)))^{-1/2}v_{r},\quad j^{\prime}+1\leq r,s\leq
j^{\prime}+d_{j},$

that satisfies (3.11), and (3.9) follows. ∎

##  4\. Maximal and Minimal Operators

In this section we introduce a weighted Hilbert space
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, define minimal and
maximal operators corresponding to system (1.3), and show that the minimal
operator is symmetric, the maximal operator is densely defined, and the
adjoint of the minimal operator is precisely the maximal operator. To clarify
the notation to follow, we will denote the domain, range, and kernel of an
operator $K$ by $\operatorname{Dom}(K)$, $\operatorname{Ran}(K)$, and
$\operatorname{Ker}(K)$, respectively. Some concepts for linear operators in
Hilbert spaces are first introduced; see [50].

######  Definition 4.1.

Let $X$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$, and
let $K:\operatorname{Dom}(K)\subset X\rightarrow X$ be a linear operator.

  1. (i)

$K$ is said to be densely defined if $\operatorname{Dom}(K)$ is dense.

  2. (ii)

$K$ is said to be Hermitian if it is formally self-adjoint, i.e., $\langle
Kf,g\rangle=\langle f,Kg\rangle$ for all $f,g\in\operatorname{Dom}(K)$.

  3. (iii)

$K$ is said to be symmetric if it is Hermitian and densely defined.

  4. (iv)

Let $K$ be a densely defined linear operator. The adjoint operator $K^{*}$ of
$K$ is defined as $\operatorname{Dom}(K^{*})=\\{g\in X:\text{the
functional}\;f\mapsto\langle g,Kf\rangle\;\text{is continuous
on}\;\operatorname{Dom}(K)\\}$ and $\langle K^{*}g,f\rangle=\langle
g,Kf\rangle$ for all $f\in\operatorname{Dom}(K)$ and
$g\in\operatorname{Dom}(K^{*})$.

  5. (v)

For given $\lambda\in\mathbb{C}$, the subspace
$\operatorname{Ran}\left(\overline{\lambda}-K\right)^{\perp}$ is called the
defect space of $K$ and $\lambda$, and
$d(\lambda)=\dim\operatorname{Ran}\left(\overline{\lambda}-K\right)^{\perp}$
is called the defect index of $K$ and $\lambda$.

If $K$ is Hermitian, then $d(\lambda)$ is constant in the upper and lower half
planes, respectively. Denote $d_{+}=d(i)$ and $d_{-}=d(-i)$. Then $d_{+}$ and
$d_{-}$ are called the positive and negative defect indices of $K$,
respectively. Further, if $K$ is densely defined, then
$\overline{\operatorname{Ran}K}\oplus\operatorname{Ker}K^{*}=X$. Hence, if $K$
is symmetric, then
$\operatorname{Ran}\left(\overline{\lambda}-K\right)^{\perp}=\operatorname{Ker}(\lambda-K^{*})$.
It follows that, for the symmetric operator $K$,
$\operatorname{Ran}(-i-K)^{\perp}=\operatorname{Ker}(i-K^{*})$ and
$\operatorname{Ran}(i-K)^{\perp}=\operatorname{Ker}(i+K^{*})$, whereby
$d_{+}=\dim\operatorname{Ker}(i-K^{*})$ and
$d_{-}=\dim\operatorname{Ker}(i+K^{*})$.

We now introduce the following linear spaces. On the time scale half line, let

$L^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):=\left\\{y:[\rho(t_{0}),\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{2d}:\;y\;\text{is
integrable on}\;[\rho(t_{0}),\infty)_{\mathbb{T}}\right\\}$

and let

$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):=\left\\{y\in
L^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):\int_{\rho(t_{0})}^{\infty}\big{(}(\Upsilon
y)^{*}W\Upsilon y\big{)}(t)\nabla t<\infty\right\\}$

for the partial left-shift operator $\Upsilon$ given in (1.4), with inner
product given by

(4.1) $\langle y,z\rangle:=\int_{\rho(t_{0})}^{\infty}\big{(}(\Upsilon
z)^{*}W\Upsilon y\big{)}(t)\nabla t,$

where the weight function $W$ is the $2d\times 2d$ nonnegative Hermitian left-
dense continuous matrix satisfying (1.5). In a similar manner, define on the
finite-length interval the linear space

(4.2)
$L^{1}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):=\left\\{y:[\rho(t_{0}),b]_{\mathbb{T}}\rightarrow\mathbb{C}^{2d}:\;y\;\text{is
integrable on}\;[\rho(t_{0}),b]_{\mathbb{T}}\right\\},$

and let

$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):=\left\\{y\in
L^{1}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):\int_{\rho(t_{0})}^{b}\big{(}(\Upsilon
y)^{*}W\Upsilon y\big{)}(t)\nabla t<\infty\right\\}$

be the space with weighted inner product $\langle\cdot,\cdot\rangle_{b}$
defined in (3.9).

We will use the notation $\|y\|_{W}=(\langle y,y\rangle)^{1/2}$ for $y\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, and
$\|y\|_{b}=(\langle y,y\rangle_{b})^{1/2}$ for $y\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$. As $W$ may be singular,
the inner products for
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ and
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ may not be positive. To
account for this, we introduce the following quotient spaces. For $y,z\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, $y$ and $z$ are said
to be equal iff $\|y-z\|_{W}=0$. In this context
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is an inner product
space with inner product $\langle\cdot,\cdot\rangle$. Likewise functions
$y,z\in L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ are said to be
equal iff $\|y-z\|_{b}=0$, making
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ into an inner product
space with inner product $\langle\cdot,\cdot\rangle_{b}$.

######  Lemma 4.2.

The space $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is a
weighted Hilbert space with inner product $\langle\cdot,\cdot\rangle$ given in
(4.1), and $L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ is a weighted
Hilbert space with inner product $\langle\cdot,\cdot\rangle_{b}$ given in
(3.9). In addition,

$\dim
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)=\int_{\rho(t_{0})}^{b}\operatorname{rank}W(t)\nabla
t.$

###### Proof.

The style of proof is based on that given in the discrete case by Shi [41]. We
will show that $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is
complete, as the proof for
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$ is similar and is
omitted. First, let us consider a simpler case, namely where
$W(t)=\widetilde{W}(t)=\operatorname{diag}\\{0,W_{2}(t)\\}$ for the
$r(t)\times r(t)$ matrix $W_{2}(t)>0$, $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$
and $0\leq r(t)\leq 2d$. For convenience, set

$\widetilde{W}^{\pm 1/2}(t):=\begin{pmatrix}0&0\\\ 0&W_{2}^{\pm
1/2}(t)\end{pmatrix},$

$y(t)=\left(y_{1}^{\operatorname{T}}(t),y_{2}^{\operatorname{T}}(y)\right)^{\operatorname{T}}$
for $y_{i}\in\mathbb{C}^{d}$, $i=1,2$, and
$y(t)=\left(y^{(1)\operatorname{T}}(t),y^{(2)\operatorname{T}}(t)\right)^{\operatorname{T}}$
with $y^{(1)}(t)\in\mathbb{C}^{r(t)}$ and $y^{(2)}(t)\in\mathbb{C}^{2d-r(t)}$.
To prove completeness, assume $\\{f_{n}\\}$ is a Cauchy sequence in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, i.e.,
given $\varepsilon>0$ there exists an $N\in\mathbb{N}$ such that
$\|f_{n}-f_{m}\|_{\widetilde{W}}<\varepsilon$ for all $n,m\geq N$. Define the
functions $g_{n}:[t_{0},\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{2d}$ via

(4.3) $g_{n}(t):=\widetilde{W}^{1/2}(t)\Upsilon f_{n}(t),\qquad n\geq 1,\qquad
t\in[t_{0},\infty)_{\mathbb{T}}.$

Since $f_{n}\in
L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, by (4.3)
we have $g_{n}\in L^{2}[t_{0},\infty)=\left\\{g\in
L^{1}[t_{0},\infty):(g^{*}g)\in L^{1}[t_{0},\infty)\right\\}$, where
$L^{2}[t_{0},\infty)$ is a known Hilbert space with $L^{2}$ norm
$\|g\|_{L^{2}}=\left(\int_{t_{0}}^{\infty}(g^{*}g)(t)\nabla t\right)^{1/2}$
for $g\in L^{2}[t_{0},\infty)$; see [39, Section 4]. It follows that
$\|g_{n}-g_{m}\|_{L^{2}}=\|f_{n}-f_{m}\|_{\widetilde{W}}$. Thus $\\{g_{n}\\}$
is a Cauchy sequence in $L^{2}[t_{0},\infty)$, so by the completeness of
$L^{2}$ there exists an integrable function $g\in L^{2}[t_{0},\infty)$ such
that $\|g_{n}-g\|_{L^{2}}\rightarrow 0$ as $n\rightarrow\infty$. Set

(4.4) $f(t)=\Upsilon^{-1}(\widetilde{W}^{-1/2}g)(t),$

where $\Upsilon^{-1}$ is given in (1.8). Then $\Upsilon
f=\Upsilon(\Upsilon^{-1}\widetilde{W}^{-1/2}g)=\widetilde{W}^{-1/2}g$, so that
$f\in L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. To
see that $f_{n}$ converges to $f$ in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, note
that by (4.3) we have $g_{n}^{(1)}(t)=0$ for all
$t\in[t_{0},\infty)_{\mathbb{T}}$, whence $g^{(1)}(t)=0$ for
$t\in[t_{0},\infty)_{\mathbb{T}}$. As a result, from (4.3) and (4.4) we see
that (suppressing the $t$)

$\displaystyle\left[\Upsilon(f_{n}-f)\right]^{*}\widetilde{W}\Upsilon(f_{n}-f)$
$\displaystyle=$ $\displaystyle\left(\begin{smallmatrix}0\\\
W_{2}^{-1/2}(g_{n}^{(2)}-g^{(2)})\end{smallmatrix}\right)^{*}\left(\begin{smallmatrix}0&0\\\
0&W_{2}\end{smallmatrix}\right)\left(\begin{smallmatrix}0\\\
W_{2}^{-1/2}(g_{n}^{(2)}-g^{(2)})\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle(g_{n}^{(2)}-g^{(2)})^{*}(g_{n}^{(2)}-g^{(2)}),$

which implies that $\|f_{n}-f\|_{\widetilde{W}}=\|g_{n}-g\|_{L^{2}}$, ergo
$f_{n}\rightarrow f$ in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ as
$n\rightarrow\infty$. Consequently,
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is
complete.

For the general case, assume $\operatorname{rank}W(t)=r(t)$ for
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. As $W(t)\geq 0$ and Hermitian, there
exists a unitary matrix $U$ such that

$U^{*}(t)W(t)U(t)=\operatorname{diag}\\{0,W_{2}(t)\\}=:\widetilde{W}(t),\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}},$

where $W_{2}(t)$ is an $r(t)\times r(t)$ positive definite matrix for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. Suppose $\\{f_{n}\\}$ is a Cauchy
sequence in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, and set
$h_{n}(t):=\Upsilon^{-1}U^{*}(\Upsilon f_{n})(t)$. Then

$\displaystyle(\Upsilon h_{n})^{*}\widetilde{W}(\Upsilon h_{n})$
$\displaystyle=$ $\displaystyle(U^{*}(\Upsilon
f_{n}))^{*}\widetilde{W}U^{*}(\Upsilon f_{n})$ $\displaystyle=$
$\displaystyle(\Upsilon f_{n})^{*}U\widetilde{W}U^{*}(\Upsilon
f_{n})=(\Upsilon f_{n})^{*}W(\Upsilon f_{n}),$

so that $h_{n}\in
L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ and
$\|h_{n}-h_{m}\|_{\widetilde{W}}=\|f_{n}-f_{m}\|_{W}$, making $\\{h_{n}\\}$ a
Cauchy sequence in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. From
the discussion earlier for the simpler case, there exists a function $h\in
L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ such that
$h_{n}\rightarrow h$ in
$L^{2}_{\widetilde{W}}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ as
$n\rightarrow\infty$. If we set $f(t)=\Upsilon^{-1}U(\Upsilon h)(t)$, then
$f\in L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ with
$\|f_{n}-f\|_{W}=\|h_{n}-h\|_{\widetilde{W}}$. It follows that
$f_{n}\rightarrow f$ in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ as
$n\rightarrow\infty$, and
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is complete.

To calculate the dimension of
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$, note that for $y\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$, $y=0$ if and only if

$\|y\|^{2}_{b}=\int_{\rho(t_{0})}^{b}(\Upsilon y)^{*}(t)W(t)\Upsilon
y(t)\nabla t=0,$

which is equivalent to $W(t)\Upsilon y(t)=0$ for all
$t\in[t_{0},b]_{\mathbb{T}}$. Thus $\widetilde{W}U^{*}(\Upsilon y)(t)=0$ for
all $t\in[t_{0},b]_{\mathbb{T}}$, and $U(\Upsilon y)(t)$ has exactly $r(t)$
components taking effect on the inner product. As $U(t)$ is invertible, we
have that $\dim
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)=\int_{\rho(t_{0})}^{b}r(t)\nabla
t$, and the proof is complete. ∎

######  Remark 4.3.

The above result establishes that
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ is a Hilbert space
with weighted norm

$\|y\|_{W}^{2}=\int_{\rho(t_{0})}^{\infty}\big{(}(\Upsilon y)^{*}W(\Upsilon
y)\big{)}(t)\nabla t.$

Following [39, Section 4.1], given ${\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right)$ with
the weighted norm

(4.5) $\|y\|_{1}^{2}:=\|y\|_{W}^{2}+\|y^{\nabla}\|_{W}^{2},\quad y\in{\rm
C}_{\rm
ld}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right),$

if we define $\mathcal{H}^{1}_{W}([\rho(t_{0}),\infty)_{\mathbb{T}})\subset
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ to be the completion
of ${\rm C}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right)$
with respect to the norm $\|\cdot\|_{1}$ in (4.5), then
$\mathcal{H}^{1}_{W}([\rho(t_{0}),\infty)_{\mathbb{T}})$ is a time-scale
analogue of the usual Sobolev space $H^{1}(I)$ on a real interval $I$, and
${\rm C}_{\rm
ld}^{1}\left([\rho(t_{0}),\infty)_{\mathbb{T}},\mathbb{C}^{2d}\right)\subset\mathcal{H}^{1}_{W}([\rho(t_{0}),\infty)_{\mathbb{T}})$.

We turn now to definitions of maximal and minimal operators corresponding to
system (1.3). We use $H$ and $H_{0}$ to denote the maximal and minimal
operators over $[\rho(t_{0}),\infty)_{\mathbb{T}}$, respectively, where

$\displaystyle D(H)$ $\displaystyle:=$ $\displaystyle\left\\{y\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right):\;\text{there
exists}\;f\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)\;\text{such
that}\;\right.$ $\displaystyle\left.(\mathscr{L}y)(t)=W(t)(\Upsilon
f)(t),\;t\in[\rho(t_{0}),\infty)_{\mathbb{T}}\right\\},$ (4.6) $\displaystyle
Hy$ $\displaystyle:=$ $\displaystyle f,$ $\displaystyle D(H_{0})$
$\displaystyle:=$ $\displaystyle\left\\{y\in D(H):\;\text{there
exists}\;b\in[\rho(t_{0}),\infty)_{\mathbb{T}}\;\text{such that}\;\right.$
$\displaystyle\left.y^{\rho}(t_{0})=y(t)=0\;\text{for
all}\;t\in[b,\infty)_{\mathbb{T}}\right\\},$ (4.7) $\displaystyle H_{0}y$
$\displaystyle:=$ $\displaystyle Hy.$

In a similar manner, we use $H^{b}$ and $H^{b}_{0}$ to denote the maximal and
minimal operators over $[\rho(t_{0}),b]_{\mathbb{T}}$, respectively, where

$\displaystyle D(H^{b})$ $\displaystyle:=$ $\displaystyle\left\\{y\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right):\;\text{there exists}\;f\in
L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)\;\text{such that}\;\right.$
$\displaystyle\left.(\mathscr{L}y)(t)=W(t)(\Upsilon
f)(t),\;t\in[\rho(t_{0}),b]_{\mathbb{T}}\right\\},$ (4.8) $\displaystyle
H^{b}y$ $\displaystyle:=$ $\displaystyle f,$ $\displaystyle D(H^{b}_{0})$
$\displaystyle:=$ $\displaystyle\left\\{y\in
D(H^{b}):y^{\rho}(t_{0})=y(b)=0\right\\},$ (4.9) $\displaystyle H^{b}_{0}y$
$\displaystyle:=$ $\displaystyle H^{b}y.$

By these definitions it is clear that $H_{0}\subset H$ and $H_{0}^{b}\subset
H^{b}$.

######  Lemma 4.4.

The operators $H_{0}$ and $H^{b}_{0}$ are Hermitian.

###### Proof.

Since the proof is similar for $H^{b}_{0}$, we focus on just $H_{0}$. For any
$y,z\in D(H_{0})$, there exists $b\in[\rho(t_{0}),\infty)_{\mathbb{T}}$ such
that

(4.10) $y^{\rho}(t_{0})=z^{\rho}(t_{0})=0,\quad y(t)=z(t)=0\quad\text{for
all}\quad t\in[b,\infty)_{\mathbb{T}},$

and there exists $f,g\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ such that $H_{0}y=f$
and $H_{0}z=g$, that is to say

$(\mathscr{L}y)(t)=W(t)(\Upsilon f)(t),\quad(\mathscr{L}z)(t)=W(t)(\Upsilon
g)(t),\quad t\in[\rho(t_{0}),\infty)_{\mathbb{T}}.$

From Theorem 2.2 and (4.10) we have that

$\displaystyle\langle H_{0}y,z\rangle-\langle y,H_{0}z\rangle$
$\displaystyle=$ $\displaystyle\langle f,z\rangle-\langle y,g\rangle$
$\displaystyle=$ $\displaystyle\int_{\rho(t_{0})}^{\infty}\big{\\{}(\Upsilon
z)^{*}W(\Upsilon f)-(\Upsilon g)^{*}W(\Upsilon y)\big{\\}}(t)\nabla t$
$\displaystyle=$ $\displaystyle\int_{\rho(t_{0})}^{\infty}\big{\\{}(\Upsilon
z)^{*}\mathscr{L}y-(\mathscr{L}z)^{*}\Upsilon y\big{\\}}(t)\nabla t$
$\displaystyle=$
$\displaystyle\lim_{b\rightarrow\infty}z^{*}(t)Jy(t)\Big{|}_{\rho(t_{0})}^{b}=0.$

Therefore $\langle H_{0}y,z\rangle=\langle y,H_{0}z\rangle$, so that $H_{0}$
is Hermitian. ∎

The following lemma has a similar proof to that just completed.

######  Lemma 4.5.

The relation $\langle H^{b}_{0}y,z\rangle_{b}=\langle y,H^{b}z\rangle_{b}$
holds for all $y\in D(H^{b}_{0})$ and for all $z\in D(H^{b})$, while $\langle
H_{0}y,z\rangle=\langle y,Hz\rangle$ holds for all $y\in D(H_{0})$ and for all
$z\in D(H)$.

######  Lemma 4.6.

If (1.6) holds, then
$\operatorname{Ran}(H^{b}_{0})=\operatorname{Ker}(H^{b})^{\perp}$.

###### Proof.

Given any $f\in\operatorname{Ran}(H_{0}^{b})$, there exists $y\in
D(H_{0}^{b})$ such that $H_{0}^{b}y=f$. For each
$z\in\operatorname{Ker}(H^{b})$, it follows from the previous lemma that
$\langle f,z\rangle_{b}=\langle H_{0}^{b}y,z\rangle_{b}=\langle
y,H^{b}z\rangle_{b}=\langle y,0\rangle_{b}=0$, so that
$f\in\operatorname{Ker}(H^{b})^{\perp}$. Thus
$\operatorname{Ran}(H_{0}^{b})\subset\operatorname{Ker}(H^{b})^{\perp}$. If
$f\in\operatorname{Ker}(H^{b})^{\perp}$, then $\langle f,z\rangle_{b}=0$ for
all $z\in\operatorname{Ker}(H^{b})$. Consider the following initial value
problem:

$Jy^{\nabla}(t)=P(t)\Upsilon y(t)+W(t)\Upsilon f(t),\quad
y^{\rho}(t_{0})=0,\qquad t\in[\rho(t_{0}),b]_{\mathbb{T}}.$

By (1.6), this problem has a unique solution $y$ on
$[\rho(t_{0}),b]_{\mathbb{T}}$. Let
$\Phi(t)=(\varphi_{1},\varphi_{2},\cdots,\varphi_{2d})(t)$ be the fundamental
solution matrix of the homogeneous system

$Jx^{\nabla}(t)=P(t)\Upsilon x(t),\quad\Phi(b)=J,\qquad
t\in[\rho(t_{0}),b]_{\mathbb{T}}.$

Clearly $\varphi_{k}\in\operatorname{Ker}(H^{b})$ for $1\leq k\leq 2d$, so by
Theorem 2.2 and (3.9),

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle
f,\varphi_{k}\rangle_{b}=\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon\varphi_{k})^{*}W\Upsilon
f\big{\\}}(t)\nabla t$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon\varphi_{k})^{*}\mathscr{L}y\big{\\}}(t)\nabla
t$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\big{\\{}(\Upsilon\varphi_{k})^{*}\mathscr{L}y-(\mathscr{L}\varphi_{k})^{*}\Upsilon
f\big{\\}}(t)\nabla t$ $\displaystyle=$
$\displaystyle\varphi^{*}_{k}(b)Jy(b)-\varphi^{*}_{k}(\rho(t_{0}))Jy(\rho(t_{0}))$
$\displaystyle=$ $\displaystyle\varphi^{*}_{k}(b)Jy(b).$

Thus we have that $\Phi^{*}(b)Jy(b)=y(b)=0$, and
$\operatorname{Ker}(H^{b})^{\perp}\subset\operatorname{Ran}(H_{0}^{b})$. ∎

######  Theorem 4.7.

If (1.6) holds, then $H_{0}$ is symmetric and $H$ is densely defined.

###### Proof.

By Lemma 4.4 and the fact that $H_{0}\subset H$, it suffices to show that
$D(H_{0})$ is dense in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, i.e., that
$D(H_{0})^{\perp}=\\{0\\}$. If $f\in D(H_{0})^{\perp}$, then
$D(H^{b}_{0})\subset D(H_{0})$ for all $t\in(\rho(t_{0}),\infty)_{\mathbb{T}}$
in the sense that all the functions of $D(H^{b}_{0})$ are considered to have
been extended by zero on to $[\rho(t_{0}),\infty)_{\mathbb{T}}$. Then, for all
$t\in(\rho(t_{0}),\infty)_{\mathbb{T}}$ and for all $z\in D(H^{b}_{0})$,
$\langle f,z\rangle_{b}=\langle f,z\rangle=0$. Set $H^{b}_{0}(z)(t)=g(t)$ for
$t\in[\rho(t_{0}),b]_{\mathbb{T}}$, and let $y$ be any solution of the system

$Jy^{\nabla}(t)=P(t)\Upsilon y(t)+W(t)\Upsilon f(t),\quad
t\in[\rho(t_{0}),b]_{\mathbb{T}}.$

By Theorem 2.2, we have

$\displaystyle\langle y,g\rangle_{b}-\langle f,z\rangle_{b}$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\left\\{(\Upsilon g)^{*}W\Upsilon
y-(\Upsilon z)^{*}W\Upsilon f\right\\}(t)\nabla t$ $\displaystyle=$
$\displaystyle\int_{\rho(t_{0})}^{b}\left\\{(\mathscr{L}z)^{*}\Upsilon
y-(\Upsilon z)^{*}\mathscr{L}y\right\\}(t)\nabla t$ $\displaystyle=$
$\displaystyle-z^{*}(b)Jy(b)+z^{*}(\rho(t_{0}))Jy(\rho(t_{0}))=0.$

We then have that $\langle y,g\rangle_{b}=\langle f,z\rangle_{b}=0$. It
follows that
$y\in\operatorname{Ran}(H^{b}_{0})^{\perp}=\operatorname{Ker}(H^{b})$ by Lemma
4.6. Therefore, $H^{b}y=0$, and thus
$f\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}=0$ in
$L^{2}_{W}\left([\rho(t_{0}),b]_{\mathbb{T}}\right)$, that is $\|f\|_{b}=0$.
Since $b>\rho(t_{0})$ is arbitrary, it follows that $f=0$ in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. Consequently,
$D(H_{0})^{\perp}=\\{0\\}$. ∎

######  Theorem 4.8.

If (1.6) holds, then $H_{0}^{*}=H$.

###### Proof.

It suffices to show $D(H_{0}^{*})=D(H)$, and $H_{0}^{*}y=Hy$ for all $y\in
D(H_{0}^{*})$. If $y\in D(H)$, then $\langle y,H_{0}z\rangle=\langle
Hy,z\rangle$ for all $z\in D(H_{0})$ by Lemma 4.5. From this we see that the
functional $\langle y,H_{0}(\cdot)\rangle$ is continuous on $D(H_{0})$. Then
$y\in D(H_{0}^{*})$ by (iv) in Definition 4.1 and thus $D(H)\subset
D(H_{0}^{*})$. We now show $D(H_{0}^{*})\subset D(H)$. If $y\in D(H_{0}^{*})$,
then $y$ and $g:=H_{0}^{*}y$ are both in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. If $x$ is a
solution of the system

(4.11) $Jx^{\nabla}(t)=P(t)\Upsilon x(t)+W(t)\Upsilon g(t),$

then for each $z\in D(H_{0})$ there exists
$b\in[\rho(t_{0}),\infty)_{\mathbb{T}}$ such that $z^{\rho}(t_{0})=z(t)=0$ for
all $t\in[b,\infty)_{\mathbb{T}}$. Then
$z\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}\in D(H^{b}_{0})$. This implies from
Lemma 4.5 that

$\langle g,z\rangle=\langle g,z\rangle_{b}=\langle H^{b}x,z\rangle_{b}=\langle
x,H^{b}_{0}z\rangle_{b}=\langle x,H_{0}z\rangle.$

It follows that

$\langle y-x,H_{0}z\rangle=\langle y,H_{0}z\rangle-\langle
x,H_{0}z\rangle=\langle H_{0}^{*}y,z\rangle-\langle g,z\rangle=0.$

Additionally, $\langle y-x,H^{b}_{0}z\rangle_{b}=\langle y-x,H_{0}z\rangle=0$.
Thus
$(y-x)\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}\in\operatorname{Ran}(H^{b}_{0})^{\perp}$.
By Lemma 4.6 we have that
$(y-x)\big{|}_{[\rho(t_{0}),b]_{\mathbb{T}}}\in\operatorname{Ker}(H^{b})$
whereby $H^{b}(y-x)=0$, in other words, $\mathscr{L}(y-x)(t)=0$ for
$t\in[\rho(t_{0}),b]_{\mathbb{T}}$. This, together with (4.11), implies that

$Jy^{\nabla}(t)-P(t)\Upsilon y(t)=Jx^{\nabla}(t)-P(t)\Upsilon
x(t)=W(t)\Upsilon g(t),\quad t\in[\rho(t_{0}),b]_{\mathbb{T}}.$

Since $b\geq\rho(t_{0})$ may be chosen arbitrarily large and $y,g\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, we have that $y\in
D(H)$ and $Hy=g=H_{0}^{*}y$. As a result, $D(H_{0}^{*})\subset D(H)$, whence
$D(H_{0}^{*})=D(H)$ and $H^{*}_{0}y=Hy$ for all $y\in D(H_{0}^{*})$. ∎

##  5\. Weyl disks and their limiting set

In this section, we first construct matrix disks for system (1.3) over time-
scale intervals of finite length. These matrix disks are called Weyl disks
[16], which turn out to be nested and converge to a limiting set. This
limiting set will play a major role in the discussions of square summable
solutions of (1.3). Again we will rely heavily on the organization of the
topic as done by Shi [41] in the discrete case.

Suppose that $\theta(t,\lambda)$ and $\phi(t,\lambda)$ are $2d\times d$
matrix-valued solutions of (1.3) satisfying
$\theta(\rho(t_{0}),\lambda)=\alpha^{*}$ and
$\phi(\rho(t_{0}),\lambda)=J\alpha^{*}$, respectively, where $\alpha$
satisfies (3.2). Then we have from (3.2) that

(5.1)
$\alpha\theta(\rho(t_{0}),\lambda)=I_{d},\quad\alpha\phi(\rho(t_{0}),\lambda)=0.$

Set

(5.2) $Y(t,\lambda):=(\theta,\phi)(t,\lambda),$

so that

(5.3) $Y(\rho(t_{0}),\lambda)=(\alpha^{*},J\alpha^{*})=:\Omega.$

Then we have from (3.2) that $\Omega$ is symplectic and unitary; in other
words,

(5.4) $\Omega^{*}J\Omega=J\quad\text{and}\quad\Omega^{*}\Omega=I_{2d}.$

Therefore, $Y(\cdot,\lambda)$ is a fundamental solution matrix of (1.3) and
satisfies, by (2.6) and from (5.4), that

(5.5) $Y^{*}\left(t,\overline{\lambda}\right)JY(t,\lambda)=J,\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}},$

and thus

(5.6) $Y(t,\lambda)JY^{*}\left(t,\overline{\lambda}\right)=J,\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}}.$

######  Lemma 5.1.

Let $\alpha$ and $\beta$ satisfy (3.2) and (3.3), respectively. Then $\lambda$
is an eigenvalue of the problem (1.3) with boundary conditions (3.1) if and
only if $\det(\beta\phi(b,\lambda))=0$. Additionally, $y(t,\lambda)$ is an
eigenfunction with respect to $\lambda$ if and only if there exists
$\xi\in\mathbb{C}^{d}$ such that $y(t,\lambda)=\phi(t,\lambda)\xi$, where
$\xi\neq 0$ is a solution of the homogeneous linear algebraic system

(5.7) $\beta\phi(b,\lambda)\xi=0.$

###### Proof.

The proof is unchanged from the discrete case, see [41, Lemma 3.1] and [16,
Lemma 2.8]. ∎

In the subsequent development we will be interested in the function $\chi$
given via

(5.8) $\chi(t,\lambda,b):=Y(t,\lambda)\begin{pmatrix}I_{d}\\\
M(\lambda,b)\end{pmatrix},$

where $M(\lambda,b)$ is a $d\times d$ matrix such that
$\beta\chi(b,\lambda,b)=0$, in other words

(5.9) $\beta\theta(b,\lambda)+\beta\phi(b,\lambda)M(\lambda,b)=0.$

Using Lemma 5.1, if $\lambda\in\mathbb{C}$ is not an eigenvalue of the problem
(1.3), (3.1), then $\beta\phi(b,\lambda)$ is invertible, and from (5.9) we
have that

(5.10)
$M(\lambda,b)=-\big{(}\beta\phi(b,\lambda)\big{)}^{-1}\beta\theta(b,\lambda).$

######  Lemma 5.2.

Let $\alpha$ satisfy (3.2). Then for each $b\in[t_{1},\infty)_{\mathbb{T}}$
$($where $t_{1}$ is specified in (1.5)$)$, we have the following.

  1. (i)

$M(\lambda,b)$ is analytic on the upper and lower half planes and at all non-
eigenvalues of the problem (1.3), (3.1) on the real axis;

  2. (ii)

for all $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,

(5.11) $M^{*}\left(\overline{\lambda},b\right)=M(\lambda,b)$

and

(5.12)
$\operatorname{Im}M(\lambda,b):=\frac{M(\lambda,b)-M^{*}(\lambda,b)}{2i}\lessgtr
0\quad\text{for}\quad\operatorname{Im}\lambda\lessgtr 0.$

###### Proof.

The proof is unchanged from the discrete case, see [41, Lemma 3.2] and [16,
Lemma 2.14]. ∎

######  Lemma 5.3.

Let $\operatorname{Im}\lambda\neq 0$. If $\beta$ satisfies (3.3) and
$\chi(t,\lambda,b)$ satisfies

(5.13) $\beta\chi(b,\lambda,b)=0,$

then

(5.14) $\chi^{*}(b,\lambda,b)J\chi(b,\lambda,b)=0.$

Conversely, if $\chi(t,\lambda,b)$ satisfies (5.14) for some $d\times d$
matrix $M$, then there exists a $d\times 2d$ matrix $\beta$ satisfying (3.3)
such that (5.13) holds.

###### Proof.

See [16, Lemma 2.13]. ∎

Set

$C(M,b):=\mp
i(I_{d},M^{*})Y^{*}(b,\lambda)JY(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right),$

where “$+$” holds if $\operatorname{Im}\lambda<0$ and “$-$” holds when
$\operatorname{Im}\lambda>0$. Using the definition of $\chi$, it follows that

$C\big{(}M(\lambda,b),b\big{)}=\mp i\chi^{*}(b,\lambda,b)J\chi(b,\lambda,b).$

Consequently we have from Lemma 5.3 that $M$ satisfies the matrix equation

(5.15) $C(M,b)=0$

if and only if there exists a $d\times 2d$ matrix $\beta$ satisfying (3.3)
such that (5.13) holds, ergo (5.10) holds. Let

(5.16) $F(b,\lambda):=\mp iY^{*}(b,\lambda)JY(b,\lambda).$

Then $F(b,\lambda)$ is a $2d\times 2d$ Hermitian matrix such that

(5.17) $C(M,b)=(I_{d},M^{*})F(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right).$

For later use we block $F(b,\lambda)$ as in

(5.18) $F(b,\lambda):=\begin{pmatrix}F_{11}&F_{12}\\\
F^{*}_{12}&F_{22}\end{pmatrix}(b,\lambda),$

where $F_{mn}(b,\lambda)$ are $d\times d$ matrices for $m,n=1,2$. Then (5.15)
can be recast in the form

(5.19)
$M^{*}F_{22}(b,\lambda)M+F_{12}(b,\lambda)M+M^{*}F^{*}_{12}(b,\lambda)+F_{11}(b,\lambda)=0.$

Using Lemma 2.3, (5.3), and (5.4), we see that

$Y^{*}(b,\lambda)JY(b,\lambda)=J+2i\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla t,$

which in tandem with (5.16) yields

(5.20)
$F(b,\lambda)=\begin{cases}-iJ+2\operatorname{Im}\lambda\displaystyle\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla
t:&\operatorname{Im}\lambda>0,\\\
iJ-2\operatorname{Im}\lambda\displaystyle\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla
t:&\operatorname{Im}\lambda<0.\end{cases}$

######  Theorem 5.4.

The matrix sets $C(M,b)\leq 0$ are closed, convex, and nested in the sense
that for fixed $M$ and fixed $\lambda\in\mathbb{C}$, $C^{\nabla}(M,t)\geq 0$,
where the nabla derivative is with respect to $t$.

###### Proof.

The proof follows from (5.17) and (5.20). See also the discussion in [16,
Remark 2.16]. ∎

As noted in the discrete case [41, Section 3], which actually refers to [16,
Remark 2.16], the intersection of the matrix sets $C(M,b)\leq 0$ is a limiting
set that is nonempty, closed, and convex. In what follows here we present a
detailed analysis of the properties of $F(b,\lambda)$, which will play a vital
role in the next section as we obtain precise relationships among the rank of
the matrix radius of the limiting set, asymptotic behavior of eigenvalues of
the Weyl disks, and the number of linearly independent square summable
solutions of system (1.3). Proceeding with this in mind, from (5.16), (5.18),
and (5.20) we see that

$\displaystyle F_{11}(b,\lambda)$ $\displaystyle=$ $\displaystyle\mp
i\theta^{*}(b,\lambda)J\theta(b,\lambda)$ $\displaystyle=$ $\displaystyle\pm
2\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon\theta)^{*}(t,\lambda)W(t)\Upsilon\theta(t,\lambda)\nabla
t,$ $\displaystyle F_{22}(b,\lambda)$ $\displaystyle=$ $\displaystyle\mp
i\phi^{*}(b,\lambda)J\phi(b,\lambda)$ $\displaystyle=$ $\displaystyle\pm
2\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon\phi)^{*}(t,\lambda)W(t)\Upsilon\phi(t,\lambda)\nabla
t,$ $\displaystyle F_{12}(b,\lambda)$ $\displaystyle=$ $\displaystyle\mp
i\theta^{*}(b,\lambda)J\phi(b,\lambda)$ $\displaystyle=$ $\displaystyle\pm
iI_{d}\pm
2\operatorname{Im}\lambda\int_{\rho(t_{0})}^{b}(\Upsilon\theta)^{*}(t,\lambda)W(t)\Upsilon\phi(t,\lambda)\nabla
t.$

Assuming (1.5), we can obtain the following from (5).

######  Theorem 5.5.

For any $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
$F_{11}(b,\lambda)>0$ and $F_{22}(b,\lambda)>0$ for all $b\geq t_{1}$.
Additionally, $F_{11}(b,\lambda)$ and $F_{22}(b,\lambda)$ are non-decreasing
with respect to $b$.

Using Theorem 5.5, equation (5.19) can be stated as

(5.22)
$\left(M+F_{22}^{-1}(b,\lambda)F_{12}^{*}(b,\lambda)\right)^{*}F_{22}(b,\lambda)\left(M+F_{22}^{-1}(b,\lambda)F_{12}^{*}(b,\lambda)\right)-\left(F_{12}F_{22}^{-1}F^{*}_{12}-F_{11}\right)(b,\lambda)=0.$

######  Theorem 5.6.

For any $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
$(F_{12}F_{22}^{-1}F^{*}_{12}-F_{11})(b,\lambda)=F_{22}^{-1}\left(b,\overline{\lambda}\right)>0$.

###### Proof.

From (5.5), (5.6), and (5.16) we see that

$F(b,\lambda)JF\left(b,\overline{\lambda}\right)=Y^{*}(b,\lambda)JY(b,\lambda)JY^{*}\left(b,\overline{\lambda}\right)JY\left(b,\overline{\lambda}\right)=-J.$

The rest of the proof is identical to [41, Proposition 3.2] and is omitted. ∎

Let

(5.23)
$\mathscr{C}(b,\lambda):=-F_{22}^{-1}(b,\lambda)F_{12}^{*}(b,\lambda),\quad\mathscr{R}(b,\lambda):=F_{22}^{-1/2}(b,\lambda).$

It follows from Theorem 5.6 that (5.15), that is to say (5.22), can be recast
as

(5.24)
$(M-\mathscr{C}(b,\lambda))^{*}\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))-\mathscr{R}^{2}\left(b,\overline{\lambda}\right)=0$

or as

(5.25)
$\left\\{\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))\mathscr{R}^{-1}\left(b,\overline{\lambda}\right)\right\\}^{*}\left\\{\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))\mathscr{R}^{-1}\left(b,\overline{\lambda}\right)\right\\}=I_{d}.$

######  Remark 5.7.

If the dimension $d=1$, then (5.24) is the equation of a circle. For this
reason we call (5.15) and/or (5.24) a Weyl circle equation, and call the
matrix set $C(M,b)\leq 0$ a Weyl disk; see [16, Definition 2.11].

Notice that if
$U:=\mathscr{R}^{-1}(b,\lambda)(M-\mathscr{C}(b,\lambda))\mathscr{R}^{-1}\left(b,\overline{\lambda}\right)$,
then (5.25) can be written as $U^{*}U=I_{d}$, and $U$ is unitary. We then have
the following results.

######  Theorem 5.8.

The Weyl circle equation (5.24) and/or (5.15) can be expressed via

(5.26)
$E_{b}(\lambda):M=\mathscr{C}(b,\lambda)+\mathscr{R}(b,\lambda)U\mathscr{R}\left(b,\overline{\lambda}\right),$

and the Weyl disk $C(M,b)\leq 0$ can be expressed via

(5.27)
$\overline{E}_{b}(\lambda):M=\mathscr{C}(b,\lambda)+\mathscr{R}(b,\lambda)V\mathscr{R}\left(b,\overline{\lambda}\right),$

where $U$ is any matrix on the unit matrix circle $\partial
D=\\{U:U\in\mathbb{C}^{d\times d}\;\text{is a unitary matrix}\\}$ and $V$ is
any matrix on the unit matrix disk $D=\\{V:V\in\mathbb{C}^{d\times
d}\;\text{satisfies}\;V^{*}V\leq I_{d}\\}$.

######  Definition 5.9.

The matrix $\mathscr{C}(b,\lambda)$ is called the center, and the matrices
$\mathscr{R}(b,\lambda)$ and $\mathscr{R}\left(b,\overline{\lambda}\right)$
are called the matrix radii, respectively, of the Weyl circle (5.26) and the
Weyl disk (5.27).

######  Theorem 5.10.

For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
the matrix sequence $\\{\mathscr{R}(b,\lambda)\\}$ converges, and
$\mathscr{R}_{0}(\lambda):=\lim_{b\rightarrow\infty}\mathscr{R}(b,\lambda)\geq
0$.

###### Proof.

From Theorem 5.5, $F_{22}(b,\lambda)>0$ for all $b\geq t_{1}$, and
$F_{22}(b,\lambda)$ is non-decreasing with respect to $b$. Recall from (5.23)
that $\mathscr{R}(b,\lambda)=F_{22}^{-1/2}(b,\lambda)$, whence
$\\{\mathscr{R}(b,\lambda)\\}$ is a non-increasing sequence of positive
definite matrices. The conclusion follows from the fact that any non-
increasing sequence of Hermitian matrices that is bounded below converges to a
Hermitian matrix. ∎

######  Lemma 5.11.

For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
and $b>\tau\geq t_{1}$, there exists $V_{0}\in D$ such that

(5.28)
$\mathscr{C}(b,\lambda)-\mathscr{C}(\tau,\lambda)=\mathscr{R}(\tau,\lambda)V_{0}\mathscr{R}\left(\tau,\overline{\lambda}\right)-\mathscr{R}(b,\lambda)V_{0}\mathscr{R}\left(b,\overline{\lambda}\right).$

###### Proof.

See Shi [41, Lemma 3.4]. ∎

######  Theorem 5.12.

For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
the matrix sequence $\\{\mathscr{C}(b,\lambda)\\}$ converges, i.e. the matrix
$\mathscr{C}_{0}(\lambda):=\lim_{b\rightarrow\infty}\mathscr{C}(b,\lambda)$ is
well defined.

###### Proof.

The result follows from Theorem 5.10 and Lemma 5.11. ∎

######  Theorem 5.13.

For any given $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
the matrix circle sequence $\\{E_{b}(\lambda)\\}$ and the matrix disk sequence
$\left\\{\overline{E}_{b}(\lambda)\right\\}$ converge as $b\rightarrow\infty$,
and their limiting sets can be represented, respectively, as

$\displaystyle E_{0}(\lambda):$ $\displaystyle
M=\mathscr{C}_{0}(\lambda)+\mathscr{R}_{0}(\lambda)U\mathscr{R}_{0}\left(\overline{\lambda}\right),\quad
U\in\partial D,$ $\displaystyle\overline{E}_{0}(\lambda):$ $\displaystyle
M=\mathscr{C}_{0}(\lambda)+\mathscr{R}_{0}(\lambda)V\mathscr{R}_{0}\left(\overline{\lambda}\right),\quad
V\in D.$

###### Proof.

The result follows from Theorems 5.8, 5.10, and 5.12. ∎

######  Remark 5.14.

Since $\mathscr{R}_{0}(\lambda)$ and
$\mathscr{R}_{0}\left(\overline{\lambda}\right)$ may be singular, the set
$\overline{E}_{0}(\lambda)$ may be a reduced matrix disk. We see that
$\overline{E}_{0}(\lambda)$ contains only one element if
$\mathscr{R}_{0}(\lambda)=0$ or
$\mathscr{R}_{0}\left(\overline{\lambda}\right)=0$, and it contains interior
points if and only if $\mathscr{R}_{0}(\lambda)$ and
$\mathscr{R}_{0}\left(\overline{\lambda}\right)$ are both invertible. Although
the limiting sets $E_{0}(\lambda)$ and $\overline{E}_{0}(\lambda)$ may be a
reduced matrix circle and a reduced matrix disk, respectively, we still give
the following definition for convenience.

######  Definition 5.15.

The matrix $\mathscr{C}_{0}(\lambda)$ is called the center, and the matrices
$\mathscr{R}_{0}(\lambda)$ and
$\mathscr{R}_{0}\left(\overline{\lambda}\right)$ are called the matrix radii
of the limiting sets $E_{0}(\lambda)$ and $\overline{E}_{0}(\lambda)$,
respectively.

######  Theorem 5.16.

Assume (1.5). For any given $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$ and for each $M\in\overline{E}_{0}(\lambda)$,
if $\operatorname{Im}\lambda\lessgtr 0$ then $\operatorname{Im}M\lessgtr 0$.

###### Proof.

Assume that $\operatorname{Im}\lambda\neq 0$ and
$M\in\overline{E}_{0}(\lambda)$. Set

(5.29) $\chi(t,\lambda):=Y(t,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right).$

Then from (5.20) we have that

(5.30)
$\displaystyle\int_{\rho(t_{0})}^{b}(\Upsilon\chi)^{*}(t,\lambda)W(t)\Upsilon\chi(t,\lambda)\nabla
t$ $\displaystyle=$ $\displaystyle(I_{d},M^{*})\int_{\rho(t_{0})}^{b}(\Upsilon
Y)^{*}(t,\lambda)W(t)\Upsilon Y(t,\lambda)\nabla
t\left(\begin{smallmatrix}I_{d}\\\ M\end{smallmatrix}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2|\operatorname{Im}\lambda|}(I_{d},M^{*})\left(F(b,\lambda)\pm
iJ\right)\left(\begin{smallmatrix}I_{d}\\\ M\end{smallmatrix}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2|\operatorname{Im}\lambda|}(I_{d},M^{*})F(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right)\pm\frac{1}{|\operatorname{Im}\lambda|}\operatorname{Im}M,$

for $\operatorname{Im}\lambda\lessgtr 0$. Because the sets $C(M,b)\leq 0$ are
nested, $\overline{E}_{0}(\lambda)$ is a subset of $C(M,b)\leq 0$ for any
$b\geq t_{1}$. Consequently we have from (5.17) that

$(I_{d},M^{*})F(b,\lambda)\left(\begin{smallmatrix}I_{d}\\\
M\end{smallmatrix}\right)\leq 0.$

This in tandem with (5.30) implies that for $b\geq t_{1}$ we have

(5.31)
$\int_{\rho(t_{0})}^{b}(\Upsilon\chi)^{*}(t,\lambda)W(t)\Upsilon\chi(t,\lambda)\nabla
t\leq\pm\frac{1}{|\operatorname{Im}\lambda|}\operatorname{Im}M,\quad\operatorname{Im}\lambda\lessgtr
0.$

The result then follows from the above relation and the assumed definiteness
condition (1.5). ∎

##  6\. Square summable solutions

We will call $y(\cdot,\lambda)$ a square summable solution of (1.3) if it is a
solution of (1.3) in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. In this section we
will make a connection between square summable solutions of (1.3) and the
elements of the limiting set $\overline{E}_{0}(\lambda)$ from Theorem 5.13,
obtaining precise relationships among the rank of the matrix radius
$\mathscr{R}_{0}(\lambda)=\lim_{b\rightarrow\infty}F_{22}^{-1/2}(b,\lambda)$
(see Theorem 5.10) of the limiting set $\overline{E}_{0}(\lambda)$, the number
of linearly independent square summable solutions of (1.3), and the asymptotic
behavior of the eigenvalues of the matrix radius $F_{22}(b,\lambda)$ of the
Weyl disk $\overline{E}_{b}(\lambda)$ from (5.27). Given the structure and
notation established in the previous sections that generalizes the discrete
results in Shi [41], the proofs of the following results are omitted, as there
is no change necessary from [41, Section 4] except for minor notational
adjustments.

######  Theorem 6.1.

For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$ and for
each $M\in\overline{E}_{0}(\lambda)$, all the columns of $\chi(\cdot,\lambda)$
are in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, where $\chi$
is given in (5.29).

######  Corollary 6.2.

For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$, system
(1.3) has at least $d$ linearly independent square summable solutions.

######  Theorem 6.3.

For
$\mathscr{R}_{0}(\lambda)=\lim_{b\rightarrow\infty}F_{22}^{-1/2}(b,\lambda)$,
set

$r(\lambda):=\operatorname{rank}\mathscr{R}_{0}(\lambda),\quad\operatorname{Im}\lambda\neq
0,$

and let
$k=d+\min\left\\{r(\lambda),r\left(\overline{\lambda}\right)\right\\}$. Then
system (1.3) has at least $k$ linearly independent square summable solutions.

By Theorem 5.5, for each $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$ and for any $b\geq t_{1}$,
$F_{22}(b,\lambda)>0$. If $\mu_{j}(b)$ are the eigenvalues of
$F_{22}(b,\lambda)$ for $1\leq j\leq d$, then $\mu_{j}(b)>0$ for $1\leq j\leq
d$, and they can be arranged as
$\mu_{1}(b)\leq\mu_{2}(b)\leq\cdots\leq\mu_{d}(b)$. We have the following
result.

######  Theorem 6.4.

For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$,
$\operatorname{rank}\mathscr{R}_{0}(\lambda)=r(\lambda)$ if and only if

$\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j},\quad 1\leq j\leq r(\lambda),$

are finite and positive, and

$\lim_{b\rightarrow\infty}\mu_{j}(b)=\infty,\quad r(\lambda)+1\leq j\leq d.$

In addition,
$\gamma_{1}^{-1/2},\gamma_{2}^{-1/2},\cdots,\gamma_{r(\lambda)}^{-1/2}$ are
the $r(\lambda)$ positive eigenvalues of $\mathscr{R}_{0}(\lambda)$.

######  Lemma 6.5.

For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$, system
(1.3) has exactly $d+l$ linearly independent square summable solutions if and
only if there exists a $d\times l$ matrix $\Lambda$ with
$\operatorname{rank}\Lambda=l$ such that $\phi(\cdot,\lambda)\Lambda\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, and
$\eta\in\operatorname{Ran}\Lambda=\left\\{\Lambda
v:v\in\mathbb{C}^{l}\right\\}$ if $\phi(\cdot,\lambda)\eta\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\eta\in\mathbb{C}^{d}$.

######  Theorem 6.6.

If $\operatorname{rank}\mathscr{R}_{0}(\lambda)=r(\lambda)$ for
$\operatorname{Im}\lambda\neq 0$, then system (1.3) has exactly $d+r(\lambda)$
linearly independent square summable solutions and thus, $r(\lambda)$ is
independent of the coefficient matrix $\alpha$ of the left boundary condition
in (3.1).

######  Theorem 6.7.

For each $\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$, system
(1.3) has exactly $d+r(\lambda)$ linearly independent square summable
solutions if and only if $\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j}$ are
finite and positive for $1\leq j\leq r(\lambda)$, and
$\lim_{b\rightarrow\infty}\mu_{j}(b)=\infty$ for $r(\lambda)+1\leq j\leq d$.
In addition,
$\gamma_{1}^{-1/2},\gamma_{2}^{-1/2},\cdots,\gamma_{r(\lambda)}^{-1/2}$ are
the $r(\lambda)$ positive eigenvalues of $\mathscr{R}_{0}(\lambda)$.

##  7\. Classification of singular linear Hamiltonian nabla systems

In this section, we introduce the defect index $d(\lambda)$ of the minimal
operator $H_{0}$ (defined in Section $4$) and $\lambda$. We establish a
precise correspondence between $d(\lambda)$ and the number of linearly
independent square summable solutions of (1.3). Based on this correspondence,
we show that the defect indices $d_{\pm}$ of $H_{0}$ are not less than $d$. In
addition, we obtain a precise correspondence between $d(\lambda)$ and
$\operatorname{rank}\mathscr{R}_{0}(\lambda)$. Moreover, we discuss the defect
index problem for the special case where $P(t)$ and $W(t)$ are both real, and
the largest defect index problem for the general case. Building on the above
results, we present a suitable classification for singular Hamiltonian nabla
systems by using the positive and negative defect indices of $H_{0}$. Lastly,
we derive several equivalent conditions on the limit circle and the limit
point cases. The proofs of the first few results below carry over from the
discrete case unchanged [41, Section 5].

######  Theorem 7.1.

For all $\lambda\in\mathbb{C}$, the defect index $d(\lambda)$ of the minimal
operator $H_{0}$ and $\lambda$ is equal to the number of linearly independent
square summable solutions of system (1.3).

######  Remark 7.2.

As $H_{0}$ is Hermitian by Lemma 4.4, the defect index $d(\lambda)$ of $H_{0}$
is constant in the upper and lower half planes, respectively. Set

(7.1) $d_{+}=d(i),\quad d_{-}=d(-i);$

these are called the positive and negative defect indices of the minimal
operator, respectively. The following result then follows directly from
Corollary 6.2 and Theorem 7.1.

######  Theorem 7.3.

The number of linearly independent square summable solutions of system (1.3)
in the upper half plane is $d_{+}$, and in the lower half plane is $d_{-}$.
These numbers are independent of $\lambda$, with $d_{\pm}\geq d$.

######  Theorem 7.4.

The rank of $\mathscr{R}_{0}(\lambda)$ is equal to a constant $r_{+}$ for all
$\lambda$ with $\operatorname{Im}\lambda>0$, and equal to a constant $r_{-}$
for all $\lambda$ with $\operatorname{Im}\lambda<0$. Moreover, these ranks
satisfy the equations

(7.2) $d_{+}=d+r_{+},\qquad d_{-}=d+r_{-}.$

######  Lemma 7.5.

For any $\lambda_{0}\in\mathbb{C}$, the fundamental matrix solution
$\Phi(\cdot,\lambda_{0})$ of (1.3)${}_{\lambda_{0}}$ with initial condition
$\Phi(\rho(t_{0}),\lambda_{0})=I_{2d}$ satisfies

(7.3) $\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)=1$

for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$.

###### Proof.

By the initial condition, (7.3) holds at $t=\rho(t_{0})$. If
$t\in[t_{0},\infty)_{\mathbb{T}}$ is a left-scattered point, then from (2.3)
we have

(7.4)
$\left(I_{2d}-\nu(t)\mathcal{S}(t,\lambda_{0})\right)\Phi(t,\lambda_{0})=\Phi^{\rho}(t,\lambda_{0}).$

Consequently by (2.5) and (7.4) we have
$\det\big{(}\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\big{)}=\det\big{(}\Phi^{\rho*}(t,\lambda_{0})\Phi^{\rho}(t,\lambda_{0})\big{)}$,
so that

$\left[\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)\right]^{\nabla}(t)=0$

if $t$ is a left-scattered point.

By Liouville’s formula on time scales [22], we have

(7.5)
$\det\Phi(t,\lambda_{0})=\hat{e}_{q}(t,\rho(t_{0}))\det\Phi(\rho(t_{0}),\lambda_{0})=\hat{e}_{q}(t,\rho(t_{0})),\quad
t\in[\rho(t_{0}),\infty)_{\mathbb{T}},$

where
$q(t)=\lambda_{1}\oplus_{\nu}\lambda_{2}\oplus_{\nu}+\cdots+\oplus_{\nu}\lambda_{2d}$
for eigenvalues $\lambda_{1},\cdots,\lambda_{2d}$ of
$\mathcal{S}(\cdot,\lambda)$ given in (2.3), and
$x=\hat{e}_{q}(\cdot,\rho(t_{0}))$ is the nabla exponential function [15,
Chapter 3] that uniquely solves the initial value problem

$x^{\nabla}(t)=q(t)x(t),\quad x^{\rho}(t_{0})=1.$

It follows that
$\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)=\hat{e}_{(q\oplus_{\nu}q^{*})}(t,\rho(t_{0}))$.
Suppose $t\in[t_{0},\infty)_{\mathbb{T}}$ is a left-dense point. Then
$\nu(t)=0$,
$(q\oplus_{\nu}q^{*})=(q+q^{*})=\operatorname{tr}(\mathcal{S}+\mathcal{S}^{*})=0$
from (2.3), and

$\det\left(\Phi^{*}(t,\lambda_{0})\Phi(t,\lambda_{0})\right)=\hat{e}_{(q+q^{*})}(t,\rho(t_{0}))=\hat{e}_{0}(t,\rho(t_{0}))\equiv
1.$

Therefore (7.3) holds for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. ∎

######  Theorem 7.6 (The Largest Defect Index Theorem).

If there exists $\lambda_{0}\in\mathbb{C}$ such that all the solutions of
(1.3)${}_{\lambda_{0}}$ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, then all solutions
of (1.3)λ are in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$,
for any $\lambda\in\mathbb{C}$.

###### Proof.

Assume that all solutions of (1.3)${}_{\lambda_{0}}$ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\lambda_{0}\in\mathbb{C}$. Given any $\lambda\in\mathbb{C}$, let
$\Phi(\cdot,\lambda)$ be the fundamental matrix solution of (1.3)λ, that is
$\Phi(\cdot,\lambda)$ solves (1.3)λ, or (2.1)λ, respectively, and satisfies
$\Phi(\rho(t_{0}),\lambda)=I_{2d}$; note that (7.3) then holds. As
$\Phi(t,\lambda)$ and $\Phi\left(t,\lambda_{0}\right)$ are both invertible,
there exists an invertible matrix $X(t,\lambda)$ such that

(7.6) $\Phi(t,\lambda)=\Phi\left(t,\lambda_{0}\right)X(t,\lambda).$

We will show that $X(t,\lambda)$ is bounded for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. Using (7.6), the nabla product rule,
and the simple useful time-scale formula $X^{\rho}=X-\nu X^{\nabla}$, we see
that

$\displaystyle\Phi^{\nabla}(t,\lambda)=\Phi^{\rho}\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda)+\Phi^{\nabla}\left(t,\lambda_{0}\right)X(t,\lambda),$
(7.7)
$\displaystyle(\Upsilon\Phi)(t,\lambda)=(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)-\nu(t)\operatorname{diag}\\{0,I_{d}\\}\Phi^{\rho}\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda),$

for $\Upsilon$ in (1.4). From (7.7) and the fact that $\Phi(\cdot,\lambda)$
and $\Phi(\cdot,\lambda_{0})$ are fundamental solution matrices for (1.3)λ and
(1.3)${}_{\lambda_{0}}$, respectively, we arrive at

(7.8) $X^{\nabla}(t,\lambda)=Q(t,\lambda)X(t,\lambda),$

where we have taken

(7.9) $\displaystyle Q(t,\lambda)$ $\displaystyle=$
$\displaystyle(\lambda-\lambda_{0})Z^{-1}(t,\lambda)(\Upsilon\Phi)^{*}(t,\lambda_{0})W(t)(\Upsilon\Phi)(t,\lambda_{0}),$
(7.10) $\displaystyle Z(t,\lambda)$ $\displaystyle=$
$\displaystyle(\Upsilon\Phi)^{*}(t,\lambda_{0})\begin{pmatrix}0&-I_{d}+\nu(t)A^{*}(t)\\\
I_{d}&\nu(t)(B(t)+\lambda W_{2}(t))\end{pmatrix}\Phi^{\rho}(t,\lambda_{0}).$

The multiplier $(\Upsilon\Phi)^{*}(t,\lambda_{0})$ appears in (7.9) and (7.10)
via (1.8), and will help in the sequel with the analysis on $Q(t,\lambda)$.
First we focus on $Z(t,\lambda)$. From (1.3)${}_{\lambda_{0}}$ we have that

(7.11)
$\Phi^{\rho}(t,\lambda_{0})=\begin{pmatrix}I_{d}-\nu(t)A(t)&-\nu(t)(B(t)+\lambda_{0}W_{2}(t))\\\
0&I_{d}\end{pmatrix}(\Upsilon\Phi)(t,\lambda_{0}).$

If we substitute (7.11) into (7.10), we see that

(7.12) $\displaystyle Z(t,\lambda)$ $\displaystyle=$
$\displaystyle(\lambda-\lambda_{0})\int_{\rho(t)}^{t}(\Upsilon\Phi)^{*}(s,\lambda_{0})\operatorname{diag}\\{0,W_{2}(s)\\}(\Upsilon\Phi)(s,\lambda_{0})\nabla
s$
$\displaystyle+(\Upsilon\Phi)^{*}(t,\lambda_{0})\begin{pmatrix}0&-I_{d}+\nu(t)A^{*}(t)\\\
I-\nu(t)A(t)&0\end{pmatrix}(\Upsilon\Phi)(t,\lambda_{0}),$

where we have used the time-scale formula
$\nu(t)f(t)=\int_{\rho(t)}^{t}f(s)\nabla s$ in the first line of (7.12). As
all solutions of (1.3)${}_{\lambda_{0}}$ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$,
$\Phi(\cdot,\lambda_{0})\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, whence

(7.13)
$V(\lambda_{0}):=\int_{\rho(t_{0})}^{\infty}(\Upsilon\Phi)^{*}(s,\lambda_{0})W(s)(\Upsilon\Phi)(s,\lambda_{0})\nabla
s<\infty.$

Consequently the first term on the right-hand side of (7.12) tends to zero as
$t\rightarrow\infty$ in the time scale. Let us denote by $\Psi$ the second
term on the right-hand side of (7.12). From (7.11) we have

(7.14)
$\Psi(t)=\Phi^{\rho*}(t,\lambda_{0})J\Phi^{\rho}(t,\lambda_{0})+2i\operatorname{Im}\lambda_{0}\int_{\rho(t)}^{t}(\Upsilon\Phi)^{*}(s,\lambda_{0})\operatorname{diag}\\{0,W_{2}(s)\\}(\Upsilon\Phi)(s,\lambda_{0})\nabla
s,$

where we have used the time-scale formula
$\nu(t)f(t)=\int_{\rho(t)}^{t}f(s)\nabla s$ again; the second term on the
right-hand side of (7.14) goes to 0 as $t\rightarrow\infty$ in the time scale
since $\Phi(\cdot,\lambda_{0})\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. By Lemma 2.3 and the
initial condition for $\Phi(\cdot,\lambda_{0})$ we see that

$\Phi^{\rho*}(t,\lambda_{0})J\Phi^{\rho}(t,\lambda_{0})=J+2i\operatorname{Im}\lambda_{0}\int_{\rho(t_{0})}^{\rho(t)}(\Upsilon\Phi)^{*}(s,\lambda_{0})W(s)(\Upsilon\Phi)(s,\lambda_{0})\nabla
s.$

It then follows from (7.13) and (7.14) that

$\lim_{t\rightarrow\infty}\Psi(t)=J+2i\operatorname{Im}\lambda_{0}V(\lambda_{0}),$

so that

(7.15)
$\lim_{t\rightarrow\infty}Z(t,\lambda)=J+2i\operatorname{Im}\lambda_{0}V(\lambda_{0}).$

From (7.10) and (7.11) we see that $Z(t,\lambda)$ is invertible for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$; we need to show that
$Z^{-1}(t,\lambda)$ is bounded for all
$t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$. From (7.11) we have that

$\det(\Upsilon\Phi)^{*}(t,\lambda_{0})=\det\Phi^{\rho*}(t,\lambda_{0})\det
E^{*}(t),$

so that from (7.10) we obtain

$\displaystyle\det Z(t,\lambda)$ $\displaystyle=$
$\displaystyle\det(\Upsilon\Phi)^{*}(t,\lambda_{0})\det\left(I_{d}-\nu(t)A^{*}(t)\right)\det\Phi^{\rho}(t,\lambda_{0})$
$\displaystyle=$
$\displaystyle\det\Phi^{\rho*}(t,\lambda_{0})\det\Phi^{\rho}(t,\lambda_{0})=1$

since $\Phi(\cdot,\lambda_{0})$ satisfies (7.3). As a result,

(7.16) $Z^{-1}(t,\lambda)=(\det
Z(t,\lambda))^{-1}\operatorname{adj}Z(t,\lambda),$

where $\operatorname{adj}Z(t,\lambda)$ is the adjugate matrix of
$Z(t,\lambda)$. Moreover, from (7.15) we see that
$\operatorname{adj}Z(t,\lambda)$ is bounded on
$[\rho(t_{0}),\infty)_{\mathbb{T}}$, whence $Z^{-1}(t,\lambda)$ is as well by
(7.16). Let $c\in\mathbb{R}$ be a positive constant such that

(7.17)
$\|Z^{-1}(t,\lambda)\|:=\left(\sum_{k=1}^{2d}\sum_{j=1}^{2d}|z_{jk}(t)|\right)^{1/2}\leq
c,\quad t\in[\rho(t_{0}),\infty)_{\mathbb{T}}.$

Now we will show that

(7.18) $\int_{\rho(t_{0})}^{\infty}\|Q(t,\lambda)\|_{1}\nabla
t<\infty,\quad\|Q(t,\lambda)\|_{1}:=\sup_{\|\xi\|=1}\|Q(t,\lambda)\xi\|,$

where $Q(\cdot,\lambda)$ is given in (7.9). From (7.13) it follows that all
the diagonal entries of the expression

(7.19) $(\Upsilon\Phi)^{*}(t,\lambda_{0})W(t)(\Upsilon\Phi)(t,\lambda_{0})$

are nonnegative and absolutely summable over
$[\rho(t_{0}),\infty)_{\mathbb{T}}$. By referring to the nonnegativity of
(7.19), the absolute value of each non-diagonal entry of (7.19) is less than
or equal to the sum of the two diagonal entries that lie exactly in the same
column and row as the non-diagonal entry does. As a result, each non-diagonal
entry of (7.19) is also absolutely summable over
$[\rho(t_{0}),\infty)_{\mathbb{T}}$. Thus, it follows that

(7.20)
$\int_{\rho(t_{0})}^{\infty}\|(\Upsilon\Phi)^{*}(t,\lambda_{0})W(t)(\Upsilon\Phi)(t,\lambda_{0})\|\nabla
t<\infty.$

Consequently from (7.9), (7.17), and (7.20) we have that
$\int_{\rho(t_{0})}^{\infty}\|Q(t,\lambda)\|\nabla t<\infty$, so that (7.18)
follows.

We are ready to show that $X(t,\lambda)$ is bounded on
$[\rho(t_{0}),\infty)_{\mathbb{T}}$. For a solution of (7.8) to exist, we need
the coefficient matrix $Q$ to be $\nu-$regressive, in other words we need to
show that $I_{2d}-\nu(t)Q(t)$ is invertible for all
$t\in[t_{0},\infty)_{\mathbb{T}}$. From (2.2), (7.9), (7.10), and (7.11) we
have that

$I_{2d}-\nu(t)Q(t)=Z^{-1}(t,\lambda)(\Upsilon\Phi)^{*}(t,\lambda_{0})\begin{pmatrix}-(\lambda-\lambda_{0})\nu(t)W_{1}(t)&-E^{*-1}(t)\\\
E^{-1}(t)&0\end{pmatrix}(\Upsilon\Phi)(t,\lambda_{0});$

since $\Phi(\cdot,\lambda_{0})$ is a fundamental matrix, by (7.11) again we
see that every matrix on the right-hand side here is invertible, making
$I_{2d}-\nu(t)Q(t)$ invertible for all $t\in[t_{0},\infty)_{\mathbb{T}}$.
Therefore

$X(t,\lambda)=\hat{e}_{Q(\cdot,\lambda)}(t,\rho(t_{0}))X(\rho(t_{0}),\lambda)\stackrel{{\scriptstyle\eqref{shi5.4}}}{{=}}\hat{e}_{Q(\cdot,\lambda)}(t,\rho(t_{0}))$

is a well-defined matrix, and thus

$\|X(t,\lambda)\|_{1}\leq\exp\left\\{\int_{\rho(t_{0})}^{t}\|Q(s,\lambda)\|_{1}\nabla
s\right\\}.$

Combining this with (7.18) we conclude that $\|X(t,\lambda)\|_{1}$ is bounded
for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$.

Let us now show that all solutions of (1.3)λ are in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. From the second
line of (7.7) and (7.11) we have that

$\displaystyle(\Upsilon\Phi)^{*}(t,\lambda)W(t)(\Upsilon\Phi)(t,\lambda)$
$\displaystyle=$ $\displaystyle
X^{*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)W(t)(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)$
$\displaystyle-\nu(t)X^{*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda)$
$\displaystyle-\nu(t)X^{\nabla*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)$
$\displaystyle+(\nu(t))^{2}X^{\nabla*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X^{\nabla}(t,\lambda);$

using the simple formula $\nu X^{\nabla}=X-X^{\rho}$ again, we simplify this
to

$\displaystyle(\Upsilon\Phi)^{*}(t,\lambda)W(t)(\Upsilon\Phi)(t,\lambda)$
$\displaystyle=$ $\displaystyle
X^{*}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{W_{1}(t),0\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X(t,\lambda)$
$\displaystyle+X^{*\rho}(t,\lambda)(\Upsilon\Phi)^{*}\left(t,\lambda_{0}\right)\operatorname{diag}\\{0,W_{2}(t)\\}(\Upsilon\Phi)\left(t,\lambda_{0}\right)X^{\rho}(t,\lambda).$

From the boundedness of $\|X(t,\lambda)\|_{1}$ and (7.20), we see that

$\int_{\rho(t_{0})}^{\infty}\|(\Upsilon\Phi)^{*}(t,\lambda)W(t)(\Upsilon\Phi)(t,\lambda)\|\nabla
t<\infty,$

putting $\Phi(\cdot,\lambda)\in
L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$. ∎

######  Remark 7.7.

Given the above results, we are able to generalize the remainder of [41,
Section 5] verbatim, as the proofs are unchanged. For completeness we include
the main results here.

######  Definition 7.8.

Let $d_{\pm}$ be the positive and negative defect indices of $H_{0}$, where
$H_{0}$ is the minimal operator corresponding to system (1.3) defined in
(4.7). Then the dynamic Hamiltonian nabla operator $\mathscr{L}$ given in
(2.7) is said to be in the limit $(d_{+},d_{-})$ case at $t=\infty$. In the
special case of $d_{+}=d_{-}=d$, $\mathscr{L}$ is said to be in the limit
point case $(l.p.c.)$ at $t=\infty$, and in the other special case of
$d_{+}=d_{-}=2d$, $\mathscr{L}$ is said to be in the limit circle case
$(l.c.c.)$ at $t=\infty$.

######  Remark 7.9.

It is clear that there may be at most $1+d^{2}$ cases for the singular dynamic
Hamiltonian nabla system (1.3) of degree $d$ by the largest defect index
theorem (Theorem 7.6) and by using the fact that $d\leq d_{\pm}\leq 2d$.
However, in the special case of $d=1$, the classification is simple just like
the formal self-adjoint second-order scalar difference operators; in other
words, $\mathscr{L}$ is either in $l.p.c.$ or in $l.c.c.$ at $t=\infty$ by
using the largest defect index theorem.

######  Theorem 7.10.

Assume (7.3). Then the following nine statements are equivalent.

  1. (i)

$\mathscr{L}$ is in $l.c.c.$ at $t=\infty$;

  2. (ii)

system (1.3)λ has $2d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;

  3. (iii)

$\mathscr{R}_{0}(\lambda)$ is invertible for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;

  4. (iv)

the limiting set $\overline{E}_{0}(\lambda)$ has nonempty interior for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;

  5. (v)

$\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j}$ is finite and positive for
$1\leq j\leq d$, where $\mu_{j}(b)(1\leq j\leq d)$ are eigenvalues of
$F_{22}(b,\lambda)$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;

  6. (vi)

system (1.3)${}_{\lambda_{0}}$ has $2d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  7. (vii)

$\mathscr{R}_{0}(\lambda_{0})$ is invertible for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  8. (viii)

the limiting set $\overline{E}_{0}(\lambda_{0})$ has nonempty interior for
some $\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  9. (ix)

$\lim_{b\rightarrow\infty}\mu_{j}(b)=\gamma_{j}$ is finite and positive for
$1\leq j\leq d$, where $\mu_{j}(b)(1\leq j\leq d)$ are eigenvalues of
$F_{22}(b,\lambda_{0})$ for some $\lambda_{0}\in\mathbb{C}$ with
$\operatorname{Im}\lambda_{0}\neq 0$.

Similarly, the following equivalent conditions on the limit point case can be
concluded by Theorems 6.4 and 7.4.

######  Theorem 7.11.

Assume (7.3). Then the following seven statements are equivalent:

  1. (i)

$\mathscr{L}$ is in $l.p.c.$ at $t=\infty$;

  2. (ii)

system (1.3)λ has $d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;

  3. (iii)

$\mathscr{R}_{0}(\lambda)=0$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;

  4. (iv)

$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda)$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;

  5. (v)

systems (1.3)${}_{\lambda_{0}}$ and (1.3)${}_{\overline{\lambda}_{0}}$ have
exactly $d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$, respectively, for
some $\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  6. (vi)

$\mathscr{R}_{0}(\lambda_{0})=\mathscr{R}_{0}\left(\overline{\lambda}_{0}\right)=0$
for some $\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  7. (vii)

$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda_{0})$ and the smallest eigenvalue of
$F_{22}\left(b,\overline{\lambda}_{0}\right)$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$.

If all the coefficients of system (1.3)λ are real, we have the following
results by Theorems 7.6 and 7.9.

######  Corollary 7.12.

If $P(t)$ and $W(t)$ are real for all $t\in[\rho(t_{0}),\infty)_{\mathbb{T}}$,
then following nine statements are equivalent.

  1. (i)

$\mathscr{L}$ is in $l.p.c.$ at $t=\infty$;

  2. (ii)

system (1.3)λ has exactly $d$ linearly independent solutions in
$L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;

  3. (iii)

$\mathscr{R}_{0}(\lambda)=0$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda\neq 0$;

  4. (iv)

the limiting set $E_{0}(\lambda)$ contains only one element for all
$\lambda\in\mathbb{C}$ with $\operatorname{Im}\lambda\neq 0$;

  5. (v)

$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda)$ for all $\lambda\in\mathbb{C}$ with
$\operatorname{Im}\lambda>0$;

  6. (vi)

system (1.3)${}_{\lambda_{0}}$ has exactly $d$ linearly independent solutions
in $L^{2}_{W}\left([\rho(t_{0}),\infty)_{\mathbb{T}}\right)$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  7. (vii)

$\mathscr{R}_{0}(\lambda_{0})=0$ for some $\lambda_{0}\in\mathbb{C}$ with
$\operatorname{Im}\lambda_{0}\neq 0$;

  8. (viii)

the limiting set $E_{0}(\lambda_{0})$ contains only one element for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$;

  9. (ix)

$\lim_{b\rightarrow\infty}\mu_{1}(b)=\infty$, where $\mu_{1}(b)$ is the
smallest eigenvalue of $F_{22}(b,\lambda_{0})$ for some
$\lambda_{0}\in\mathbb{C}$ with $\operatorname{Im}\lambda_{0}\neq 0$.

As a consequence of Theorems 6.6, 7.4, and 7.6, Corollary 7.12 holds in the
special case of $d=1$, no matter if the coefficients of (1.3)λ are real or
complex.

######  Corollary 7.13.

If $d=1$, then the equivalent statements $\rm{(i)}-\rm{(ix)}$ in Corollary
7.12 hold.

######  Remark 7.14.

Much of the theory of Weyl and Titchmarsh remains that can be extended to time
scales, such as $M(\lambda)$ theory in the limit point case [41, Section 6],
asymptotic expansion of Weyl-Titchmarsh matrices and Green’s matrices [16],
and so on, leaving the future of the subject open to interested researchers.

##  8\. alternative form

In this section we introduce a possible alternative form for this theory on
Sturmian time scales. For example, instead of system (1.3), consider the
alternative system

(8.1) $J(\Upsilon y)^{\Delta}(t)=\Big{(}\lambda W(t)+P(t)\Big{)}y(t),\quad
t\in[t_{0},\infty)_{\mathbb{T}},\quad
J=\left(\begin{smallmatrix}0_{n}&-I_{n}\\\
I_{n}&0_{n}\end{smallmatrix}\right),$

for the same block matrices $W$ and $P$, where we have the delta derivative
and $\Upsilon y$ on the left-hand side for $\Upsilon$ in (1.4), $y$ on the
right-hand side, and where this time we assume

(8.2) $E_{2}(t):=\Big{(}I_{n}+\mu(t)A^{*}(t)\Big{)}^{-1}$

exists instead of (2.2). System (8.1) may also be viewed as a generalization
of (1.1) and (1.2). For (8.1) we have the integration by parts formula
(compare with Theorem 2.2)

$\int_{a}^{b}\left[z^{*}J(\Upsilon y)^{\Delta}-\left(J(\Upsilon
z)^{\Delta}\right)^{*}y\right](t)\Delta t=(\Upsilon z)^{*}(b)J(\Upsilon
y)(b)-(\Upsilon z)^{*}(a)J(\Upsilon y)(a).$

Moreover, the scalar product in (4.1) is now replaced by a standard weighted
product without shifts given by

(8.3) $(x,y)_{W}:=\int_{t_{0}}^{\infty}y^{*}(t)W(t)x(t)\Delta t,\quad x,y\in
L_{W}^{2}([t_{0},\infty)_{\mathbb{T}}).$

Additionally, as in (2.3) we may rewrite (8.1) as the system

(8.4) $(\Upsilon y)^{\Delta}(t)=\mathcal{K}(t,\lambda)(\Upsilon
y)(t),\quad\mathcal{K}(\cdot,\lambda):=-J(\lambda W+P)\widehat{H},$

where on $[t_{0},\infty)_{\mathbb{T}}$ we use $E_{2}=(I_{n}+\mu A^{*})^{-1}$
and

(8.5) $\widehat{H}:=\begin{pmatrix}I_{n}&0_{n}\\\ \mu E_{2}(C-\lambda
W_{1})&E_{2}\end{pmatrix},\quad\text{with}\quad-J(\lambda
W+P)=\begin{pmatrix}A&\lambda W_{2}+B\\\ C-\lambda W_{1}&-A^{*}\end{pmatrix},$

since $E_{2}A^{*}=A^{*}E_{2}$ and $I-\mu A^{*}E_{2}=E_{2}$. Directly from the
definition of $\mathcal{K}(\cdot,\lambda)$ in (8.4) we have that

$I_{2n}+\mu(t)\mathcal{K}(t,\lambda)=\begin{pmatrix}I_{n}+\mu(t)A(t)&\mu(t)(\lambda
W_{2}(t)+B(t))\\\ 0_{n}&I_{n}\end{pmatrix}\widehat{H}(t),$

so that $I_{2n}+\mu\mathcal{K}(\cdot,\lambda)$ is invertible by (8.2),
$\mathcal{K}(\cdot,\lambda)$ is regressive, and the matrix equation $(\Upsilon
y)^{\Delta}=\mathcal{K}(\cdot,\lambda)\Upsilon y$ with initial condition
$(\Upsilon y)(t_{0})=y_{0}$ has a unique solution $\Upsilon y$ on
$[t_{0},\infty)_{\mathbb{T}}$. It follows that an initial value problem
involving (8.1) has a unique solution $y$ in

$\left\\{y=(y_{1},y_{2})^{\operatorname{T}}\Big{|}\;y_{1},y_{2}^{\rho}:[t_{0},\infty)_{\mathbb{T}}\rightarrow\mathbb{C}^{n}\;\text{are
delta differentiable}\right\\}.$

In summary, to unify (1.1) and (1.2) on Sturmian time scales, systems
equivalent to (1.3) or (8.1) must be used to account for the shifts in the
discrete case [41]. For such systems to admit the existence and uniqueness of
solutions to initial value problems, an integration by parts formula, and a
matrix weighted scalar product in a Hilbert space, the Sturmian assumption
(1.7) is essential.

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