On Solutions of Systems of Differential Equations on Half-Line with Summable
Coefficients

Maria Kuznetsova111National Research Saratov State University, Astrakhanskaya
Street 83, Saratov, 410012, Russia. e-mail<EMAIL_ADDRESS>
Abstract. We consider a system of differential equations and obtain its
solutions with exponential asymptotics and analyticity with respect to the
spectral parameter. Solutions of such type have importance in studying
spectral properties of differential operators. Here, we consider the system of
first-order differential equations on a half-line with summable coefficients,
containing a nonlinear dependence on the spectral parameter. We obtain
fundamental systems of solutions with analyticity in certain sectors, in which
it is possible to apply the method of successive approximations. We also
construct non-fundamental systems of solutions with analyticity in a large
sector, including two previously considered neighboring sectors. The obtained
results admit applications in studying inverse spectral problems for the
higher-order differential operators with distribution coefficients.

Key words: systems of differential equations, fundamental systems of
solutions, summable coefficients, asymptotic formulae, nonlinear dependence on
a parameter

2010 Mathematics Subject Classification: 34E05, 34A30, 34A45

This is an extended version of the paper accepted for publication in
Lobachevskii Journal of Mathematics. The present version differs by the
detailed proofs of Lemma 1, Theorem 1, and Theorem 4. We also added Section 5
and Appendix B, that are absent in the journal version.

##  1 Introduction

In this paper, we study the system of differential equations on the half-line
$x\geq 0$

$\mathrm{\bf y}^{\prime}=\big{(}\lambda
F(x)+A(x)+C(x,\lambda)\big{)}\mathrm{\bf y},\quad\mathrm{\bf
y}=[y_{j}(x)]_{j=1}^{n},$ (1.1)

where $\lambda\in\mathbb{C}_{0}$ is the spectral parameter,
$\mathbb{C}_{0}:=\mathbb{C}\setminus\\{0\\}.$ Here, $F,$ $A,$ and $C$ are
matrix functions of order $n$ satisfying the following conditions.

  1. I.

$F(x)=\rho(x)B,$ where $\rho\in L[0,T)$ for any $T>0$ and $\rho(x)>0$ a.e.,
while $B$ is a constant diagonal matrix:
$B=\mathrm{diag}\\{b_{j}\\}_{j=1}^{n},$ $b_{j}\in\mathbb{C}$ for
$j=\overline{1,n}.$

  2. II.

$A(x)=[a_{jk}(x)]_{j,k=1}^{n},$ where $a_{jk}\in L[0,\infty)$ for
$j,k=\overline{1,n}.$

  3. III.

$C(x,\lambda)=[c_{jk}(x,\lambda)]_{j,k=1}^{n},$ where each component
$c_{jk}(x,\lambda)$ is a holomorphic mapping of $\lambda\in\mathbb{C}_{0}$ to
$L[0,\infty),$ and

$\|C(\cdot,\lambda)\|_{L[0,\infty)}:=\max_{j,k=\overline{1,n}}\|c_{jk}(\cdot,\lambda)\|_{L[0,\infty)}\to
0,\quad\lambda\to\infty.$ (1.2)

Solutions of system (1.1) are considered in the class of vector functions
$\mathrm{\bf y}=[y_{j}(x)]_{j=1}^{n}$ such that $y_{j}\in AC[0,T]$ for any
$T>0,$ $j=\overline{1,n}.$

We aim to obtain solutions of system (1.1) possessing certain asymptotics and
analyticity with respect to the spectral parameter. Fundamental systems of
solutions (FSS) with the mentioned properties is an important tool in the
spectral theory of differential operators, see [2, 1, 4, 6, 3, 5]. Herewith,
construction of FSS for the $n$-th order differential equation with the
spectral parameter

$u^{(n)}+p_{1}(x)u^{(n-1)}+\ldots+p_{n}(x)u=\lambda^{n}\rho^{n}(x)u$ (1.3)

can be reduced to construction of FSS for first-order system (1.1), where

$F(x)=\rho(x)\mathrm{diag}\,\\{b_{j}\\}_{j=1}^{n},\quad
b_{j}=\exp\Big{(}\frac{2\pi ij}{n}\Big{)},\;j=\overline{1,n},\quad
C(x,\lambda)=\sum_{\nu=1}^{n-1}C_{\nu}(x)\lambda^{-j},$ (1.4)

while the elements of $A(x)$ are proportional to $p_{1}(x).$

Systems of first-order differential equations under various assumptions on the
coefficients were studied in [7, 1, 8, 11, 16, 3, 10, 12, 9, 13, 17, 15, 14]
and other works. We distinguish the classical case when, in notations of
system (1.1), the coefficients of $A$ are absolutely continuous, see [7, 1,
8]. In this case, for obtaining the FSS with certain asymptotics and
analyticity with respect to the spectral parameter, one can apply the standard
method of successive approximations, that requires the related integral
operators to be contraction mappings. Then, the FSS are obtained for a
sufficiently large $|\lambda|\geq\lambda_{0}>0,$ separately for $\lambda$ in
each closed sector $\overline{\Gamma_{\kappa}}$ for which the values
$\\{b_{j}\\}_{j=1}^{n}$ can be renumbered so that

$\mathrm{Re}\,(\lambda b_{1})\geq\mathrm{Re}\,(\lambda
b_{2})\geq\ldots\geq\mathrm{Re}\,(\lambda
b_{n}),\quad\lambda\in\overline{\Gamma_{\kappa}}.$ (1.5)

Here, we study system (1.1) in the non-classical case when the coefficients of
$A$ are only summable. It is more difficult, because, in general, the
corresponding integral operators are not contraction mappings even for large
$\lambda\in\overline{\Gamma_{\kappa}}.$ To obtain the FSS in this case, one
should apply a modified method of successive approximations [10, 12, 13, 11],
proving by rigorous estimates that the squares of the integral operators are
contraction mappings. The mentioned works [10, 12, 13, 11] concern systems of
differential equations on a finite interval. In contrast to them, we study
system (1.1) on the half-line $x\geq 0.$

The non-classical case is crucial for studying $n$-th order differential
equations (1.3) with distribution coefficients, see [18, 19, 6]. The
regularization approach [18, 19] allows interpreting such equations as systems
(1.1) with summable coefficients, wherein the matrices $F$ and $C$ have form
(1.4). In relation with this, in [12], there were constructed the FSS of
systems (1.1) on a finite interval under conditions similar to I–III. This
result was used for obtaining the FSS of equations with distribution
coefficients, those, subsequently, were applied in studying spectral
properties of the corresponding differential operators, see [6, 20, 21].

As far as we know, construction of the FSS in the case of a half-line was
considered in detail only in [17], wherein the results were obtained for
absolutely continuous coefficients of $A(x).$ That construction was applied
for studying inverse spectral problems, see [5, 9]. Here, we study the more
difficult non-classical case. Moreover, we use the specifics of the half-line
and for $\lambda\in\overline{\Gamma_{\kappa}},$ construct a family of the FSS
having exponential asymptotics for $x\geq\alpha$ and determined for
$|\lambda|\geq\lambda_{\alpha}>0,$ where $\alpha\geq 0$ is a parameter.
Imposing the condition

$\exists\varphi(\alpha)>0,\;\alpha\geq 0\colon\ \varphi(\alpha)\to
0,\;\sup_{|\lambda|\geq\varphi(\alpha)}\|C(\cdot,\lambda)\|_{L[\alpha,\infty)}\to
0,\quad\alpha\to\infty,$ (1.6)

we get that $\lambda_{\alpha}\to 0$ as $\alpha\to\infty.$ Thus, we obtain the
FSS analytic at $\lambda\in{\Gamma_{\kappa}}$ arbitrarily close to $0.$
Condition (1.6) is quite general to include the matrices $C$ in (1.4) with
$\|C_{\nu}\|_{L[0,\infty)}<\infty,$ $\nu=\overline{1,n-1},$ that are obtained
after regularization of $n$-th order equations with distribution coefficients,
see Proposition 3.

Note that the unboundedness of $x\geq 0$ does not allow us consideration of
shifted sectors used in [12, 2, 13], containing the boundaries between
neighboring sectors $\Gamma_{\kappa}.$ Instead of this, we introduce large
sectors $\Omega_{m}$ including two neighboring sectors $\Gamma_{\kappa}$ and
construct non-fundamental systems of solutions analytic at
$\lambda\in\Omega_{m}.$ The properties of these systems are sufficient for
solving the inverse spectral problems considered in [5, 21] in the case of a
half-line.

The paper is organized as follows. In Section 2, we introduce necessary
objects and justify application of the method of successive approximations,
see Lemma 2. In Section 3, we obtain the FSS with the needed properties,
reducing (1.1) to a system of integral equations, see Theorem 2. Under
additional conditions, we prove that the residual members from asymptotics of
Theorem 2 are square summable with respect to $\lambda,$ see Theorem 3. In
Section 4, we construct the non-fundamental systems of solutions analytic at
$\lambda$ from the large sector, see Theorem 4. In Section 5, we give
application of our results to a second order equation. Appendix A contains the
detailed proof of auxiliary Theorem 1. In Appendix B, we provide the basic
facts on holomorphic mappings and obtain Proposition 5 that is used in the
proof of Lemma 2.

The main results of the paper are Theorems 2–4. They can be applied in
studying inverse spectral problems for the higher-order operators with
distribution coefficients.

##  2 Preliminaries

Introduce the function $p(x)=\int_{0}^{x}\rho(t)\,dt$ and the matrix
$D(x)=[d_{jk}(x)]_{j,k=1}^{n},$ where

$d_{jk}(x)=\left\\{\begin{array}[]{cc}a_{jk}(x),&b_{j}=b_{k},\\\ 0,&b_{j}\neq
b_{k},\end{array}\right.\quad j,k=\overline{1,n},\quad x\geq 0.$

For the parameter $\alpha\geq 0,$ consider the matrix-function
$M_{\alpha}(x)=[m_{jk}(x)]_{j,k=1}^{n}$ of order $n$ that is the solution of
the initial-value problem

$M^{\prime}_{\alpha}(x)=D(x)M_{\alpha}(x),\;x\geq 0,\quad
M_{\alpha}(\alpha)=I,$ (2.1)

where $I$ is the identity matrix. Here and below, we denote the dependence on
$\alpha$ by the last lower index and omit it for brevity if the obtained
notation has two or more indices.

The existence and the uniqueness of $M_{\alpha}(x)$ along with some its
properties is ensured by the following proposition, being a consequence of the
results obtained in [12].

######  Proposition 1.

There exists a unique solution $M_{\alpha}(x)=[m_{jk}(x)]_{j,k=1}^{n}$ of
initial-value problem (2.1) whose components belong to $AC[0,\infty).$ It has
the following properties.

  1. 1.

For each $x\geq 0,$ the matrix $M_{\alpha}(x)$ is invertible, and the
components of the matrix $M^{-1}_{\alpha}(x)=[\tilde{m}_{jk}(x)]_{j,k=1}^{n}$
belong to $AC[0,\infty).$

  2. 2.

If $b_{j}\neq b_{k}$ for $j,k\in\overline{1,n},$ then
$m_{jk}\equiv\tilde{m}_{jk}\equiv 0.$

  3. 3.

$\max_{j,k=\overline{1,n}}\sup_{x\geq\alpha}|m_{jk}(x)|\leq
e^{a},\quad\max_{j,k=\overline{1,n}}\sup_{x\geq\alpha}|\tilde{m}_{jk}(x)|\leq
e^{a},\quad a:=n\|A\|_{L[0,\infty)}.$ (2.2)

  4. 4.

The matrices $M_{\alpha}(x)$ and $B$ commute.

  5. 5.

If the numbers $b_{j},$ $j=\overline{1,n},$ are distinct, then we have
$M_{\alpha}(x)=\mathrm{diag}\\{\exp(\int_{\alpha}^{x}a_{jj}(t)\,dt)\\}_{j=1}^{n}.$

Introduce the matrices $Q_{\alpha}(x)=[q_{jk}(x)]_{j,k=1}^{n}$ and
$R_{\alpha}(x,\lambda)=[r_{jk}(x,\lambda)]_{j,k=1}^{n}$ as follows:

$Q_{\alpha}(x)=M_{\alpha}^{-1}(x)(A(x)-D(x))M_{\alpha}(x),\quad
R_{\alpha}(x,\lambda)=M_{\alpha}^{-1}(x)C(x,\lambda)M_{\alpha}(x),$

where $x\geq 0$ and $\lambda\in\mathbb{C}_{0}.$ Using property 2 of
Proposition 1, it is easy to see that if $b_{j}=b_{k}$ for
$j,k\in\overline{1,n},$ then $q_{jk}\equiv 0.$ By (2.2), we also have

$\|Q_{\alpha}\|_{L[\alpha,\infty)}\leq
e^{2a}\|A-D\|_{L[\alpha,\infty)},\quad\|R_{\alpha}(\cdot,\lambda)\|_{L[\alpha,\infty)}\leq
e^{2a}\gamma_{\alpha}(\lambda),\quad\alpha\geq 0,\;\lambda\in\mathbb{C}_{0},$
(2.3)

where we put
$\gamma_{\alpha}(\lambda)=\|C(\cdot,\lambda)\|_{L[\alpha,\infty)}.$ Condition
(1.2) yields that $\gamma_{\alpha}(\lambda)\to 0$ uniformly on $\alpha\geq 0$
as $\lambda\to\infty.$ As well, for any fixed $\alpha\geq 0,$ the function
$\gamma_{\alpha}(\lambda)$ is continuous on $\mathbb{C}_{0},$ since the
elements of $C(x,\lambda)$ are holomorphic mappings of
$\lambda\in\mathbb{C}_{0}$ to $L[0,\infty).$

Further, we need to study the behavior of certain values depending
additionally on the parameters $k\in\overline{1,n}$ and $\omega\in\mathbb{C}.$
Let $\Lambda\subset\mathbb{C}_{0}$ be a domain such that

$\mathrm{Re}\,(\lambda
b_{j})\geq\mathrm{Re}\,(\lambda\omega)\geq\mathrm{Re}\,(\lambda b_{l}),\quad
j=\overline{1,k-1},\;l=\overline{k,n},\quad\lambda\in\overline{\Lambda}.$
(2.4)

For $j,l\in\overline{1,n}$ and $\alpha\geq 0,$ consider the following
functions of $s,x\geq\alpha$ and $\lambda\in\overline{\Lambda}:$

$\left.\begin{array}[]{c}\displaystyle\nu_{jl}(s,x,\lambda)=\int_{t_{1}}^{t_{2}}q_{jl}(t)\exp\big{(}\lambda
g_{jl}(s,x,t)\big{)}\,dt,\\\\[8.53581pt]
\displaystyle\varkappa_{jl}(s,x,\lambda)=\int_{t_{1}}^{t_{2}}r_{jl}(t,\lambda)\exp\big{(}\lambda
g_{jl}(s,x,t)\big{)}\,dt,\quad\lambda\neq 0,\end{array}\right\\}$ (2.5)

where $g_{jl}(s,x,t):=(b_{l}-\omega)(p(t)-p(s))+(b_{j}-\omega)(p(x)-p(t))$ and

$(t_{1},t_{2}):=\left\\{\begin{array}[]{cc}(x,s),&j,l<k,\\\
\big{(}\max\\{x,s\\},\infty\big{)},&j<k\leq l,\\\
\big{(}\alpha,\min\\{x,s\\}\big{)},&l<k\leq j,\\\ (s,x),&k\leq
j,l.\end{array}\right.$ (2.6)

In (2.5) and below, we agree that in the case $t_{1}\geq t_{2},$ an integral
$\int_{t_{1}}^{t_{2}}$ is considered to be $0.$ In the opposite case
$t_{1}<t_{2},$ for the variable of integration $t$ in (2.5), we have that
$t\geq s\Leftrightarrow l\geq k$ and $t\leq x\Leftrightarrow j\geq k.$ Taking
into account (2.4) and that $p(x)$ is increasing, we obtain the inequality

$\mathrm{Re}\,\big{(}\lambda g_{jl}(s,x,t)\big{)}\leq 0,\quad
t_{1}<t<t_{2},\quad s,x\geq\alpha,\;\lambda\in\overline{\Lambda}.$ (2.7)

In particular, it yields that the integrals in (2.5) are well-defined as the
Lebesgue integrals.

######  Lemma 1.

For $\lambda\in\overline{\Lambda},$ denote
$\displaystyle\theta_{\alpha}(\lambda)=\sup_{\begin{subarray}{c}j,l\in\overline{1,n};\\\
s,x\geq\alpha\end{subarray}}|\nu_{jl}(s,x,\lambda)|.$ If $\lambda\to\infty,$
then we have ${\theta_{\alpha}(\lambda)\to 0}$ uniformly on $\alpha.$ As well,
the following estimates hold:

$\sup_{\begin{subarray}{c}j,l=\overline{1,n};\\\
s,x\geq\alpha\end{subarray}}|\varkappa_{jl}(s,x,\lambda)|\leq\gamma_{\alpha}(\lambda)e^{2a},\quad\theta_{\alpha}(\lambda)\leq
ae^{2a},\quad\lambda\in\overline{\Lambda}\setminus\\{0\\}.$ (2.8)

###### Proof.

Inequalities (2.8) easily follow from (2.3) and (2.7). Let us prove
$\theta_{\alpha}(\lambda)\to 0$ as $\overline{\Lambda}\ni\lambda\to\infty$
uniformly on $\alpha\geq 0.$ It is equivalent to that for any $\varepsilon>0$
and $j,l\in\overline{1,n},$ there exists $\lambda_{*}>0$ such that

$|\nu_{jl}(s,x,\lambda)|<\varepsilon,\quad
s,x\geq\alpha,\quad|\lambda|\geq\lambda_{*},$ (2.9)

and $\lambda_{*}$ does not depend on $\alpha.$ If $b_{j}=b_{l},$ then
$\nu_{jl}=0,$ and (2.9) is obvious. Let $b_{j}\neq b_{l};$ for definiteness,
we assume $j,l<k$ (the other cases in (2.6) are proceeded analogously). Then,
we can consider only $s>x$ (otherwise $\nu_{jl}=0$), and

$\nu_{jl}(s,x,\lambda)=\int_{x}^{s}q_{jl}(t)\exp(\lambda
g_{jl}\big{(}s,x,t)\big{)}\,dt.$

Let $T>0$ be such that $\|q_{jl}\|_{L[T,\infty)}<\frac{\varepsilon}{2}.$ If
$x\geq T,$ we immediately have

$|\nu_{jl}(s,x,\lambda)|\leq\int_{x}^{s}|q_{jl}(t)|\,dt\leq\int_{T}^{\infty}|q_{jl}(t)|\,dt<\frac{\varepsilon}{2}.$

If $x<T,$ then

$|\nu_{jl}(s,x,\lambda)|\leq\Bigg{|}\int_{\min\\{s,T\\}}^{s}q_{jl}(t)e^{\lambda
g_{jl}(s,x,t)}\,dt\Bigg{|}+|I_{1}|,\quad
I_{1}:=\int_{x}^{\min\\{s,T\\}}q_{jl}(t)e^{\lambda g_{jl}(s,x,t)}\,dt.$

Here, the first summand either equals $0$ (if $\min\\{s,T\\}=s$) or does not
exceed $\frac{\varepsilon}{2}$ (if $\min\\{s,T\\}=T$). In any case, we have

$|\nu_{jl}(s,x,\lambda)|<|I_{1}|+\frac{\varepsilon}{2}.$ (2.10)

Changing the variable $t$ on $\xi=p(t)$ in the integral $I_{1},$ we obtain

$I_{1}=\int_{l_{1}}^{l_{2}}f_{jl}(\xi)e^{\lambda
g_{jl}(s,x,p^{-1}(\xi))}\,d\xi,\quad l_{1}:=p(x),\;l_{2}:=p(\min\\{s,T\\}),$
(2.11)

where $f_{jl}(\xi)=\frac{q_{jl}(p^{-1}(\xi))}{\rho(p^{-1}(\xi))},$ while
$p^{-1}$ is the inverse function to $p,$ and $[l_{1},l_{2}]\subseteq[0,p(T)].$
Note that $\int_{0}^{p(T)}|f_{jl}(\xi)|\,d\xi=\int_{0}^{T}|q_{jl}(t)|\,dt,$
which yields $f_{jl}\in L[0,p(T)].$ By this reason, there exists a
continuously differentiable function $\tilde{f}_{jl}\in C^{(1)}[0,p(T)]$ such
that

$\|f_{jl}-\tilde{f}_{jl}\|_{L[0,p(T)]}\leq\frac{\varepsilon}{4}.$

Using this inequality and that $\big{|}e^{\lambda
g_{jl}(s,x,p^{-1}(\xi))}\big{|}\leq 1$ as $\xi\in[l_{1},l_{2}],$ we arrive at

$|I_{1}|\leq\int_{l_{1}}^{l_{2}}|f_{jl}(\xi)-\tilde{f}_{jl}(\xi)|\,d\xi+|I_{2}|\leq\frac{\varepsilon}{4}+|I_{2}|,\quad
I_{2}:=\int_{l_{1}}^{l_{2}}\tilde{f}_{jl}(\xi)e^{\lambda
g_{jl}(s,x,p^{-1}(\xi))}\,d\xi.$ (2.12)

Note that
$g_{jl}(s,x,p^{-1}(\xi))=(b_{l}-b_{j})\xi+p(x)(b_{j}-\omega)-p(s)(b_{l}-\omega),$
and

$\frac{\partial e^{\lambda
g_{jl}(s,x,p^{-1}(\xi))}}{\partial\xi}=(b_{l}-b_{j})\lambda e^{\lambda
g_{jl}(s,x,p^{-1}(\xi))}.$

Applying integration in parts to $I_{2},$ we obtain

$I_{2}=\frac{1}{\lambda(b_{l}-b_{j})}\left(\tilde{f}_{jl}(l_{2})e^{\lambda
g_{jl}(s,x,p^{-1}(l_{2}))}-\tilde{f}_{jl}(l_{1})e^{\lambda
g_{jl}(s,x,p^{-1}(l_{1}))}-\int_{l_{1}}^{l_{2}}\tilde{f}^{\prime}_{jl}(\xi)e^{\lambda
g_{jl}(s,x,p^{-1}(\xi))}\,d\xi\right).$

Denote
$M_{jl}^{\varepsilon}=2\displaystyle\sup_{\xi\in[0,p(T)]}|\tilde{f}_{jl}(\xi)|+\|\tilde{f}_{jl}\|_{L[0,p(T)]}.$
The obtained equality yields
$|I_{2}|\leq\frac{M_{jl}^{\varepsilon}}{|\lambda||b_{l}-b_{j}|},$ and for
$|\lambda|\geq\lambda_{*}=\frac{4M_{jl}^{\varepsilon}}{\varepsilon|b_{l}-b_{j}|},$
we have the estimate $|I_{2}|\leq\frac{\varepsilon}{4}.$ Combining it with
(2.10) and (2.12), we obtain (2.9). ∎

Now, we prove a lemma on the solvability of a system of integral equations,
applying the method of successive approximations. We will use this lemma in
the proofs of Theorems 2 and 4, assigning different values to $k,$ $\omega$
and $\Lambda.$

######  Lemma 2 (the main lemma).

Let the parameters $k\in\overline{1,n},$ $\alpha\geq 0,$ and
$\omega\in\mathbb{C}$ be fixed, and let a domain
$\Lambda\subset\mathbb{C}_{0}$ be such that (2.4) holds. Denote by ${\bf
BC}_{n}$ the Banach space of the vector functions $\mathrm{\bf
z}(x)=[z_{j}(x)]_{j=1}^{n}$ whose components are bounded and continuous on
$[\alpha,\infty),$ equipped with the norm

$\|\mathrm{\bf z}\|_{{\bf
BC}_{n}}=\max_{j=\overline{1,n}}\sup_{x\geq\alpha}|z_{j}(x)|.$

We determine the operator ${\cal V}_{k}(\lambda)$ that acts on $\mathrm{\bf
z}(x)=[z_{j}(x)]_{j=1}^{n}\in{\bf BC}_{n}$ as follows:

${\cal V}_{k}(\lambda){\bf z}:=[f_{j}(x)]_{j=1}^{n},\quad
f_{j}(x)=\left\\{\begin{array}[]{cc}\displaystyle-\sum_{l=1}^{n}\int_{x}^{\infty}v_{jl}(t,\lambda)e^{\lambda(b_{j}-\omega)(p(x)-p(t))}z_{l}(t)\,dt,&j=\overline{1,k-1},\\\
\displaystyle\sum_{l=1}^{n}\int_{\alpha}^{x}v_{jl}(t,\lambda)e^{\lambda(b_{j}-\omega)(p(x)-p(t))}z_{l}(t)\,dt,&j=\overline{k,n},\end{array}\right.$
(2.13)

where for $j,l=\overline{1,n},$ we put
$v_{jl}(x,\lambda):=q_{jl}(x)+r_{jl}(x,\lambda).$ Then, for
$\lambda\in\overline{\Lambda}\setminus\\{0\\},$ the operator ${\cal
V}_{k}(\lambda)$ is a linear bounded operator in ${\bf BC}_{n}$ with the
following properties.

1) There exists $\lambda_{\alpha}>0$ such that

$\|{\cal V}^{2}_{k}(\lambda)\|_{{\bf BC}_{n}\to{\bf
BC}_{n}}<\frac{1}{2},\quad\lambda\in\overline{\Lambda^{\alpha}},\quad\Lambda^{\alpha}:=\\{\lambda\in\Lambda\colon|\lambda|>\lambda_{\alpha}\\},$
(2.14)

where we denoted ${\cal V}^{\eta}_{k}(\lambda):=\underbrace{{\cal
V}_{k}(\lambda)\ldots{\cal V}_{k}(\lambda)}_{\eta\text{ times}}.$ Moreover,
$\lambda_{\alpha}\to 0$ as $\alpha\to\infty.$

2) For any $\lambda\in\overline{\Lambda^{\alpha}}$ and $\bf{w}\in{\bf
BC}_{n},$ the equation

$\mathrm{\bf z}={\bf w}+{\cal V}_{k}(\lambda)\mathrm{\bf z}$ (2.15)

has a unique solution $\mathrm{\bf z}\in{\bf BC}_{n},$ which satisfies the
estimates

$\|\mathrm{\bf z}\|_{{\bf BC}_{n}}\leq N_{\alpha}\|\mathrm{\bf w}\|_{{\bf
BC}_{n}},\quad\|\mathrm{\bf z}-\mathrm{\bf w}\|_{{\bf BC}_{n}}\leq
N_{\alpha}\|{\cal V}_{k}(\lambda){\bf w}\|_{{\bf BC}_{n}}.$ (2.16)

Here and below, $N_{\alpha}$ denotes different constants depending only on
$\alpha.$

3) Let $\mathrm{\bf w}$ be a mapping of $\lambda$ to ${\bf BC}_{n}$ that is
holomorphic in $\Lambda^{\alpha}$ and continuous on
$\overline{\Lambda^{\alpha}}.$ Then, the solution of (2.15) $\mathrm{\bf z},$
being a mapping of $\lambda$ to ${\bf BC}_{n},$ possesses the same properties.

###### Proof.

For the variable of integration $t$ in (2.13), by (2.4), we have
$(p(x)-p(t))\mathrm{Re}\,\big{(}\lambda(b_{j}-\omega)\big{)}\leq 0.$ This
inequality along with (2.3) yield that ${\cal V}_{k}(\lambda)$ is a bounded
operator in ${\bf BC}_{n}$ with the estimate

$\|{\cal V}_{k}(\lambda)\|_{{\bf BC}_{n}\to{\bf BC}_{n}}\leq
ne^{2a}(\gamma_{\alpha}(\lambda)+a).$ (2.17)

1) Let us consider the operator ${\cal V}^{2}_{k}(\lambda).$ For an arbitrary
$\mathrm{\bf z}=[z_{j}(x)]_{j=1}^{n}\in{\bf BC}_{n},$ put $\mathrm{\bf
f}=[f_{j}(x)]_{j=1}^{n}:={\cal V}_{k}^{2}\mathrm{\bf z}.$ By the definition,
for $\lambda\in\overline{\Lambda}\setminus\\{0\\}$ and $j=\overline{1,n},$ we
have

$f_{j}(x)=\sum_{l,m=1}^{n}\pm\int_{\tau_{j}(x)}^{\sigma_{j}(x)}v_{jl}(t,\lambda)e^{\lambda(b_{j}-\omega)(p(x)-p(t))}\int_{\tau_{l}(t)}^{\sigma_{l}(t)}v_{lm}(s,\lambda)e^{\lambda(b_{l}-\omega)(p(t)-p(s))}z_{m}(s)ds\,dt,$

where
$\big{(}\tau_{j}(x),\sigma_{j}(x)\big{)}:=\left\\{\begin{array}[]{cc}(x,\infty),&j=\overline{1,k-1},\\\\[2.84526pt]
(\alpha,x),&j=\overline{k,n},\end{array}\right.$ and for each summand, one of
two signs $+$ or $-$ should be chosen instead of $\pm.$ Changing the
integration order, we obtain

$f_{j}(x)=\sum_{m=1}^{n}\sum_{l=1}^{n}\pm\int_{\alpha}^{\infty}v_{lm}(s,\lambda)\big{(}\varkappa_{jl}(s,x,\lambda)+\nu_{jl}(s,x,\lambda)\big{)}z_{m}(s)\,ds,$

and

$\|\mathrm{\bf f}\|_{{\bf BC}_{n}}\leq\|\mathrm{\bf z}\|_{{\bf
BC}_{n}}\left(\sup_{\begin{subarray}{c}j,l=\overline{1,n};\\\
s,x\geq\alpha\end{subarray}}\big{|}\varkappa_{jl}(s,x,\lambda)|+\theta_{\alpha}(\lambda)\right)\sum_{l,m=1}^{n}\int_{\alpha}^{\infty}|v_{lm}(s,\lambda)|ds.$

Put
$K_{\alpha}:=\displaystyle\sup_{|\lambda|\geq\varphi(\alpha)}\gamma_{\alpha}(\lambda)+\|A-D\|_{L[\alpha,\infty)},$
where $\varphi(\alpha)$ is taken from (1.6). Then, the latter inequality along
with (2.3) and (2.8) yield

$\|{\cal V}^{2}_{k}(\lambda)\|_{{\bf BC}_{n}\to{\bf
BC}_{n}}<n^{2}e^{2a}K_{\alpha}\big{(}e^{2a}\gamma_{\alpha}(\lambda)+\theta_{\alpha}(\lambda)\big{)},\quad\lambda\in\overline{\Lambda},\;|\lambda|\geq\varphi(\alpha).$
(2.18)

By Lemma 1 and condition (1.2), we have
$e^{2a}\gamma_{\alpha}(\lambda)+\theta_{\alpha}(\lambda)\to 0$ as
$\lambda\to\infty$ uniformly on $\alpha\geq 0.$ Then, there exists
$\lambda_{\alpha}\geq\varphi(\alpha)$ such that (2.14) holds.

Note that condition (1.6) yields $K_{\alpha}\to 0$ as $\alpha\to\infty.$
Applying the inequalities $\gamma_{\alpha}(\lambda)\leq K_{\alpha}$ and
$\theta_{\alpha}(\lambda)\leq ae^{2a}$ to (2.18), for a sufficiently large
$\alpha,$ we arrive at

$\|{\cal V}^{2}_{k}(\lambda)\|_{{\bf BC}_{n}\to{\bf
BC}_{n}}<n^{2}e^{4a}K_{\alpha}\big{(}a+K_{\alpha}\big{)}<\frac{1}{2},\quad\lambda\in\overline{\Lambda},\;|\lambda|\geq\varphi(\alpha).$

Thus, for a sufficiently large $\alpha,$ we can put
$\lambda_{\alpha}:=\varphi(\alpha)\to 0$ as $\alpha\to\infty.$

2) Now, we solve equation (2.15) by the method of successive approximations.
Consider the series

$\mathrm{\bf z}=\mathrm{\bf w}+\sum_{\eta=0}^{\infty}{\cal
V}^{2\eta+1}_{k}(\lambda)\mathrm{\bf w}+\sum_{\eta=0}^{\infty}{\cal
V}^{2\eta+2}_{k}(\lambda)\mathrm{\bf
w},\quad\lambda\in\overline{\Lambda^{\alpha}}.$ (2.19)

By (2.14), this series converges in ${\bf BC}_{n}$ and gives us the unique
solution of (2.15). Using also (2.17), it is easy to obtain estimates (2.16).

3) One can show that under our assumptions on $\mathrm{\bf w},$ the mapping
${\cal V}_{k}(\lambda)\mathrm{\bf w}$ is continuous on
$\overline{\Lambda^{\alpha}}$ and holomorphic in $\Lambda^{\alpha}$ (for the
proof of the holomorphy property, see Proposition 5 in Appendix B). By
induction, each summand in (2.19) is continuous on
$\overline{\Lambda^{\alpha}}$ and holomorphic in $\Lambda^{\alpha}.$ The
values $\gamma_{\alpha}(\lambda)$ and $\|\mathrm{\bf w}(\cdot,\lambda)\|_{{\bf
BC}_{n}}$ are continuous functions of $\lambda.$ Then, by (2.14) and (2.17),
the series in (2.19) converges uniformly on $\lambda$ from every compact
subset of $\overline{\Lambda^{\alpha}}.$ The required statement follows from
the fact that the uniform limit of continuous mappings is also continuous and
from Theorem F in Appendix B. ∎

In what follows, we need the following auxiliary statement.

######  Theorem 1.

Assume additionally that the elements of the matrix $A-D$ belong to
$L_{2}[\alpha,\infty)$ and
$\displaystyle\mathop{\mathrm{essinf}}_{x\geq\alpha}\,\rho(x)>0.$ Then, for an
arbitrary half-line $\Sigma\subset\overline{\Lambda},$ we have
$\theta_{\alpha}\in L_{2}(\Sigma).$

Its proof is based on the technique offered in [13, Sect. 7], wherein a
similar statement was proved in the case of a finite interval (see also [22]).
However, the case of a half-line studied here brings several differences. For
this reason, as well as for the sake of completeness, we provide the proof in
Appendix A.

##  3 Fundamental systems of solutions (FSS)

Let $\mathrm{\bf y}_{1},\mathrm{\bf y}_{2},\ldots,\mathrm{\bf y}_{m}$ be a set
of $m\geq 1$ solutions of system (1.1) with some fixed
$\lambda\in\mathbb{C}_{0}.$ It is convenient to represent this set as the
$n\times m$ matrix $Y(x,\lambda)$ whose $k$-th column coincides with
$\mathrm{\bf y}_{k},$ $k=\overline{1,m}.$ We call the matrix $Y(x,\lambda)$
system of solutions of (1.1).

Let a system of solutions $Y(x,\lambda)$ consist of $m=n$ columns. If its
columns are linearly independent on $x\in[0,\infty),$ $Y(x,\lambda)$ is called
FSS of (1.1). It is known that criterion of FSS is $\det Y(x,\lambda)\neq 0$
for some $x\geq 0.$

In this section, we obtain fundamental systems of solutions
$Y_{\alpha}(x,\lambda)$ depending on the parameter $\alpha\geq 0.$ They are
constructed individually for $\lambda$ belonging to special sectors. Consider
the set of the lines $\mathrm{Re}\,(\lambda b_{j})=\mathrm{Re}\,(\lambda
b_{l}),$ where $j,l=\overline{1,n}$ and $b_{j}\neq b_{l}$ (see [12, 3]). These
lines split the plane $\lambda\in\mathbb{C}$ into sectors of the form
$\Gamma_{\kappa}=\\{\lambda\in\mathbb{C}_{0}\colon\beta_{\kappa-1}<\arg\lambda<\beta_{\kappa}\\}.$
For each such sector $\Gamma_{\kappa},$ one can renumber
$\\{b_{j}\\}_{j=1}^{n}$ so that (1.5) holds. Obviously, there is a finite
number $J$ of the sectors $\Gamma_{\kappa},$ $\kappa=\overline{1,J},$ they are
non-overlapping, and
$\mathop{\cup}_{\kappa=1}^{J}\overline{\Gamma_{\kappa}}=\mathbb{C}.$

Let us consider $\lambda\in\overline{\Gamma_{\kappa}}$ with fixed
$\kappa\in\overline{1,J}$ and proceed to a numeration satisfying (1.5). Note
that an arbitrary numeration of $\\{b_{j}\\}_{j=1}^{n}$ can be achieved by
permutations of rows and columns of the objects in (1.1). In fact, the
mentioned permutations do not influence the formulations of Theorems 2 and 3.

######  Theorem 2.

For any $\alpha\geq 0,$ there exists $\lambda_{\alpha}>0$ such that for
$\lambda\in\overline{\Gamma^{\alpha}_{\kappa}},$
${\Gamma}^{\alpha}_{\kappa}:=\\{\lambda\in\Gamma_{\kappa}\colon|\lambda|>\lambda_{\alpha}\\},$
there exists a FSS of (1.1)
$Y_{\alpha}(x,\lambda)=[y_{jk}(x,\lambda)]_{j,k=1}^{n}$ with the following
properties.

  1. 1.

For $j,k=\overline{1,n}$ and $\lambda\in\overline{\Gamma^{\alpha}_{\kappa}},$
uniformly on $x\geq\alpha,$ we have

$y_{jk}(x,\lambda)=e^{\lambda
b_{k}(p(x)-p(\alpha))}\big{(}m_{jk}(x)+s_{jk}(x,\lambda)\big{)},\quad
s_{jk}(x,\lambda)=o(1),\quad\lambda\to\infty,$ (3.1)

where $m_{jk}(x)$ are the elements of the matrix $M_{\alpha}$ determined by
(2.1).

  2. 2.

For a fixed $x\geq 0,$ the functions $y_{jk}(x,\lambda),$
$j,k=\overline{1,n,}$ are continuous on $\overline{\Gamma_{\kappa}^{\alpha}}$
and analytic in $\Gamma_{\kappa}^{\alpha}.$

  3. 3.

$y_{jk}(\alpha,\lambda)=\delta_{jk}$ for $k=\overline{1,n}$ and
$j=\overline{k,n},$ where $\delta_{jk}$ is the Kronecker delta.

Moreover, $\lambda_{\alpha}\to 0$ as $\alpha\to\infty.$

In the proof, we need the following proposition, which is a consequence of
Proposition 2 and Corollary 1 in [12].

######  Proposition 2.

For $\alpha>0$ and $\lambda\in\mathbb{C}_{0},$ consider the initial-value
problem for system (1.1) on the finite segment $x\in[0,\alpha]$ under the
initial condition $\mathrm{\bf y}(\alpha,\lambda)=[w_{j}(\lambda)]_{j=1}^{n}.$
Then, the initial-value problem has a unique solution $\mathrm{\bf
y}(x,\lambda)=[y_{j}(x,\lambda)]_{j=1}^{n}$ with the components
$y_{j}(\cdot,\lambda)\in AC[0,\alpha].$ Moreover, if the functions
$w_{l}(\lambda),$ $l=\overline{1,n},$ are analytic in (continuous on)
$\Lambda\subseteq\mathbb{C}_{0},$ then for a fixed $x\in[0,\alpha],$ the
functions $y_{j}(x,\lambda),$ $j=\overline{1,n},$ are also analytic in
(continuous on) $\Lambda.$

###### Proof of Theorem 2.

First, we construct the needed solutions of (1.1) for $x\geq\alpha,$ then we
extend them on $[0,\alpha]$ applying Proposition 2. For $k=\overline{1,n}$ and
$x\geq\alpha,$ let us represent the $k$-th column of the matrix function
$Y_{\alpha}(x,\lambda)$ as

$[y_{jk}(x,\lambda)]_{j=1}^{n}=M_{\alpha}(x)\mathrm{\bf
z}_{k}(x,\lambda)e^{\lambda b_{k}(p(x)-p(\alpha))},\quad\mathrm{\bf
z}_{k}(x,\lambda)=[z_{jk}(x,\lambda)]_{j=\overline{1,n}}.$ (3.2)

Substituting $\mathrm{\bf y}=[y_{jk}(x,\lambda)]_{j=1}^{n}$ into (1.1), we
arrive at the equation

$\mathrm{\bf z}^{\prime}_{k}(x,\lambda)=\lambda r(x)(B\mathrm{\bf
z}_{k}(x,\lambda)-b_{k}\mathrm{\bf
z}_{k}(x,\lambda))+(Q_{\alpha}(x)+R_{\alpha}(x,\lambda))\mathrm{\bf
z}_{k}(x,\lambda),\quad x\geq\alpha.$ (3.3)

Now, we construct a solution of (3.3) $\mathrm{\bf z}_{k}\in{\bf BC}_{n}$
satisfying the conditions

$\left\\{\begin{array}[]{cc}\displaystyle\lim_{x\to\infty}z_{jk}(x,\lambda)=0,&j=\overline{1,k-1},\\\\[5.69054pt]
z_{jk}(\alpha,\lambda)=\delta_{jk},&j=\overline{k,n}.\end{array}\right.$ (3.4)

Integrating (3.3) under conditions (3.4), for each fixed $k\in\overline{1,n},$
we obtain the equation

$\mathrm{\bf z}_{k}=\mathrm{\bf e}_{k}+{\cal V}_{k}(\lambda)\mathrm{\bf
z}_{k},\quad\mathrm{\bf e}_{k}:=[\delta_{jk}]_{j=1}^{n},$ (3.5)

where the operator ${\cal V}_{k}(\lambda)$ is determined in Lemma 2 with
$\omega:=b_{k}$ and $\Lambda:=\Gamma_{\kappa}.$ We can apply Lemma 2, since
(1.5) yields (2.4). Then, for $\lambda\in\overline{\Gamma_{\kappa}^{\alpha}},$
equation (3.5) has a unique solution $\mathrm{\bf z}_{k}\in{\bf BC}_{n},$ and

$\|\mathrm{\bf z}_{k}-\mathrm{\bf e}_{k}\|_{{\bf BC}_{n}}\leq
N_{\alpha}\|\mathrm{\bf z}^{1}_{k}\|_{{\bf BC}_{n}},\quad\mathrm{\bf
z}^{1}_{k}:={\cal V}_{k}(\lambda)\mathrm{\bf e}_{k}.$ (3.6)

Introduce operators ${\cal R}_{k}(\lambda)$ and ${\cal Q}_{k}(\lambda)$
analogously to ${\cal V}_{k}(\lambda)$ in (2.13), replacing each component
$v_{jl}(t,\lambda)$ by $r_{jl}(t,\lambda)$ and $q_{jl}(t),$ respectively. By
the same way as (2.17), we obtain that $\|{\cal R}_{k}(\lambda)\|_{{\bf
BC}_{n}\to{\bf BC}_{n}}\leq ne^{2a}\gamma_{\alpha}(\lambda).$ We also have
${\cal Q}_{k}\mathrm{\bf e}_{k}=[\nu_{jk}(\alpha,x,\lambda)]_{j=1}^{n},$ and,
consequently, $\|{\cal Q}_{k}\mathrm{\bf e}_{k}\|_{{\bf
BC}_{n}}\leq\theta_{\alpha}(\lambda).$ Then, since ${\cal
V}_{k}(\lambda)={\cal R}_{k}(\lambda)+{\cal Q}_{k}(\lambda),$ we obtain

$\|\mathrm{\bf z}^{1}_{k}\|_{{\bf BC}_{n}}\leq\|{\cal
R}_{k}(\lambda)\mathrm{\bf e}_{k}\|_{{\bf BC}_{n}}+\|{\cal Q}_{k}\mathrm{\bf
e}_{k}\|_{{\bf BC}_{n}}\leq
ne^{2a}\gamma_{\alpha}(\lambda)+\theta_{\alpha}(\lambda).$

Using (3.6) and the latter inequality, we arrive at

$\|\mathrm{\bf z}_{k}-\mathrm{\bf e}_{k}\|_{{\bf BC}_{n}}\leq
N_{\alpha}(\gamma_{\alpha}(\lambda)+\theta_{\alpha}(\lambda)),\quad\lambda\in\overline{\Gamma_{\kappa}^{\alpha}}.$
(3.7)

Constructing the matrix
$Y_{\alpha}(x,\lambda)=[y_{jk}(x,\lambda)]_{j,k=1}^{n}$ with the components
get by (3.2), we obtain a system of solutions of (1.1) for $x\geq\alpha,$
possessing the needed properties 2 and 3. Property 3 yields that
$Y_{\alpha}(x,\lambda)$ is a FSS. From (2.2), (3.2) and (3.7) it follows that
formulae (3.1) hold with

$|s_{jk}(x,\lambda)|\leq
N_{\alpha}(\gamma_{\alpha}(\lambda)+\theta_{\alpha}(\lambda)),\quad
j,k=\overline{1,n},\;x\geq\alpha,$ (3.8)

wherein, by Lemma 1 and condition (1.2), the right side tends to $0$ as
$\lambda\to\infty.$ Thus, property 1 is also established.

It remains to extend each column vector $[y_{jk}(x,\lambda)]_{j=1}^{n}$ on
$x\in[0,\alpha]$ by the solution of system (1.1) under the initial condition
$\mathrm{\bf y}(\alpha,\lambda)=[y_{jk}(\alpha,\lambda)]_{j=1}^{n}.$ By virtue
of Proposition 2, the needed solution exists and keeps property 2. ∎

Further, we give a complement to Theorem 2 under additional assumptions on the
matrices $A(x)$ and $C(x,\lambda),$ that concerns properties of the residual
members $s_{jk}(x,\lambda)$ in (3.1).

######  Theorem 3.

Assume that the elements of the matrix $A-D$ belong to $L_{2}[\alpha,\infty),$
$\displaystyle\mathop{\mathrm{essinf}}_{x\geq\alpha}\,\rho(x)>0,$ and
$\gamma_{\alpha}(\lambda)=O(\lambda^{-1})$ for
$|\lambda|\geq\lambda_{\alpha}.$ Then, for an arbitrary half-line
$\Sigma\subset\overline{\Gamma_{\kappa}^{\alpha}}$ and the values
$s_{jk}(x,\lambda)$ in (3.1), we have

$\sup_{x\geq\alpha}|s_{jk}(x,\lambda)|\in L_{2}(\Sigma),\quad
j,k=\overline{1,n}.$

The statement of the theorem is a simple corollary of inequality (3.8) and
Theorem 1.

##  4 Solutions analytic in large sectors

In this section, we restrict ourselves to the case when
$\\{b_{j}\\}_{j=1}^{n}$ is the set of all $n$-th roots of unity, i.e. up to a
numeration, $b_{j}=\exp(\frac{2\pi ij}{n}),$ $j=\overline{1,n}.$ Remind that
exactly this case realizes after reducing $n$-th order equation (1.3) to
system (1.1).

If $n>2,$ the sectors $\Gamma_{\kappa}$ introduced in Section 3 have the form

$\Gamma_{\kappa}=\left\\{\lambda\in{\mathbb{C}}_{0}\colon\frac{\pi(\kappa-1)}{n}<\arg\lambda<\frac{\pi\kappa}{n}\right\\},\quad\kappa=\overline{1,2n}.$
(4.1)

If $n=2,$ then we have two sectors: the right half-plane
$\Gamma_{1}=\\{\lambda\in\mathbb{C}\colon\mathrm{Re}\,\lambda>0\\}$ and the
left half-plane
$\Gamma_{2}=\\{\lambda\in\mathbb{C}\colon\mathrm{Re}\,\lambda<0\\}.$ Further,
we consider the case $n>2.$ The results of this section will be valid for
$n=2$ as well, if one takes formula (4.1) as the definition of the sectors
$\Gamma_{\kappa},$ $\kappa=\overline{1,4}.$ However, in this case, Theorem 4
is weaker than Theorem 2.

For $m\in\overline{2,n},$ consider a large sector including two neigboring
sectors $\Gamma_{\kappa}:$

$\Omega_{m}=\left\\{\lambda\in\mathbb{C}\colon\arg\lambda\in\Big{(}[(-1)^{m-1}-1]\frac{\pi}{2n};\
[(-1)^{m-1}+3]\frac{\pi}{2n}\Big{)}\right\\}.$

It is clear that
$\overline{\Omega_{m}}=\overline{\Gamma_{1}}\cup\overline{\Gamma_{\sigma}},$
where $\sigma=2n$ if $m$ is even and $\sigma=2$ if $m$ is odd. In what
follows, we fix the numeration of $\\{b_{j}\\}_{j=1}^{n}$ for which (1.5)
holds in $\kappa=1:$

$b_{2s+1}=e^{\frac{2\pi
is}{n}},\;s=\overline{0,\left\lfloor\frac{n-1}{2}\right\rfloor};\quad
b_{2p}=e^{-\frac{2\pi
ip}{n}},\;p=\overline{1,\left\lfloor\frac{n}{2}\right\rfloor}.$ (4.2)

For this fixed numeration and $\lambda$ from the neighboring sector
$\Gamma_{\sigma},$ some inequalities in (1.5) become incorrect. One can see
that

$\mathrm{Re}\,(\lambda b_{j})\geq\mathrm{Re}\,(\lambda
b_{m+1})\geq\mathrm{Re}\,(\lambda b_{l}),\quad
j=\overline{1,m-1},\;l=\overline{m,n},\quad\lambda\in\overline{\Gamma_{\sigma}},$
(4.3)

where and below we put $b_{n+1}:=b_{n}$ in the case $m=n.$

Now, we obtain non-fundamental system of solutions of (1.1) with analytic
dependence on the spectral parameter from the large sector $\Omega_{m}.$

######  Theorem 4.

Let $\alpha\geq 0,$ $m\in\overline{2,n},$ and
$\Omega_{m}^{\alpha}:=\\{\lambda\in\Omega_{m}\colon|\lambda|>\lambda_{\alpha}\\}.$
For $\lambda\in\overline{\Omega_{m}^{\alpha}},$ there exists a system of
solutions of (1.1)
$U_{\alpha}(x,\lambda)=[u_{jk}(x,\lambda)]_{\begin{subarray}{c}j=\overline{1,n},\\\
k=\overline{m,n}\end{subarray}}$ possessing the following properties.

  1. 1.

For $j=\overline{1,n}$ and $k=\overline{m,n},$ uniformly on $x\geq\alpha,$ we
have

$u_{jk}(x,\lambda)=\left\\{\begin{array}[]{cc}O\left(e^{\lambda
b_{m}(p(x)-p(\alpha))}\right),&\lambda\in\overline{\Gamma^{\alpha}_{1}},\\\
O\left(e^{\lambda
b_{m+1}(p(x)-p(\alpha))}\right),&\lambda\in\overline{\Gamma_{\sigma}^{\alpha}}.\end{array}\right.$

  2. 2.

For a fixed $x\geq 0,$ the functions $u_{jk}(x,\lambda),$ $j=\overline{1,n},$
$k=\overline{m,n},$ are continuous on $\overline{\Omega^{\alpha}_{m}}$ and
analytic in $\Omega^{\alpha}_{m}.$

  3. 3.

$u_{jk}(\alpha,\lambda)=\delta_{jk},$ $j,k=\overline{m,n}.$

In the proof, we need a lemma.

######  Lemma 3.

Consider the number $\omega=b_{m}\exp\big{(}\frac{(-1)^{m}\pi i}{n}\big{)}$
and the sector

$\Lambda=\left\\{\lambda\in\mathbb{C}_{0}\colon\arg\lambda\in\Big{(}\frac{(-1)^{m-1}\pi}{2n};\frac{(-1)^{m-1}\pi}{2n}+\frac{\pi}{n}\Big{)}\right\\},$
(4.4)

see Figure 1. The following inequalities hold:

$\mathrm{Re}\,(\lambda
b_{j})\geq\mathrm{Re}\,(\lambda\omega)\geq\mathrm{Re}\,(\lambda b_{l}),\quad
j=\overline{1,m-1},\;l=\overline{m,n},\quad\lambda\in\overline{\Lambda}.$
(4.5)

$m$ is odd

$m$ is even

Figure 1: The sectors $\Gamma_{1},$ $\Gamma_{\sigma},$ and $\Lambda.$

###### Proof.

It is sufficient to consider only $\lambda$ such that $|\lambda|=1.$ Then,
$\lambda\in\overline{\Lambda}$ if and only if $\lambda=e^{i\tau}$ for
$\tau\in\Big{[}\frac{(-1)^{m-1}\pi}{2n};\frac{(-1)^{m-1}\pi}{2n}+\frac{\pi}{n}\Big{]}.$
For any fixed $z\in\mathbb{C}_{0},$ ${\mathrm{R}e}\,(\lambda z)$ runs over a
closed segment as $\lambda$ runs over $\overline{\Lambda}$ and $|\lambda|=1.$
Let us find these segments for $z\in\\{\omega,b_{1},\ldots,b_{n}\\}.$

For definiteness, we assume that $m$ is odd; the case of even $m$ is proceeded
analogously. By (4.2), for
$s=\overline{0,\left\lfloor\frac{n-1}{2}\right\rfloor},$ we have

${\mathrm{R}e}\,(\lambda b_{2s+1})={\mathrm{R}e}\,e^{i\big{(}\tau+\frac{2\pi
s}{n}\big{)}}=\cos\Big{(}\tau+\frac{2\pi s}{n}\Big{)},$ $\tau+\frac{2\pi
s}{n}\in\big{[}(4s+1)\zeta;\,(4s+3)\zeta\big{]}\subset\left[\zeta;\pi+\zeta\right],\quad\zeta:=\frac{\pi}{2n}.$

Since the cosine function decreases on $[\zeta;\,\pi]$ and increases on
$[\pi;\,\pi+\zeta]$ (the latter segment appears only for $s=\frac{n-1}{2}$ if
$n$ is odd), we have

${\mathrm{R}e}\,(\lambda
b_{2s+1})\in\left[\cos\big{(}\min\\{(4s+3)\zeta,\pi\\}\big{)};\,\cos\big{(}(4s+1)\zeta\big{)}\right],\quad
s=\overline{0,\left\lfloor\frac{n-1}{2}\right\rfloor}.$ (4.6)

Analogously, we obtain

${\mathrm{R}e}\,(\lambda
b_{2p})\in\left[\cos\big{(}(4p-1)\zeta\big{)};\,\cos\big{(}(4p-3)\zeta\big{)}\right],\quad
p=\overline{1,\left\lfloor\frac{n}{2}\right\rfloor},$ (4.7)
${\mathrm{R}e}\,(\lambda\omega)\in\left[\cos\big{(}(4k+1)\zeta\big{)};\,\cos\big{(}\max\\{(4k-1)\zeta,0\\}\big{)}\right],\quad
k=\frac{m-1}{2}.$ (4.8)

One can see that the segments in (4.8) can intersect the segments in
(4.6)–(4.7) only at the boundary points. Then,

${\mathrm{R}e}\,(\lambda\omega)\geq{\mathrm{R}e}\,(\lambda
b_{2s+1})\Leftrightarrow\cos\big{(}(4s+1)\zeta\big{)}\pi\leq\cos\big{(}(4k+1)\zeta\big{)}\Leftrightarrow
s\geq k\Leftrightarrow 2s+1\geq m.$
${\mathrm{R}e}\,(\lambda\omega)\geq{\mathrm{R}e}\,(\lambda
b_{2p})\Leftrightarrow\cos\big{(}(4p-3)\zeta\big{)}\pi\leq\cos\big{(}(4k+1)\zeta\big{)}\Leftrightarrow
p-1\geq k\Leftrightarrow 2p\geq m.$

Thus, we have proved (4.5) for all possible values $j=2s+1$ or $j=2p.$ ∎

###### Proof of Theorem 4..

As in the proof of Theorem 2, applying substitutions (3.2) for each fixed
$k=\overline{m,n},$ we obtain system (3.3) with respect to the column vector
$\mathrm{\bf z}_{k}=[z_{jk}(x,\lambda)]_{j=1}^{n}.$ Integrating it under the
conditions

$\left\\{\begin{array}[]{cc}\displaystyle\lim_{x\to\infty}z_{jk}(x,\lambda)=0,&j=\overline{1,m-1},\\\
z_{jk}(\alpha,\lambda)=\delta_{jk},&j=\overline{m,n},\end{array}\right.$ (4.9)

we arrive at the system of integral equations

$z_{jk}(x,\lambda)=\left\\{\begin{array}[]{cc}\displaystyle-\sum_{l=1}^{n}\int_{x}^{\infty}v_{jl}(t,\lambda)e^{\lambda(b_{j}-b_{k})(p(x)-p(t))}z_{lk}(t,\lambda)\,dt,&j=\overline{1,m-1},\\\\[5.69054pt]
\displaystyle\delta_{jk}+\sum_{l=1}^{n}\int_{\alpha}^{x}v_{jl}(t,\lambda)e^{\lambda(b_{j}-b_{k})(p(x)-p(t))}z_{lk}(t,\lambda)\,dt,&j=\overline{m,n}.\end{array}\right.$
(4.10)

Let us study solvability of this system in the class of vector functions
$\mathrm{\bf z}_{k}=[z_{jk}(x,\lambda)]_{j=1}^{n},$ $x\geq\alpha,$ whose
components belong to $AC[\alpha,T]$ for any $T>\alpha.$

First, we consider $\lambda\in\overline{\Lambda},$ where $\Lambda$ is given in
(4.4), see Figure 1. Substituting $\mathrm{\bf z}_{k}=e^{\lambda(\omega-
b_{k})(p(x)-p(\alpha))}\mathrm{\bf\tilde{z}}_{k}$ into (4.10), we obtain an
equation with respect to $\mathrm{\bf\tilde{z}}_{k}:$

$\mathrm{\bf\tilde{z}}_{k}=\mathrm{\bf w}_{k}+{\cal
V}_{m}(\lambda)\mathrm{\bf\tilde{z}}_{k},\quad\mathrm{\bf
w}_{k}:=e^{\lambda(b_{k}-\omega)(p(x)-p(\alpha))}\mathrm{\bf e}_{k},$ (4.11)

where ${\cal V}_{m}(\lambda)$ is determined in Lemma 2, while $\mathrm{\bf
e}_{k}$ is given in (3.5). Inequalities (4.5) yield that $\|\mathrm{\bf
w}_{k}\|_{\bf BC_{n}}\leq 1$ and assure applicability of Lemma 2. By virtue of
this lemma, for $\lambda\in\overline{\Lambda^{\alpha}},$ there exists a unique
solution of (4.11) $\mathrm{\bf\tilde{z}}_{k}\in{\bf BC}_{n}.$ Consequently,
for $\lambda\in\overline{\Lambda^{\alpha}},$ there exists a solution of (4.10)
$\mathrm{\bf z}^{0}_{k}(\lambda)=e^{\lambda(\omega-
b_{k})(p(x)-p(\alpha))}\mathrm{\bf\tilde{z}}_{k}$ such that

$\mathrm{\bf
z}^{0}_{k}=[z^{0}_{jk}(x,\lambda)]_{j=1}^{n},\quad|z^{0}_{jk}(x,\lambda)|\leq
N_{\alpha}\left|e^{\lambda(\omega-b_{k})(p(x)-p(\alpha))}\right|,\quad
x\geq\alpha,\;\lambda\in\overline{\Lambda^{\alpha}},\quad j=\overline{1,n},$
(4.12)

with some $N_{\alpha}>0.$ Moreover, for each fixed $x\geq\alpha,$ the
components $z^{0}_{jk}(x,\lambda),$ $j=\overline{1,n},$ are analytic in
$\Lambda^{\alpha}$ and continuous on $\overline{\Lambda^{\alpha}}.$ Note that
the solution of (4.10) satisfying (4.12) is unique, otherwise equation (4.11)
would have at least two solutions from ${\bf BC}_{n}.$

Considering $\Gamma_{1}$ instead of $\Lambda$ and $b_{m}$ instead of $\omega,$
we can apply the same reasoning as in the previous paragraph, because, by the
numeration of $\\{b_{j}\\}_{j=1}^{n},$ inequalities (4.5) hold with
$\omega=b_{m}$ and $\Lambda=\Gamma_{1}.$ Then, for
$\lambda\in\overline{\Gamma_{1}^{\alpha}},$ there exists a solution of (4.10)
$\mathrm{\bf z}^{1}_{k}(\lambda)=[z^{1}_{jk}(x,\lambda)]_{j=1}^{n}$ satisfying

$|z^{1}_{jk}(x,\lambda)|\leq
N_{\alpha}\big{|}e^{\lambda(b_{m}-b_{k})(p(x)-p(\alpha))}\big{|},\quad
x\geq\alpha,\;\lambda\in\overline{\Gamma_{1}^{\alpha}},\quad
j=\overline{1,n}.$ (4.13)

For each fixed $x\geq\alpha,$ its components are continuous on
$\overline{\Gamma_{1}^{\alpha}}$ and analytic in $\Gamma_{1}^{\alpha}.$

Put $\mathrm{H}_{1}=\Lambda^{\alpha}\cap\Gamma_{1}^{\alpha}$ and
$\Xi_{1}=\Lambda^{\alpha}\cup\Gamma_{1}^{\alpha}.$ Now, we prove that
$\mathrm{\bf z}^{0}_{k}(\lambda)=\mathrm{\bf z}^{1}_{k}(\lambda)$ for
$x\geq\alpha$ and $\lambda\in\overline{\mathrm{H}_{1}}.$ By (4.5), we have

$\big{|}e^{\lambda(b_{m}-b_{k})(p(x)-p(\alpha))}\big{|}\leq\left|e^{\lambda(\omega-
b_{k})(p(x)-p(\alpha))}\right|,\quad
x\geq\alpha,\;\lambda\in\overline{\mathrm{H}_{1}}.$

Then, in the overlapping part of two sectors
$\lambda\in\overline{\mathrm{H}_{1}},$ the solution of (4.10) $\mathrm{\bf
z}^{1}_{k}(\lambda)$ satisfies (4.12). However, for
$\lambda\in\overline{\Lambda^{\alpha}},$ such solution is unique. By this
reason, $\mathrm{\bf z}^{0}_{k}(\lambda)$ and $\mathrm{\bf
z}^{1}_{k}(\lambda)$ coincide for $\lambda\in\overline{\mathrm{H}_{1}}.$ This
means that for $\lambda\in\overline{\Xi_{1}},$ we can consider a unified
solution $\mathrm{\bf z}_{k}(\lambda)=[z_{jk}(x,\lambda)]_{j=1}^{n}$ putting
$\mathrm{\bf z}_{k}(\lambda)=\mathrm{\bf z}^{0}_{k}(\lambda)$ for
$\lambda\in\overline{\Lambda^{\alpha}}$ and $\mathrm{\bf
z}_{k}(\lambda)=\mathrm{\bf z}^{1}_{k}(\lambda)$ for
$\lambda\in\overline{\Gamma_{1}^{\alpha}}\setminus\overline{\mathrm{H}_{1}}.$
For each fixed $x\geq\alpha,$ the components $z_{jk}(x,\lambda),$
$j=\overline{1,n,}$ are continuous on $\overline{\Xi_{1}}$ and analytic in
$\Xi_{1},$ being analytic continuations.

Analogously, using (4.3), for $\lambda\in\overline{\Gamma_{\sigma}^{\alpha}},$
we construct a solution of (4.10) $\mathrm{\bf
z}^{\sigma}_{k}=[z^{\sigma}_{jk}(x,\lambda)]_{j=1}^{n}$ satisfying

$|z^{\sigma}_{jk}(x,\lambda)|\leq
N_{\alpha}\big{|}e^{\lambda(b_{m+1}-b_{k})(p(x)-p(\alpha))}\big{|},\quad
x\geq\alpha,\;\lambda\in\overline{\Gamma_{\sigma}^{\alpha}},\quad
j=\overline{1,n}.$ (4.14)

For each fixed $x\geq\alpha,$ its components are continuous on
$\overline{\Gamma_{\sigma}^{\alpha}}$ and analytic in
$\Gamma_{\sigma}^{\alpha}.$ Put
$\mathrm{H}_{\sigma}=\Lambda^{\alpha}\cap\Gamma_{\sigma}^{\alpha}$ and
$\Xi_{\sigma}=\Lambda^{\alpha}\cup\Gamma_{\sigma}^{\alpha}.$ Obviously,
$\Xi_{1}\cup\Xi_{\sigma}=\Omega_{m}^{\alpha}.$ Applying the reasoning similar
to the previous paragraph, we obtain that $\mathrm{\bf
z}^{\sigma}_{k}(\lambda)=\mathrm{\bf z}^{0}_{k}(\lambda)$ for
$\lambda\in\overline{\mathrm{H}_{\sigma}}.$ Thus, the solution $\mathrm{\bf
z}^{\sigma}_{k}(\lambda)$ extends $\mathrm{\bf z}_{k}$ from the overlapping
part $\lambda\in\overline{\mathrm{H}_{\sigma}}$ on
$\lambda\in\overline{\Xi_{\sigma}}.$ Putting $\mathrm{\bf
z}_{k}(\lambda)=\mathrm{\bf z}^{\sigma}_{k}(\lambda)$ for
$\lambda\in\overline{\Gamma_{\sigma}^{\alpha}}\setminus\overline{\mathrm{H}_{\sigma}},$
we obtain a unified solution $\mathrm{\bf
z}_{k}=[z_{jk}(x,\lambda)]_{j=1}^{n},$ whose components are continuous on
$\overline{\Omega_{m}^{\alpha}}$ and analytic in $\Omega_{m}^{\alpha}$ for
each fixed $x\geq\alpha.$

Consider

$u_{jk}(x,\lambda)=\exp\left(\int_{\alpha}^{x}a_{jj}(t)\,dt+\lambda
b_{k}(p(x)-p(\alpha))\right)z_{jk}(x,\lambda),\quad
j=\overline{1,n},\;k=\overline{m,n}.$

Then, the matrix
$U_{\alpha}(x,\lambda)=[u_{jk}(x,\lambda)]_{\begin{subarray}{c}j=\overline{1,n},\\\
k=\overline{m,n}\end{subarray}}$ is a system of solutions of (1.1) for
$x\geq\alpha.$ Using (4.9), (4.13), and (4.14), it is easy to obtain
properties 1 and 3. Analyticity and continuity of $z_{jk}$ yield property 2
for $x\geq\alpha.$ It remains to extend each column vector
$[u_{jk}(x,\lambda)]_{j=1}^{n}$ on $x\in[0,\alpha]$ by the solution of system
(1.1) under the initial condition $\mathrm{\bf
y}(\alpha,\lambda)=[u_{jk}(\alpha,\lambda)]_{j=1}^{n}$ and apply Proposition
2. ∎

In Theorem 4, we obtained the system of solutions $U_{\alpha}(x,\lambda)$
consisting of $n-m+1$ vector functions, which is non-fundamental. However, for
$\lambda\in\overline{\Gamma^{\alpha}_{1}}$ or
$\lambda\in\overline{\Gamma^{\alpha}_{\sigma}},$ it can be supplemented with
the first $m-1$ columns of the system $Y_{\alpha}(x,\lambda)$ constructed in
Theorem 2. Then, by properties 3 of Theorems 2 and 4, we obtain a FSS.

######  Remark 1.

One can consider system (1.1) on the line $x\in\mathbb{R}$ under the
conditions similar to I–III, substituting $[0,\infty)$ by $\mathbb{R}$ and
assuming that $\rho\in L(-T,T)$ for any $T>0.$ For finite
$\alpha\in\mathbb{R},$ the results of Theorems 2–4 are carried to the case of
the line, since we can construct the solutions with the needed properties for
$x\geq\alpha$ and extend them on $x\in[-T,\alpha]$ for any $T>0.$ Note that
the line has other specifics concerning the statement of inverse spectral
problems. Inverse scattering problems considered in [3, 4] require the
solutions determined by the behavior at $x\to\infty,$ that differ from the
solutions constructed here.

##  5 Application to second-order equations with distribution potentials

In this section, we consider application of our results to the following
second-order equation with respect to a function $u:$

$-u^{\prime\prime}(x)+q(x)u(x)+zp_{0}(x)u(x)=z^{2}u(x),\quad x\geq 0,$ (5.1)

where $z\in\mathbb{C}$ is the spectral parameter. We assume that for some
$\sigma\in L\cap L_{2}[0,\infty),$ $q=\sigma^{\prime}$ in the sense of
distributions; then, $q$ is a distribution potential. Let also $p_{0}\in L\cap
L_{2}[0,\infty).$ In the particular case $p_{0}=0$ equality (5.1) turns into
the Sturm–Liouville equation with a distribution potential, see [23]. Further,
we reduce (5.1) to first-order system (1.1) with $n=2.$ Then, we apply
Theorems 2 and 3 to obtain FSS of (5.1). For equation (5.1) with distribution
potential $q,$ construction of other solutions was considered in [25, 24].

First, we apply a regularization approach offered in [23]. Introduce the
quasi-derivative

$u^{[1]}=u^{\prime}-\sigma u.$ (5.2)

Imposing the restrictions $u,u^{[1]}\in AC[0,T)$ for any $T>0,$ we can rewrite
(5.1) in the regularized form

$-(u^{[1]})^{\prime}-\sigma u^{[1]}-\sigma^{2}u+zp_{0}(x)u=z^{2}u,\quad x\geq
0,\;z\in\mathbb{C},$ (5.3)

where the left side belongs to $L[0,T)$ for any $T>0.$ Equalities (5.2) and
(5.3) should be treated together as a first-order system with respect to the
functions $u$ and $u^{[1]}.$

Consider the spectral parameter $\lambda=zi\in\mathbb{C}_{0}$ and the vector-
function $\mathrm{\bf v}=[v_{j}]_{j=1}^{2},$ where $v_{1}=u$ and
$v_{2}=\frac{u^{[1]}}{\lambda}.$ Then, (5.2) and (5.3) are equivalent to the
system

$\mathrm{\bf v}^{\prime}=\lambda\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\mathrm{\bf v}+\begin{pmatrix}\sigma&0\\\
-p_{0}i&-\sigma\end{pmatrix}\mathrm{\bf v}-\begin{pmatrix}0&0\\\
\frac{\sigma^{2}}{\lambda}&0\end{pmatrix}\mathrm{\bf v}.$ (5.4)

It remains to bring this system to form (1.1). For this purpose, consider the
matrices

$\Theta=\begin{pmatrix}1&1\\\
1&-1\end{pmatrix},\quad\Theta^{-1}=\frac{1}{2}\begin{pmatrix}1&1\\\
1&-1\end{pmatrix}.$

Making substitution $\mathrm{\bf y}=\Theta^{-1}\mathrm{\bf v},$ for
$\lambda\in\mathbb{C}_{0},$ we arrive at the equivalent system

$\mathrm{\bf y}^{\prime}=\lambda\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\mathrm{\bf
y}+\begin{pmatrix}\frac{-ip_{0}}{2}&\sigma-\frac{ip_{0}}{2}\\\
\sigma+\frac{ip_{0}}{2}&\frac{ip_{0}}{2}\end{pmatrix}\mathrm{\bf
y}+\frac{\sigma^{2}}{2\lambda}\begin{pmatrix}-1&-1\\\
1&1\end{pmatrix}\mathrm{\bf y}.$ (5.5)

It is easy to see that (5.5) has form (1.1) with matrices $A,$ $C,$ and $F$
satisfying assumptions I–III, while $n=2$ and $\rho(x)\equiv 1.$ As well, the
matrix $C=\frac{\sigma^{2}}{2\lambda}\begin{pmatrix}-1&-1\\\ 1&1\end{pmatrix}$
satisfies condition (1.6). To prove this, we verify condition (1.6) for the
matrices $C$ of a more general form.

######  Proposition 3.

Let $C(x,\lambda)=\sum_{k=1}^{N}C_{k}(x)\lambda^{-k},$ where $N>0$ and
$C_{k},$ $k=\overline{1,N},$ are arbitrary matrices of order $n$ with the
elements from $L[0,\infty).$ Then, the matrix $C$ satisfies (1.6).

###### Proof.

For any $\alpha>0,$ put $\displaystyle
K_{\alpha}=\sum_{k=1}^{N}\|C_{k}\|_{L[\alpha,\infty)}.$ It is clear that
$K_{\alpha}\to 0$ as $\alpha\to\infty.$ Then, to satisfy (1.6), it is
sufficient to take
$\varphi(\alpha)=\max\\{\alpha^{-1},\sqrt[2N]{K_{\alpha}}\\}.$ Indeed, we have

$\sup_{|\lambda|\geq\varphi(\alpha)}\|C\|_{L[\alpha,\infty)}\leq\sum_{k=1}^{N}\sup_{|\lambda|\geq\varphi(\alpha)}|\lambda^{-k}|\|C_{k}\|_{L[\alpha,\infty)}\leq\sum_{k=1}^{N}\varphi^{-k}(\alpha)\|C_{k}\|_{L[\alpha,\infty)}.$

One can see that $\varphi(\alpha)\to 0$ as $\alpha\to\infty,$ and
$\varphi(\alpha)<1$ for a sufficiently large $\alpha.$ Then, for such
$\alpha,$

$\sup_{|\lambda|\geq\varphi(\alpha)}\|C\|_{L[\alpha,\infty)}\leq\sum_{k=1}^{N}\varphi^{-N}(\alpha)\|C_{k}\|_{L[\alpha,\infty)}=K_{\alpha}\varphi^{-N}(\alpha).$

If $K_{\alpha}=0,$ then $\|C\|_{L[\alpha,\infty)}=0\leq\sqrt{K_{\alpha}}$ for
any $\lambda\in\mathbb{C}_{0}.$ If $K_{\alpha}>0,$ then
$\varphi^{N}(\alpha)\geq\sqrt{K_{\alpha}}$ and
$\displaystyle\sup_{|\lambda|\geq\varphi(\alpha)}\|C\|_{L[\alpha,\infty)}\leq
K_{\alpha}/\sqrt{K_{\alpha}}=\sqrt{K_{\alpha}}.$ In any case,
$\displaystyle\sup_{|\lambda|\geq\varphi(\alpha)}\|C\|_{L[\alpha,\infty)}\leq\sqrt{K_{\alpha}}\to
0$ as $\alpha\to\infty.$ ∎

Thus, we can apply Theorem 2 to system (5.5). We have $b_{1}=1,$ $b_{2}=-1,$
and there are two sectors: the right-half plane
$\Gamma_{1}=\\{\lambda\in\mathbb{C}\colon{\mathrm{R}e}\,\lambda>0\\}$ and the
left-half plane
$\Gamma_{2}=\\{\lambda\in\mathbb{C}\colon{\mathrm{R}e}\,\lambda<0\\}.$ For
$\alpha\geq 0$ and $\lambda\in\overline{\Gamma_{1}^{\alpha}},$ we obtain that
system (5.5) has a FSS $Y_{\alpha}=[y_{jk}(x,\lambda)]_{j,k=1}^{2},$ whose
components are continuous functions of
$\lambda\in\overline{\Gamma_{1}^{\alpha}}$ and analytic functions of
$\lambda\in{\Gamma_{1}^{\alpha}}.$ The following asymptotics hold:

$y_{jk}(x,\lambda)=\exp\left(b_{k}\lambda(x-\alpha)\right)\left(\delta_{jk}\exp\left((-1)^{k}i\int_{\alpha}^{x}\frac{p_{0}(t)}{2}\,dt\right)+s_{jk}(x,\lambda)\right),$
(5.6)

where $s_{jk}(x,\lambda)=o(1)$ as $\lambda\to\infty$ uniformly on
$x\geq\alpha,$ and $j,k=1,2.$ Since $\frac{ip_{0}}{2},\sigma\in
L_{2}[0,\infty)$ and $\|C\|_{L[0,\infty)}=O(\lambda^{-1}),$ applying Theorem
3, we obtain

$\sup_{x\geq\alpha}|s_{jk}(x,\lambda)|\in L_{2}(\Sigma),$ (5.7)

where $\Sigma\subset\overline{\Gamma_{1}^{\alpha}}$ is an arbitrary half-line.

Consider the matrix-function $V_{\alpha}=\Theta Y_{\alpha},$
$V_{\alpha}=:[v_{jk}(x,\lambda)]_{j,k=1}^{2},$ that is a FSS of system (5.4).
Putting $u_{1}:=v_{11}$ and $u_{2}:=v_{12},$ from (5.4), we have that $u_{1}$
and $u_{2}$ are solutions of equation (5.1), while $u^{[1]}_{1}=\lambda
v_{21}$ and $u^{[1]}_{2}=\lambda v_{22}.$ For any $T>0$ and $k=1,2,$ we have
$u_{k}(\cdot,\lambda),u_{k}^{[1]}(\cdot,\lambda)\in AC[0,T),$ since
$v_{jk}(\cdot,\lambda)\in AC[0,T).$ The system $\\{u_{1},u_{2}\\}$ of
solutions of equation (5.1) is fundamental in the sense that the vectors
$\Big{[}u_{1},\;u^{[1]}_{1}\Big{]}^{t}$ and
$\Big{[}u_{2},\;u^{[1]}_{2}\Big{]}^{t}$ are linearly independent on $x\geq 0.$

Proceed to $z=-i\lambda,$ then $\lambda\in\overline{\Gamma_{1}}$ if and only
if $z\in\overline{\mathbb{C}_{-}},$ where
$\mathbb{C}_{-}=\\{z\in\mathbb{C}\colon\mathrm{Im}\,z<0\\}.$ Using (5.6) and
(5.7), we arrive at the following theorem.

######  Theorem 5.

For $\alpha\geq 0,$ there exists $\lambda_{\alpha}>0$ such that for
$z\in\overline{\mathbb{C}_{-}^{\alpha}},$
$\mathbb{C}_{-}^{\alpha}:=\\{z\in\mathbb{C}_{-}\colon|z|>\lambda_{\alpha}\\},$
there exists a FSS of (5.1) $\\{u_{1}(x,z),u_{2}(x,z)\\}$ with the following
properties.

  1. 1.

For $k=1,2,$ we have the representation

$u_{k}(x,z)=\exp\big{(}{(-1)^{k+1}zi(x-\alpha)}\big{)}\left(\exp\left({(-1)^{k}i\int_{\alpha}^{x}\frac{p_{0}(t)}{2}\,dt}\right)+\tilde{s}_{1k}(x,z)\right),$
$u^{[1]}_{k}(x,z)=(-1)^{k+1}zi\exp\big{(}{(-1)^{k+1}zi(x-\alpha)}\big{)}\left(\exp\left({(-1)^{k}i\int_{\alpha}^{x}\frac{p_{0}(t)}{2}\,dt}\right)+\tilde{s}_{2k}(x,z)\right),$

where for $j=1,2,$ $\tilde{s}_{jk}(x,z)=o(1)$ uniformly on $x\geq\alpha$ as
$z\to\infty.$ Moreover,

$\sup_{x\geq\alpha}|\tilde{s}_{jk}(x,\cdot)|\in L_{2}(\Sigma),\quad j,k=1,2,$

where $\Sigma\subset\overline{\mathbb{C}_{-}^{\alpha}}$ is a half-line.

  2. 2.

For a fixed $x\geq 0,$ the functions $u_{jk}(x,z),$ $j,k=\overline{1,n},$ are
continuous on $\overline{\mathbb{C}_{-}^{\alpha}}$ and analytic in
$\mathbb{C}_{-}^{\alpha}.$

Moreover, $\lambda_{\alpha}\to 0$ as $\alpha\to\infty.$

An analogous theorem can be obtained for the values
$z\in\overline{\mathbb{C}_{+}},$
$\mathbb{C}_{+}:=\\{z\in\mathbb{C}\colon\mathrm{Im}\,z>0\\}.$ However, the
difference between $z\in\mathbb{C}_{+}$ and $z\in\mathbb{C}_{-}$ is
insignificant, since one case reduces to the other case by the substitution
$z=-\tilde{z}.$

If $p_{0}=0,$ Theorem 5 gives the result for the Sturm–Liouville equation on
the half-line with a distribution potential. It agrees with the formulae for
FSS of $n$-th order equations given in [21, 17] in the particular case $n=2.$
We can also apply Theorem 5 to equations on a finite interval $x\in[0,T],$
putting $\sigma(x)=0$ for $x>T.$ In this case, we obtain the FSS of the
Sturm–Liouville equation on a finite interval similar to the ones provided in
[12, Corollary 2].

Funding. This work was supported by the Russian Science Foundation (project
no. 21-71-10001), https://rscf.ru/en/project/21-71-10001/.

## References

  * [1] J. Tamarkin, “Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions”, Math. Zeitschr. 27 (1), 1–54 (1928).
  * [2] M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969; Ungar, New York, 1967, 1968).
  * [3] R. Beals, P. Deift, and C. Tomei, Direct and Inverse Scattering on the Line (AMS, Providence, 1988).
  * [4] G. Freiling and V. A. Yurko, Inverse Sturm–Liouville Problems and Their Applications (NOVA Science Publishers, New York, 2001).
  * [5] V. A. Yurko, “Inverse problems of spectral analysis for differential operators and their applications”, J. Math. Sci. 98 (3), 319–426 (2000).
  * [6] N. P. Bondarenko, “Inverse spectral problems for arbitrary-order differential operators with distribution coeffcients”, Mathematics 9 (22), 2989 (2021).
  * [7] G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear diferential equations containing a parameter”, Trans. Amer. Math. Soc. 9 (2), 219–231 (1908).
  * [8] G. D. Birkhoff and R. E. Langer, “The boundary problems and developments associated with a system of ordinary differential equations of the first order”, Proc. Am. Acad. Arts Sci. 58, 49–128 (1923).
  * [9] V. A. Yurko, “An inverse problem for systems of differential equations with nonlinear dependence on the spectral parameter”, Diff. Equat. 33 (3), 388–394 (1997).
  * [10] V. S. Rykhlov, “Asymptotical formulas for solutions of linear differential systems of the first order”, Results Math. 36 (3–4), 342–353 (1999).
  * [11] A. I. Vagabov, “On the asymptotics with respect to a parameter of solutions of differential systems with coefficients in the class $L_{q}$”, Diff. Equat. 46 (1), 17–23 (2010).
  * [12] A. M. Savchuk and A. A. Shkalikov, “Asymptotic analysis of solutions of ordinary differential equations with distribution coeffcients”, Sb. Math. 211 (11), 1623–1659 (2020).
  * [13] A. M. Savchuk and I. V. Sadovnichaya, “Spectral analysis of one-dimensional Dirac system with summable potential and Sturm–Liouville operator with distribution coefficients”, Contemp. Math. Fund. Direct. 66 (3), 373–530 (2020) [in Russian].
  * [14] A. A. Shkalikov, “Regular spectral problems of hyperbolic type for a system of first-order ordinary differential equations”, Math. Notes 110 (5), 806–810 (2021).
  * [15] A. P. Kosarev and A. A. Shkalikov, “Asymptotics in the spectral parameter for solutions of $2\times 2$ systems of ordinary differential equations”, Math. Notes 114 (4), 472–488 (2023).
  * [16] M. M. Malamud and L. L. Oridoroga “On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations”, J. Funct. Anal. 263, 1939–1980 (2012).
  * [17] V. A. Yurko, “Asymptotics of solutions of differential equations with a spectral parameter”, arXiv — a free preprint service. https://arxiv.org/abs/2204.07505. Accessed 2024.
  * [18] K. A. Mirzoev and A. A. Shkalikov, “Differential operators of even order with distribution coefficients”, Math. Notes 99, 779–784 (2016).
  * [19] K. A. Mirzoev and A. A. Shkalikov, “Ordinary differential operators of odd order with distribution coefficients”, arXiv — a free preprint service [in Russian]. https://arxiv.org/abs/1912.03660. Accessed 2024.
  * [20] N. P. Bondarenko, “Spectral data asymptotics for the higher-order differential operators with distribution coefficients”, J. Math. Sci. 266, 794–815 (2022).
  * [21] N. P. Bondarenko, “Linear differential operators with distribution coefficients of various singularity orders”, Math. Meth. Appl. Sci. 46 (6), 6639–6659 (2023).
  * [22] A. M. Savchuk, “The Calderon–Zygmund operator and its relation to asymptotic estimates for ordinary differential operators”, Contemp. Math. Fund. Direct. 63 (4), 689–702 (2017) [in Russian].
  * [23] A. M. Savchuk and A. A. Shkalikov, “Sturm–Liouville operators with singular potentials”, Math. Notes 66 (6), 741–753 (1999).
  * [24] N. I. Pronska, “ Asymptotics of eigenvalues and eigenfunctions of energy-dependent Sturm–Liouville equations”, Mat. Stud. 40 (1), 38–52 (2013).
  * [25] R. O. Hryniv and S. S. Manko, “Inverse scattering on the half-line for energy-dependent Schrodinger equations”, Inverse Problems 36 (9), 095002 (2020).
  * [26] L. Grafakos, Modern Fourier Analysis (Springer, New York, 2009).
  * [27] J. Garnett, Bounded Analytic Functions (Springer, New York, 2007).
  * [28] J. Conway, Functions of One Complex Variable (Second ed.; Springer-Verlag, New York, 1973).
  * [29] S. B. Chae, Holomorphy and Calculus in Normed Spaces (Marcel Dekker Inc., New York, 1985).

##  Appendix A Appendix: proof of Theorem 1

First, we introduce auxiliary values and prove necessary theorems for them.
Let $f\in L[0,\infty)$ be some fixed function. Denote
$\mathbb{C}_{+}:=\\{\lambda\in\mathbb{C}\colon\mathrm{Im}\,\lambda>0\\}$ and

$\psi(s,x,\lambda):=\int_{\min\\{s,x\\}}^{\max\\{s,x\\}}f(t)e^{i\lambda|x-t|}\,dt,\quad\Psi(\lambda):=\sup_{x,s\geq
0}|\psi(s,x,\lambda)|,\quad\lambda\in\overline{\mathbb{C}_{+}}.$ (A.1)

Note that the integrals in (A.1) are well-defined, since
$\mathrm{Re}\,\big{(}i\lambda|x-t|\big{)}\leq 0$ for
$\lambda\in\overline{\mathbb{C}_{+}}.$ Obviously, $\Psi(\lambda)$ has real
non-negative values. Its definition is similar to the one of
$\theta_{\alpha}(\lambda)$ and, in fact, $\Psi(\lambda)$ will participate in
estimates for $\theta_{\alpha}(\lambda).$ Let us obtain necessary properties
of $\Psi(\lambda).$

######  Proposition 4.

The function $\Psi(\lambda)$ is uniformly continuous on
$\overline{\mathbb{C}_{+}}$ and $\Psi(\lambda)\to 0$ as $\lambda\to\infty.$

###### Proof.

First, note that for any $\zeta>0,$ there exists $T>0$ such that

$|\psi(s,x,\lambda)|\leq\zeta+|\psi(\min\\{s,T\\},\min\\{x,T\\},\lambda)|,\quad\lambda\in\overline{\mathbb{C}_{+}},\;s,x\geq
0.$ (A.2)

Indeed, it is sufficient to take $T$ such that $\|f\|_{L[T,\infty)}\leq\zeta.$
To prove (A.2), for definiteness, we assume $x>s;$ the case $s\geq x$ can be
considered analogously. If $x\leq T,$ then (A.2) is obvious. If $x>T,$ then we
have

$|\psi(s,x,\lambda)|=\left|\int_{s}^{x}f(t)e^{i\lambda(x-t)}\,dt\right|\leq\left|\int_{\max\\{T,s\\}}^{x}f(t)e^{i\lambda(x-t)}\,dt\right|+\big{|}e^{i\lambda(x-T)}\big{|}\left|\int_{s}^{T}f(t)e^{i\lambda(T-t)}\,dt\right|,$

where $\big{|}e^{i\lambda(x-T)}\big{|}\leq 1$ and the second summand is absent
if $s>T.$ Then, using the inequality $\|f\|_{L[T,\infty)}\leq\zeta,$ we obtain
$|\psi(s,x,\lambda)|\leq\zeta+|\psi(\min\\{s,T\\},T,\lambda)|,$ and (A.2) is
proved.

Now, we prove the continuity of $\Psi(\lambda).$ Let $\varepsilon>0$ be fixed
and $x_{z},s_{z}>0$ be such that

$|\psi(s_{z},x_{z},z)|+\frac{\varepsilon}{2}\geq\Psi(z),\quad
z\in\overline{\mathbb{C}_{+}}.$ (A.3)

Using (A.2), we can always choose $x_{z},s_{z}\in[0,T],$ where $T>0$ depends
on $\varepsilon.$ Let $\delta>0$ be such that

$\big{|}e^{\lambda
i\xi}-e^{zi\xi}\big{|}\leq\frac{\varepsilon}{2\|f\|_{L[0,\infty)}+1},\quad\xi\in[0,T],$
(A.4)

as soon as $\lambda,z\in\overline{\mathbb{C}_{+}}$ and $|\lambda-z|<\delta.$
Then, for such $\lambda$ and $z,$ we have

$\Psi(\lambda)-\Psi(z)\overset{(\ref{x_z})}{\geq}|\psi(s_{z},x_{z},\lambda)|-|\psi(s_{z},x_{z},z)|-\frac{\varepsilon}{2}\geq-\frac{\varepsilon}{2}-\int\left|e^{\lambda
i|t-x_{z}|}-e^{zi|t-x_{z}|}\right||f(t)|\,dt,$

wherein the integral is taken on a subinterval of $[0,T],$ and
$|t-x_{z}|\in[0,T].$ Using (A.4), we arrive at
$\Psi(\lambda)-\Psi(z)\geq-\varepsilon.$ Note that $\lambda$ and $z$ can be
swapped. Then, for any $\varepsilon>0$ there exists $\delta>0$ such that
$|\Psi(\lambda)-\Psi(z)|\leq\varepsilon$ for $|\lambda-z|<\delta,$ and the
continuity is proved.

The statement $\Psi(\lambda)\to 0$ as $\lambda\to\infty$ is proved analogously
to that $\theta_{\alpha}(\lambda)\to 0$ in Lemma 1. ∎

######  Theorem A.

If $f\in L_{2}[0,\infty),$ then $\Psi\in L_{2}(\mathbb{R}).$

###### Proof.

Consider the Carleson operator ${\cal C},$ that acts on $w\in
L_{2}(\mathbb{R})$ as follows:

${\cal
C}w(\sigma):=\displaystyle\mathop{\sup}_{N>0}\Big{|}\int_{-N}^{N}\hat{w}(s)e^{2\pi
i\sigma s}\,ds\Big{|},\quad\sigma\in\mathbb{R},$

where $\hat{w}$ is the Fourier transform of $w:$

$\displaystyle\hat{w}(s)=\mathop{\int}_{\mathbb{R}}w(x)e^{-2\pi isx}\,dx,\quad
s\in\mathbb{R}.$

It is known that ${\cal C}$ is a bounded operator in $L_{2}(\mathbb{R})$ (see,
e.g., Theorem 6.2.1 in [26]).

Put $f(t)=0$ for $t<0.$ There is defined the inverse Fourier transform of $f:$
$\check{f}(t)=\hat{f}(-t),$ $t\in\mathbb{R},$ and $\check{f}\in
L_{2}(\mathbb{R}).$ Since $({\check{f}})\hat{\vphantom{1mm}}=f,$ we have

$\mathop{\sup}_{N>0}\left|\int_{0}^{N}f(s)e^{2\pi i\sigma s}\,ds\right|={\cal
C}\check{f}(\sigma)\in L_{2}(\mathbb{R}).$ (A.5)

Note that $\Psi(\lambda)=\max\\{\Psi_{1}(\lambda),\Psi_{2}(\lambda)\\},$ where

$\Psi_{1}(\lambda):=\sup_{x\geq s\geq
0}\left|\int_{s}^{x}f(t)e^{i\lambda(x-t)}\,dt\right|,\quad\Psi_{2}(\lambda):=\sup_{s\geq
x\geq 0}\left|\int_{x}^{s}f(t)e^{i\lambda(t-x)}\,dt\right|.$

Then, it is sufficient to obtain $\Psi_{1},\Psi_{2}\in L_{2}(\mathbb{R}).$
Consider, for definiteness, $\Psi_{1}.$ For $\tau\in\mathbb{R},$ we have

$\Psi_{1}(-\tau)=\sup_{x\geq s\geq 0}\left|\int_{s}^{x}f(t)e^{i\tau
t}\,dt\right|\big{|}e^{-i\tau x}\big{|}=\sup_{x\geq s\geq
0}\left|\int_{s}^{x}f(t)e^{i\tau t}\,dt\right|.$

Applying the inequality $|\int_{s}^{x}|\leq|\int_{0}^{x}|+|\int_{0}^{s}|,$ we
arrive at

$\Psi_{1}(-\tau)\leq 2\sup_{N>0}\left|\int_{0}^{N}f(t)e^{i\tau
t}\,dt\right|=2{\cal C}\check{f}\left(\frac{\tau}{2\pi}\right),$

which along with (A.5) give us the needed statement. For $\Psi_{2},$ the
computations are analogous. ∎

Further, we need the following definition.

######  Definition 1.

A measure $\mu\geq 0$ in $\mathbb{C}_{+}$ is called Carleson measure if

$\sup_{x\in\mathbb{R},\;y>0}y^{-1}\mu(S_{x,y})<\infty,\quad
S_{x,y}:=\big{\\{}\lambda\in\mathbb{C}\colon\mathrm{Re}\,\lambda\in(x,x+y),\;\mathrm{Im}\,\lambda\in(0,y)\big{\\}}.$

In particular, if $\mu$ is the standard Lebesgue measure on a half-line
$\tilde{\Sigma}\subset\mathbb{C}_{+}$ that equals $0$ in the other part of the
plane $\mathbb{C}_{+},$ then $\mu$ is a Carleson measure.

For a measure $\mu\geq 0$ in $\mathbb{C}_{+},$ denote by
$L_{2}(\mathbb{C}_{+},\mu)$ the set of measurable functions $h(z),$
$z\in\mathbb{C}_{+},$ for which the Lebesgue integral
$\int_{\mathbb{C}_{+}}|h(z)|^{2}\,d\mu$ is finite.

######  Theorem B.

Under the conditions of Theorem A, for any Carleson measure $\mu,$
$\Psi\in{L_{2}(\mathbb{C}_{+},\mu)}.$

To prove Theorem B, we need several auxiliary statements.

######  Theorem C (Theorem 5.6 in [27]).

Let $g$ be an arbitrary function from $L_{2}(\mathbb{R})$ and let $\Phi(z),$
$z\in\mathbb{C}_{+},$ be its Poisson integral:

$\Phi(x+iy)=\mathop{\int}_{\mathbb{R}}\frac{y}{\pi([x-t]^{2}+y^{2})}g(t)\,dt,\quad
y>0,\;x\in\mathbb{R}.$ (A.6)

Then, for any Carleson measure $\mu,$ we have $\Phi\in
L_{2}(\mathbb{C}_{+},\mu).$

######  Theorem D (Corollary 3.2 in [27]).

Let $g$ be a bounded and uniformly continuous function on $\mathbb{R}.$
Consider the function $\Phi$ defined by (A.6) and extend it on
$x\in\mathbb{R}$ by the formula $\Phi(x)=g(x).$ Then, the function $\Phi$ is
harmonic in $\mathbb{C}_{+}$ and continuous on $\overline{\mathbb{C}_{+}}.$

######  Definition 2 ([28, 27]).

Let $w\colon\Omega\to\mathbb{R}$ be a continuous function, where
$\Omega\subseteq\mathbb{C}$ is a domain. For $\lambda\in\mathbb{C}$ and
$\delta>0,$ denote
$B_{\delta}(\lambda)=\\{z\in\mathbb{C}\colon|z-\lambda|<\delta\\}.$ The
function $w$ is called subharmonic if

$w(\lambda)\leq\frac{1}{2\pi}\int_{0}^{2\pi}w(\lambda+\delta e^{it})\,dt$

whenever $\overline{B_{\delta}(\lambda)}\subset\Omega.$ Clearly, every
harmonic function is subharmonic.

######  Theorem E (Maximum Principle, see §3.3 in [28]).

Let $w_{1}$ and $w_{2}$ be real-valued continuous functions on a domain
$\Omega\subseteq\mathbb{C}.$ Denote $\partial_{\infty}\Omega=\partial\Omega$
if $\Omega$ is bounded and
$\partial_{\infty}\Omega=\partial\Omega\cup\\{\infty\\}$ if $\Omega$ is
unbounded. Assume that $w_{1}$ and $-w_{2}$ are subharmonic and that

$\mathop{\lim\sup}_{\begin{subarray}{c}z\to\lambda,\\\
z\in\Omega\end{subarray}}w_{1}(z)\leq\mathop{\lim\inf}_{\begin{subarray}{c}z\to\lambda,\\\
z\in\Omega\end{subarray}}w_{2}(z),\quad\lambda\in\partial_{\infty}\Omega.$

Then, $w_{1}(z)\leq w_{2}(z)$ for $z\in\Omega.$

###### Proof of Theorem B.

Applying Gauss’ mean value theorem to $\psi(s,x,\lambda)$ in
$\lambda\in\mathbb{C}_{+}$ and proceeding to the supremum by $x,s\geq 0,$ we
obtain

$\Psi(\lambda)\leq\frac{1}{2\pi}\int_{0}^{2\pi}\Psi(\lambda+\delta
e^{it})\,dt,\quad\lambda\in\mathbb{C}_{+},\quad\delta\in(0,\mathrm{Im}\,\lambda).$

Then, the function $\Psi$ is subharmonic in $\mathbb{C}_{+}.$

Define $\Phi(z)$ for $z\in\mathbb{C}_{+}$ by (A.6) with $g=\Psi$ and put
$\Phi(z)=\Psi(z)$ for $z\in\mathbb{R}.$ Proposition 4 yields that $\Psi(z)$ is
uniformly continuous and bounded on $\mathbb{R}.$ Then, by Theorem D, the
function $\Phi$ is harmonic on $\mathbb{C}_{+}$ and continuous on
$\overline{\mathbb{C}_{+}}.$ Since $\displaystyle\lim_{z\to\infty}\Psi(z)=0,$
$\Phi(z)\geq 0,$ and for any $x\in\mathbb{R},$ $\displaystyle\lim_{z\to
x}\Psi(z)=\displaystyle\lim_{z\to x}\Phi(z),$ we can apply Theorem E with
$w_{1}=\Psi$ and $w_{2}=\Phi.$ Then, we arrive at
$\Psi(\lambda)\leq\Phi(\lambda)$ for $\lambda\in\mathbb{C}_{+}.$

By virtue of Theorem A, we also have $\Psi\in L_{2}(\mathbb{R}).$ Applying
Theorem C, we obtain $\Phi\in L_{2}(\mathbb{C}_{+},\mu),$ and, consequently,
$\Psi\in L_{2}(\mathbb{C}_{+},\mu).$ ∎

Now, we are ready to prove Theorem 1.

###### Proof of Theorem 1.

For each pair of indices $j,l=\overline{1,n},$ we prove that
$\displaystyle\sup_{x,s\geq\alpha}|\nu_{jl}(s,x,\lambda)|\in L_{2}(\Sigma).$
We can exclude from consideration the pairs with $b_{l}=b_{j},$ because for
them $\nu_{jl}\equiv q_{jl}\equiv 0.$ For the rest values $j,l$ and
$\lambda\in\overline{\Lambda},$ under condition (2.4), we obtain the estimate

$\sup_{x,s\geq\alpha}|\nu_{jl}(s,x,\lambda)|\leq\Psi_{jl}(\tilde{\lambda}),\quad\overline{\mathbb{C}_{+}}\ni\tilde{\lambda}=\left\\{\begin{array}[]{cc}-i\lambda(b_{j}-b_{l}),&l<j,\\\
i\lambda(b_{j}-b_{l}),&j<l,\end{array}\right.$ (A.7)

where $\Psi_{jl}(\lambda)$ is constructed as the function $\Psi(\lambda)$ in
(A.1) with
$f(\xi):=\frac{q_{jl}(p_{\alpha}^{-1}(\xi))}{\rho(p_{\alpha}^{-1}(\xi))},$
while $p_{\alpha}^{-1}$ is the inverse function to
$p_{\alpha}(x):=p(x)-p(\alpha).$ Note that under our assumptions on $a_{jl}$
and $\rho,$ $f\in L\cap L_{2}[0,\infty).$

To prove (A.7), one should consider all possible positions of $j$ and $l$ with
respect to the fixed $k\in\overline{1,n}$ as listed in (2.6). For
definiteness, let $j,l<k;$ the other cases are proceeded analogously (see [22,
Theorem 3.1]). Then, we can assume $x<s$ (otherwise $\nu_{jl}\equiv 0$).
Consider the subcase $j<l.$ We can rewrite

$\nu_{jl}(s,x,\lambda)=e^{\lambda(b_{l}-\omega)(p_{\alpha}(x)-p_{\alpha}(s))}\int_{x}^{s}q_{jl}(t)e^{\tilde{\lambda}i(p_{\alpha}(t)-p_{\alpha}(x))}\,dt,\quad\big{|}e^{\lambda(b_{l}-\omega)(p_{\alpha}(x)-p_{\alpha}(s))}\big{|}\overset{(\ref{omega
numer})}{\leq}1.$

After change of the variables $t=p_{\alpha}^{-1}(\xi),$
$x=p_{\alpha}^{-1}(\zeta),$ and $s=p_{\alpha}^{-1}(\eta),$ we arrive at

$\sup_{s,x\geq\alpha}|\nu_{jl}(s,x,\lambda)|\leq\sup_{\eta>\zeta\geq
0}\left|\int_{\zeta}^{\eta}f(\xi)e^{\tilde{\lambda}i(\xi-\zeta)}\,d\xi\right|\leq\Psi_{jl}(\tilde{\lambda}).$

Consider the subcase $l<j,$ in which we use the representation

$\nu_{jl}(s,x,\lambda)=e^{\lambda(b_{j}-\omega)(p_{\alpha}(x)-p_{\alpha}(s))}\int_{x}^{s}q_{jl}(t)e^{\tilde{\lambda}i(p_{\alpha}(s)-p_{\alpha}(t))}\,dt,\quad\big{|}e^{\lambda(b_{j}-\omega)(p_{\alpha}(x)-p_{\alpha}(s))}\big{|}\overset{(\ref{omega
numer})}{\leq}1.$

After change of the variables $t=p_{\alpha}^{-1}(\xi),$
$x=p_{\alpha}^{-1}(\zeta),$ and $s=p_{\alpha}^{-1}(\eta),$ we arrive at

$\sup_{s,x\geq\alpha}|\nu_{jl}(s,x,\lambda)|\leq\sup_{\eta>\zeta\geq
0}\left|\int_{\zeta}^{\eta}f(\xi)e^{\tilde{\lambda}i(\eta-\xi)}\,d\xi\right|\leq\Psi_{jl}(\tilde{\lambda}).$

Thus, we have obtained (A.7). Denote by ${\tilde{\Sigma}}$ the image of the
half-line $\Sigma$ after action of the linear mapping
$\lambda\to\tilde{\lambda}.$ Then,
${\tilde{\Sigma}}\subset\overline{\mathbb{C}_{+}}$ is a half-line, and it
remains to prove that $\Psi_{ij}(\tilde{\lambda})\in L_{2}({\tilde{\Sigma}}).$
If $\tilde{\Sigma}\subset\mathbb{R},$ then the desired statement follows from
Theorem A. If $\tilde{\Sigma}\subset\mathbb{C}_{+},$ then we use the
representation
$\int_{\tilde{\Sigma}}|\Psi_{ij}(\tilde{\lambda})|^{2}\,d\tilde{\lambda}=\int_{\mathbb{C}_{+}}|\Psi_{ij}(\tilde{\lambda})|^{2}\,d\mu,$
where $\mu$ is a Carleson measure, and apply Theorem B. ∎

##  Appendix B Appendix: holomorphic mappings

Here, we provide some basic notions and theorems from the theory of
holomorphic mappings, see [29]. We also obtain Proposition 5 that is used for
proving the holomorphy of the function (2.19) constructed in Lemma 2.

Let $\bf X$ be a complex Banach space with the norm $\|\cdot\|_{\bf X}$ and
let $\Omega\subseteq\mathbb{C}$ be a domain.

######  Definition 3.

A mapping $G\colon\Omega\to\bf X$ is called holomorphic in $\Omega$ if it is
differentiable, i.e.

$\forall\lambda\in\Omega\ \exists\dot{G}(\lambda)\in{\bf
X}\colon\quad\lim_{\xi\to\lambda}\left\|\frac{G(\xi)-G(\lambda)}{\xi-\lambda}-\dot{G}(\lambda)\right\|_{\bf
X}=0.$

In this case, the mapping $\dot{G}\colon\Omega\to\bf X$ is called the
derivative of $G.$

It is well-known that holomorphy is a stronger property than continuity (a
mapping $G\colon\Omega\to\bf X$ is continuous if $\|G(\xi)-G(\lambda)\|_{\bf
X}\to 0$ as $\xi\to\lambda\in\Omega$). Holomorphic mappings are the objects
that generalize holomorphic functions, and in general, they have similar
properties, see [29]. Obviously, a linear combination of holomorphic mappings
is a holomorphic mapping.

The following theorem is an analogue of the statement that the uniform limit
of holomorphic functions is also holomorphic.

######  Theorem F (9D in [29]).

Let $G_{n}\colon\Omega\to{\bf X},$ $n\geq 0,$ be a sequence of holomorphic
mappings. If $G_{n}$ converges uniformly on each compact subset of $\Omega$ to
the mapping $G\colon\Omega\to{\bf X},$ then $G$ is holomorphic.

We also need two theorems that have no analogues in classical complex
analysis.

######  Theorem G (12C in [29]).

Let $G\colon\Omega\to{\bf X},$ where ${\bf X}$ is the Cartesian product of
complex Banach spaces ${\bf X}_{1},\ldots,{\bf X}_{m}.$ Let
$G_{k}\colon\Omega\to{\bf X}_{k}$ be the coordinate map of $G,$
$k=\overline{1,m}.$ Then, $G$ is holomorphic in $\Omega$ if and only if
$G_{k}$ is holomorphic in $\Omega,$ $k=\overline{1,m}.$

######  Theorem H (9.12 in [29]).

Denote by ${\bf X}^{*}$ the set of all linear bounded functionals on ${\bf
X}.$ A mapping $G\colon\Omega\to{\bf X}$ is holomorphic if and only if for any
$f\in{\bf X}^{*},$ $f(G(\lambda))$ is a holomorphic function in $\Omega.$

In particular, these theorems yield that if $\mathrm{\bf
z}=[z_{j}(x,\lambda)]_{j=1}^{n}$ is a holomorphic mapping of
$\lambda\in\Omega\to{\bf BC}_{n},$ then for $j\in\overline{1,n}$ and any fixed
$x\geq\alpha,$ $z_{j}(x,\lambda)$ is a holomorphic function in $\Omega.$ The
analyticity of the components $y_{jk}(x,\lambda)$ and $u_{jk}(x,\lambda)$ in
Theorems 2 and 4 will follow from this fact.

We aim to prove the following proposition.

######  Proposition 5.

Consider the operator ${\cal V}_{k}(\lambda)$ and the domain
$\Lambda^{\alpha}$ determined in Lemma 2. Let $\mathrm{\bf w}\in{\bf BC}_{n}$
satisfy the conditions in item 3) of Lemma 2. Then, the mapping ${\cal
V}_{k}(\lambda)\mathrm{\bf w}\colon\Lambda^{\alpha}\to{\bf BC}_{n}$ is
holomorphic.

For the proof, we need a lemma.

######  Lemma 4.

Let ${\mathrm{Z}}\subset\mathbb{C}$ and $\beta\in\mathbb{C}_{0}$ be such that
$\mathrm{Re}\,\big{(}\xi\beta\big{)}\geq 0$ for
$\xi\in\overline{{\mathrm{Z}}}.$ Then, for any $\delta>0$ there exists
$\varepsilon>0$ such that for
$\xi\in{\mathrm{Z}}_{\delta}:=\\{\xi\in{\mathrm{Z}}\colon\mathrm{dist}\,(\xi,\partial{\mathrm{Z}})\geq\delta\\},$
we have $\mathrm{Re}\,\big{(}\xi\beta\big{)}\geq\varepsilon.$

###### Proof.

First, we prove the statement in the case $\beta=1.$ We have that
${\mathrm{Z}}$ is a subset of the closed right half-plane
$\overline{\mathbb{C}_{r}}=\\{\lambda\in\mathbb{C}\colon\mathrm{Re}\,\lambda\geq
0\\}.$ Note that for arbitrary sets $A$ and $B$ in a normed space,
$\mathrm{dist}(\xi,\partial B)\geq\mathrm{dist}(\xi,\partial A)$ as soon as
$A\subseteq B$ and $\xi\in A.$ Applying this inequality to $A={\mathrm{Z}}$
and $B=\overline{\mathbb{C}_{r}},$ we obtain

$\mathrm{Re}\,\xi=\mathrm{dist}(\xi,\partial\mathbb{C}_{r})\geq\mathrm{dist}(\xi,\partial{\mathrm{Z}})\geq\delta,\quad\xi\in{\mathrm{Z}}_{\delta}.$

Thus, we arrive at the needed statement with $\varepsilon=\delta.$

The case $\beta\neq 1$ reduces to the previous one by the substitution
$\xi\beta=:\tilde{\xi}.$ ∎

###### Proof of Proposition 5..

Denote $\mathrm{\bf w}=:[w_{j}(x,\lambda)]_{j=1}^{n}$ and ${\cal
V}_{k}(\lambda)\mathrm{\bf w}=:[h_{j}(x,\lambda)]_{j=1}^{n}.$ By Theorem G, we
should prove that for each $j\in\overline{1,n},$ the component
$h_{j}(x,\lambda)$ is a holomorphic mapping of
$\lambda\in\Lambda^{\alpha}\to{\bf BC}_{1}.$ Remind that ${\bf BC}_{1}$
denotes the Banach space of the functions that are bounded and continuous on
$[\alpha,\infty)$ having the norm $\displaystyle\|\psi\|_{{\bf
BC}_{1}}=\sup_{x\geq\alpha}|\psi(x)|.$

For definiteness, assume $j<k$ (the case $j\geq k$ is analogous). Then,
denoting $\beta=b_{j}-\omega$ and
$f(t,\lambda)=-\sum_{l=1}^{n}v_{jl}(t,\lambda)w_{l}(t,\lambda),$ we have

$h_{j}(x,\lambda)=\int_{x}^{\infty}f(t,\lambda)e^{\lambda\beta(p(x)-p(t))}\,dt.$

Under our assumptions on the matrix $C(x,\lambda),$ the elements $v_{jl}$ are
holomorphic mappings of $\lambda\in\mathbb{C}_{0}\to L[0,\infty).$ By direct
differentiation, one can check that $f\colon\lambda\in\Lambda^{\alpha}\to
L[\alpha,\infty)$ is a holomorphic mapping.

By (2.4), for $\xi\in\Lambda^{\alpha},$ we have
$\mathrm{Re}\,\big{(}\xi\beta\big{)}\geq 0,$ and

$|e^{-\xi\beta}|\leq 1,\quad\xi\in\Lambda^{\alpha}.$ (B.1)

Let $\lambda\in\Lambda^{\alpha}$ be fixed and let $\delta>0$ be such that
$\overline{B_{2\delta}(\lambda)}\subset\Lambda^{\alpha},$ where we denoted
$B_{r}(\lambda)=\\{\xi\in\mathbb{C}\colon|\lambda-\xi|<r\\},$ $r>0.$ If
$\beta\neq 0,$ we can apply Lemma 4 to ${\mathrm{Z}}:=\Lambda^{\alpha}.$ Then,
since $\mathrm{dist}\,(\xi,\partial\Lambda^{\alpha})\geq\delta$ for $\xi\in
B_{\delta}(\lambda),$ we have

$\left|e^{-\xi\beta}\right|\leq e^{-\varepsilon},\quad\xi\in
B_{\delta}(\lambda),\quad\beta\neq 0,$

where $\varepsilon>0$ is the constant determined in Lemma 4 that depends on
$\delta$ and $\beta.$ Computing the maximal value of the function
$ze^{-\varepsilon z},$ $z\geq 0,$ we obtain another inequality:

$\left|ze^{-\xi\beta z}\right|\leq ze^{-\varepsilon z}\leq\frac{1}{\varepsilon
e},\quad z\geq 0,\;\xi\in B_{\delta}(\lambda),\quad\beta\neq 0.$ (B.2)

Consider the function

$\dot{h}_{j}(x,\lambda)=\int_{x}^{\infty}\dot{f}(t,\lambda)e^{\lambda\beta(p(x)-p(t))}\,dt+\beta\int_{x}^{\infty}f(t,\lambda)(p(x)-p(t))e^{\lambda\beta(p(x)-p(t))}\,dt.$

Inequalities (B.1) and (B.2) yield that $\dot{h}_{j}(x,\lambda)\in{\bf
BC}_{1},$ since $\dot{f}(\cdot,\lambda)$ and $f(\cdot,\lambda)$ belong to
$L[\alpha,\infty).$ We prove that

$\lim_{\xi\to\lambda}\left\|\frac{h_{j}(\cdot,\xi)-h_{j}(\cdot,\lambda)}{\xi-\lambda}-\dot{h}_{j}(\cdot,\lambda)\right\|_{{\bf
BC}_{1}}=0.$ (B.3)

Consider a splitting of the expression under the norm $\|\cdot\|_{{\bf
BC}_{1}}:$

$\frac{h_{j}(x,\xi)-h_{j}(x,\lambda)}{\xi-\lambda}-\dot{h_{j}}(x,\lambda)=\int_{x}^{\infty}\left[\frac{f(t,\xi)-f(t,\lambda)}{\xi-\lambda}-\dot{f}(t,\lambda)\right]e^{-\xi\beta
z(x,t)}\,dt+\\\ +\int_{x}^{\infty}\dot{f}(t,\lambda)\big{[}e^{-\xi\beta
z(x,t)}-e^{-\lambda\beta
z(x,t)}\big{]}\,dt+\int_{x}^{\infty}f(t,\lambda)\eta(x,t,\lambda,\xi)\,dt,$
(B.4)

where

$z(x,t)=p(t)-p(x)\geq 0,\quad\eta(x,t,\lambda,\xi)=\frac{e^{-\xi\beta
z(x,t)}-e^{\lambda\beta z(x,t)}}{\xi-\lambda}+\beta z(x,t)e^{-\lambda\beta
z(x,t)}.$

Using (B.1), (B.2), and the holomorphy property of $f(t,\lambda),$ one can
prove that each of the three integrals in (B.4) tends to $0$ uniformly on
$x\geq\alpha$ as $B_{\delta}(\lambda)\ni\xi\to\lambda.$ The proofs for the
first and for the second integrals are more simple than the proof for the
third one. We provide the proof for the third integral if $\beta\neq 0$
(otherwise it immediately turns $0$).

Let $\varepsilon_{0}>0$ be arbitrary and the constants $N,T>0$ be such that
$\|f(\cdot,\lambda)\|_{L[\alpha,\infty)}<N$ and
$\|f(\cdot,\lambda)\|_{L[T,\infty)}<\varepsilon_{0}\varepsilon/(4|\beta|).$
Let us find $\delta_{0}\in(0,\delta)$ such that for $\xi\in
B_{\delta_{0}}(\lambda)$ and $x\geq\alpha,$ there holds the estimate
$\left|\int_{x}^{\infty}f(t,\lambda)\eta(x,t,\lambda,\xi)\,dt\right|\leq\varepsilon_{0}.$
Consider the inequality

$\left|\int_{x}^{\infty}f(t,\lambda)\eta(x,t,\lambda,\xi)\,dt\right|\leq\\\
\int_{x}^{\max\\{x,T\\}}|f(t,\lambda)||\eta(x,t,\lambda,\xi)|\,dt+\int_{\max\\{x,T\\}}^{\infty}|f(t,\lambda)||\eta(x,t,\lambda,\xi)|\,dt.$
(B.5)

Note that $\eta(x,t,\lambda,\xi)\to 0$ uniformly on $x,t\in[\alpha,T]$ as
$\xi\to\lambda.$ Choose $\delta_{0}\in(0,\delta)$ such that for $\xi\in
B_{\delta_{0}}(\lambda)$ and $x,t\in[\alpha,T],$
$|\eta(x,t,\lambda,\xi)|\leq\varepsilon_{0}/(2N).$ If $x\leq T,$ we have

$\int_{x}^{\max\\{x,T\\}}|f(t,\lambda)||\eta(x,t,\lambda,\xi)|\,dt\leq\frac{\varepsilon_{0}}{2},\quad\xi\in
B_{\delta_{0}}(\lambda).$ (B.6)

If $x>T,$ then $\int_{x}^{x}=0,$ and (B.6) holds as well.

For $t\geq\max\\{x,T\\}$ and $\xi\in B_{\delta_{0}}(\lambda),$ we apply the
inequality

$\left|\frac{e^{-\xi\beta z(x,t)}-e^{-\lambda\beta
z(x,t)}}{\xi-\lambda}\right|\leq\max_{\nu\in\overline{B_{\delta_{0}}(\lambda)}}|\beta
z(x,t)e^{-\nu\beta z(x,t)}|,$

being a consequence of the fundamental theorem of calculus for segments in the
complex plane. Using this inequality along with (B.2), for $\xi\in
B_{\delta_{0}}(\lambda),$ we obtain

$\int_{\max\\{x,T\\}}^{\infty}|f(t,\lambda)||e(x,t,\lambda,\nu)|\,dt\leq
2|\beta|\int_{\max\\{x,T\\}}^{\infty}|f(t,\lambda)|\max_{\nu\in\overline{B_{\delta_{0}}(\lambda)}}|z(x,t)e^{-\nu\beta
z(x,t)}|\,dt\leq\\\ \leq\frac{2|\beta|}{\varepsilon
e}\int_{T}^{\infty}|f(t,\lambda)|\,dt<\frac{\varepsilon_{0}}{2}.$ (B.7)

Combining the estimates (B.5)–(B.7), we get that
$\left|\int_{x}^{\infty}f(t,\lambda)\eta(x,t,\lambda,\xi)\,dt\right|\leq\varepsilon_{0},$
$\xi\in B_{\delta_{0}}(\lambda),$ $x\geq\alpha.$ Thus, the third integral in
(B.4) tends to $0$ uniformly on $x.$ Analogously, the other two integrals in
(B.4) tend to $0$ uniformly on $x.$ Then, we arrive at relation (B.3), which
means holomorphy at $\lambda\in\Lambda^{\alpha}.$ ∎

######  Remark 2.

We consider systems (1.1) with matrices $C(x,\lambda)$ whose components are
holomorphic mappings of $\lambda\in\mathbb{C}_{0}$ to $L[0,\infty)$ under
condition (1.2). One can show that the components with the mentioned
properties are expanded into the Laurent series:

$c_{jk}(x,\lambda)=\sum_{\eta=1}^{\infty}\frac{c_{jk\eta}(x)}{\lambda^{\eta}},\quad\lambda\in\mathbb{C}_{0},\quad\|c_{jk\eta}\|_{L[0,\infty)}\leq\delta^{\eta}N_{\delta},\;\eta\geq
1,\;j,k=\overline{1,n},$ (B.8)

where $\delta>0$ can be chosen arbitrary and $N_{\delta}>0.$ In [17],
representation (B.8) was used as a restriction on the coefficients of
$C(x,\lambda).$

To prove (B.8), consider a holomorphic mapping $G\colon\mathbb{C}_{0}\to\bf X$
such that $\displaystyle\lim_{\lambda\to\infty}\|G(\lambda)\|_{\bf X}=0.$
Denote $\xi=\frac{1}{\lambda}\in\mathbb{C}_{0}$ and
$\tilde{G}(\xi)=G(\lambda).$ Then, $\tilde{G}\colon\xi\in\mathbb{C}_{0}\to\bf
X$ is holomorphic and $\displaystyle\lim_{\xi\to
0}\big{\|}\tilde{G}(\xi)\big{\|}_{\bf X}=0.$ Let us apply the criterion from
Theorem H. For any $f\in{\bf X}^{*},$ we have that
$f\big{(}\tilde{G}(\xi)\big{)}$ is a holomorphic function in $\mathbb{C}_{0}$
and $\displaystyle\lim_{\xi\to 0}f\big{(}\tilde{G}(\xi)\big{)}=0.$ Put
$f\big{(}\tilde{G}(0)\big{)}=0.$ Then, after removing singularity at $\xi=0,$
the function $f\big{(}\tilde{G}(\xi)\big{)}$ is holomorphic in $\mathbb{C}.$

By Theorem H, the mapping $\tilde{G}$ with the value $\tilde{G}(0)\equiv 0$ is
holomorphic in $\mathbb{C}.$ It is expanded into the Taylor series, see [29,
§9.16–9.17]:

$\tilde{G}(\xi)=\sum_{\eta=0}^{\infty}g_{\eta}\xi^{\eta},\quad\|g_{\eta}\|_{\bf
X}\leq\delta^{\eta}N_{\delta},\;\eta\geq 0,$

wherein $g_{0}=\tilde{G}(0)\equiv 0.$ Substituting $\xi=1/\lambda,$ for
$\tilde{G}(\xi)=G(\lambda):=c_{jk}(x,\lambda),$ we arrive at (B.8).