# On the trace formulas and completeness property of root vectors systems for
$2\times 2$ Dirac type operators

Anton A. Lunyov  Facebook, Inc.
1 Hacker Way, Menlo Park, California, 94025
United States of America<EMAIL_ADDRESS>Mark M. Malamud  Peoples
Friendship University of Russia (RUDN University)
6 Miklukho-Maklaya St., Moscow, 117198
Russian Federation<EMAIL_ADDRESS>
###### Abstract

The paper is concerned with the completeness property of the system of root
vectors of a boundary value problem for the following $2\times 2$ Dirac type
equation

$\displaystyle Ly=-iB^{-1}y^{\prime}+Q(x)y=\lambda y,\quad
y=\operatorname{col}(y_{1},y_{2}),\quad x\in[0,1],$ $\displaystyle
B=\operatorname{diag}(b_{1},b_{2}),\quad b_{1}<0<b_{2},\quad\text{and}\quad
Q\in W_{1}^{n}[0,1]\otimes\mathbb{C}^{2\times 2},$

subject to general non-regular two-point boundary conditions $Cy(0)+Dy(1)=0$.
If $b_{2}=-b_{1}=1$ this equation is equivalent to the one dimensional Dirac
equation.

We establish asymptotic expansion of the characteristic determinant of this
boundary value problem. This expansion directly yields new completeness result
for the system of root vectors of such boundary value problem with _non-
regular and even degenerate_ boundary conditions. We also present several
explicit completeness results in terms of the values $Q^{(j)}(0)$ and
$Q^{(j)}(1)$. In the case of degenerate boundary conditions and analytic
$Q(\cdot)$, the criterion of completeness property is established.

###### keywords:

Systems of ordinary differential equations , Boundary value problem ,
Characteristic determinant Asymptotic expansion , Completeness property

###### MSC:

47E05 , 34L40 , 34L10

††journal: Journal of Functional Analysis

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Asymptotic expansion of solutions to the Dirac type system
  4. 4 Asymptotic expansion of the characteristic determinant
  5. 5 General completeness results for non-regular BVP
  6. 6 Properties of the functions $\sigma_{j}^{\pm}(x)$
  7. 7 Explicit refined completeness results for non-regular BVP
  8. References

##  1 Introduction

In this paper we continue investigations from the cycle of our previous papers
[32, 15, 17, 18, 20] on spectral properties of first order $n\times n$ systems
of ODE. Here we restrict ourselves to the case of $2\times 2$ Dirac type
system of the form

$-iB^{-1}y^{\prime}+Q(x)y=\lambda y,\qquad
y=\operatorname{col}(y_{1},y_{2}),\qquad x\in[0,1],$ (1.1)

where

$B=\operatorname{diag}(b_{1},b_{2}),\quad b_{1}<0<b_{2}\quad\text{and}\quad
Q=\begin{pmatrix}0&Q_{12}\\\ Q_{21}&0\end{pmatrix}\in
L^{1}([0,1];\mathbb{C}^{2\times 2}).$ (1.2)

In the case $-b_{1}=b_{2}=1$ system (1.1) is equivalent to the classical
$2\times 2$ Dirac system.

With system (1.1) one associates, in a natural way, the maximal operator
$L=L(Q)$ acting in $L^{2}([0,1];\mathbb{C}^{2})$ on the domain

$\operatorname{dom}(L)=\\{y\in W_{1}^{1}([0,1];\mathbb{C}^{2}):Ly\in
L^{2}([0,1];\mathbb{C}^{2})\\}.$ (1.3)

Here $W_{p}^{n}[a,b]$ denotes the Sobolev space of functions $f$ having $n-1$
absolutely continuous derivatives on $[a,b]$ and satisfying $f^{(n)}\in
L^{p}[a,b]$, in particular $W_{p}^{0}[a,b]=L^{p}[a,b]$

To obtain a boundary value problem (BVP), equation (1.1) is subject to the
following boundary conditions

$U_{j}(y):=a_{j1}y_{1}(0)+a_{j2}y_{2}(0)+a_{j3}y_{1}(1)+a_{j4}y_{2}(1)=0,\quad
j\in\\{1,2\\}.$ (1.4)

Going forward we set

$A_{jk}:=\begin{pmatrix}a_{1j}&a_{1k}\\\
a_{2j}&a_{2k}\end{pmatrix}\qquad\text{and}\qquad J_{jk}:=\det(A_{jk}),\qquad
j,k\in\\{1,\ldots,{4}\\}.$ (1.5)

Denote by $L_{U}:=L_{U}(Q)$ the operator associated in
$L^{2}([0,1];\mathbb{C}^{n})$ with the BVP (1.1)–(1.4). It is defined as the
restriction of the maximal operator $L=L(Q)$ to the domain

$\operatorname{dom}(L_{U})=\\{y\in\operatorname{dom}(L):U_{1}(y)=U_{2}(y)=0\\}.$
(1.6)

The general spectral problem for $n\times n$ first order system of the form
(1.1) for the first was investigated by G. Birkhoff and R. Langer [2]. More
precisely, they introduced the concepts of _regular and strictly regular
boundary conditions_ , investigated the asymptotic behavior of eigenvalues and
eigenfunctions and proved _a pointwise convergence result_ on spectral
decompositions for the corresponding differential operator.

The first completeness result for such systems was established by V.P.
Ginzburg [6] in the case $B=I_{n}$, $Q(\cdot)=0$.

Recall that boundary conditions (1.4) are called _regular_ , if

$J_{32}=\det(A_{12}P_{+}+A_{34}P_{-})\neq 0\quad\text{and}\quad
J_{14}=\det(A_{12}P_{-}+A_{34}P_{+})\neq 0,$ (1.7)

where $P_{+}$ (resp. $P_{-}$) is the spectral projection onto positive (resp.
negative) part of the spectrum of the matrix $B=B^{*}$.

V.A. Marchenko [33] was the first to establish completeness property for the
system of root functions of the Dirac operator $L_{U}$ ($-b_{1}=b_{2}$) with
regular boundary conditions and continuous potential matrix $Q$. The last
restriction occurs because the transformation operators used for the proof was
constructed in [33] only for continuous $Q$.

Later, L.L. Oridoroga and one of the authors [32] established completeness
property for _$B$ -weakly regular_ boundary value problems for arbitrary
$n\times n$ first order systems of ODE with integrable matrix potential $Q\in
L^{1}([0,1];\mathbb{C}^{n\times n})$ (originally this result was announced in
[30] much earlier). In particular, for $2\times 2$ Dirac type system (with
$Q\in L^{1}([0,1];\mathbb{C}^{2\times 2})$ ) _the condition of $B$-weakly
regularity turns into (1.7)_ and hence ensures the completeness property for
such BVP with regular boundary conditions.

Let us also briefly mention that the Riesz basis property in
$L^{2}([0,1];\mathbb{C}^{2})$ of BVP (1.1)–(1.4) for $2\times 2$ Dirac system
with various smoothness assumptions on the potential matrix $Q$ was
investigated in numerous papers (see [3, 5, 4, 16, 37, 18, 36, 11, 12, 13, 8]
and the references therein). The first general result for non-smooth
potentials was obtained by P. Djakov and B. Mityagin [3, 5] who proved under
the assumption $Q\in L^{2}([0,1];\mathbb{C}^{2\times 2})$ that the system of
root vectors of the BVP (1.1)–(1.4) _with strictly regular boundary
conditions_ forms a Riesz basis and forms _a block Riesz basis_ whenever
boundary conditions are only regular. Note however that the methods of these
papers substantially rely on $L^{2}$-techniques (such as Parseval equality,
Hilbert-Schmidt operators, etc.) and that cannot be applied to
$L^{1}$-potentials. The most complete result on the Riesz basis property for
$2\times 2$ Dirac and Dirac-type systems, respectively, with $Q\in
L^{1}([0,1];\mathbb{C}^{2\times 2})$ and strictly regular boundary conditions
was obtained independently, by different methods and at the same time by A.M.
Savchuk and A.A. Shkalikov [37] and by the authors [16, 18]. The case of
regular boundary conditions is treated in [37] for the first time.

Let us denote by
$\Lambda_{Q}:=\Lambda_{U,Q}=\\{\lambda_{Q,n}\\}_{n\in\mathbb{Z}}:=\\{\lambda_{n}\\}_{n\in\mathbb{Z}}$
the spectrum of the operator $L_{U}(Q)$, counting multiplicity. Namely, if
$m_{a}(\lambda_{n})$ is the algebraic multiplicity of the eigenvalue
$\lambda_{n}$ of the operator $L_{U}(Q)$, then the number $\lambda_{n}$
appears in the sequence $\Lambda_{Q}$ exactly $m_{a}(\lambda_{n})$ times. It
is worth mentioning, that the algebraic multiplicity $m_{a}(\lambda_{n})$ of
the eigenvalue $\lambda_{n}$ coincides with the multiplicity of the zero
$\lambda_{n}$ of the characteristic determinant
$\Delta_{Q}(\cdot)=\Delta_{Q,U}(\cdot)$, an entire function given by the
following formula:

$\Delta_{Q}(\lambda)=J_{12}+J_{34}e^{i(b_{1}+b_{2})\lambda}\\\
+J_{32}\varphi_{11}(1,\lambda)+J_{13}\varphi_{12}(1,\lambda)+J_{42}\varphi_{21}(1,\lambda)+J_{14}\varphi_{22}(1,\lambda).$
(1.8)

where $(\varphi_{jk}(x,\lambda))_{j,k=1}^{2}:=\Phi(x,\lambda)$ is the
fundamental matrix of the system (1.1) (uniquely) determined by the initial
condition $\Phi(0,\lambda)=I_{2}$. Recall also that

$\Delta_{0}(\lambda)=J_{12}+J_{34}e^{i(b_{1}+b_{2})\lambda}+J_{32}e^{ib_{1}\lambda}+J_{14}e^{ib_{2}\lambda}$
(1.9)

is the characteristic determinant of problem (1.1)–(1.4) with $Q=0$.

Note that _boundary conditions ( 1.7) are regular if and only if the entire
function $\Delta_{0}(\cdot)$ is of maximal growth in both half-planes
$\mathbb{C}_{\pm}$_. In this connection we recall one of the key results of
our previous work [18] which states that the characteristic determinant
$\Delta_{Q}(\cdot)$ of the problem (1.1)–(1.4) with $Q(\cdot)\in
L^{1}([0,1];\mathbb{C}^{2\times 2})$ admits the following representation:

$\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=\Delta_{0}(\lambda)+\int^{1}_{0}g_{1}(t)e^{ib_{1}\lambda
t}dt+\int^{1}_{0}g_{2}(t)e^{ib_{2}\lambda t}dt,$ (1.10)

with certain functions $g_{1},g_{2}\in L^{1}[0,1]$ expressed via kernels of
the transformation operators. In turn, this formula ensures that the
determinant $\Delta_{Q}(\cdot)$ is also an entire sine-type function with the
indicator diagram $[-ib_{2},i|b_{1}|]\subset i\mathbb{R}$, i.e. has the
exponential type $|b_{1}|(b_{2})$ in $\mathbb{C}_{+}(\mathbb{C}_{-})$.
However, the converse is not true: _certain non-regular boundary conditions_
also lead to the characteristic determinant $\Delta_{Q}(\cdot)$ with the same
indicator diagram $[-ib_{2},i|b_{1}|]$ (see [32] and [17]) although the
indicator diagram of $\Delta_{0}(\cdot)$ is narrower. A stronger effect (of
the maximal growth of $\Delta_{Q}(\cdot)$) for system (1.1) with variable
matrix $B(\cdot)$ was recently discovered in [21].

Emphasize that representation (1.10) plays a crucial role in several recent
investigations. For instance, A.S. Makin [23, 24] applied representation
(1.10) to prove that a given entire function of exponential type is a
characteristic determinant of a degenerate BVP ($\Delta_{0}(\cdot)=0$) for
$2\times 2$ Dirac system if and only if this function is of Paley-Wiener
class. In [25] Makin used this representation to find explicit algebraic
conditions on a potential matrix $Q(\cdot)$ that guarantee the (non-block!)
Riesz basis property for arbitrary regular but not necessary strictly regular
$2\times 2$ Dirac operator. Recently the authors [20] applied this
representation to evaluate Lipschitz dependance of the spectral data on the
potential matrix $Q(\cdot)$ running through compact sets of
$L^{p}([0,\ell];\mathbb{C}^{2\times 2})$, $p\in[1,2]$. Moreover, in [19] we
generalized representation (1.10) to the case of $n\times n$ system of ODE and
applied it to establish the Riesz basis property in
$L^{2}([0,1];\mathbb{C}^{n})$ of the root vectors system of the corresponding
strictly regular BVP.

If the functions $g_{1},g_{2}\in W_{1}^{n}[0,1]$, one can integrate by parts
in (1.10) and use Riemann-Lebesgue Lemma to obtain the following asymptotic
expansion for $\Delta_{Q}(\lambda)$:

$\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=e^{ib_{1}\lambda}\cdot\Bigl{(}J_{32}-\sum_{k=1}^{n}\frac{g_{1}^{(k-1)}(1)}{(-ib_{1}\lambda)^{k}}+o(\lambda^{-n})\Bigr{)},\qquad\operatorname{Im}\lambda\to+\infty,$
(1.11) $\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=e^{ib_{2}\lambda}\cdot\Bigl{(}J_{14}-\sum_{k=1}^{n}\frac{g_{2}^{(k-1)}(1)}{(-ib_{2}\lambda)^{k}}+o(\lambda^{-n})\Bigr{)},\qquad\operatorname{Im}\lambda\to-\infty,$
(1.12)

(see Remark 4.2 for details).

Note in this connection that similar representation for the characteristic
determinant of the Sturm-Liouville operators was used in [29] to establish
potential-dependent completeness property of the BVP with degenerate boundary
conditions and a potential $q\in C^{n}[0,1]$. Later A.S. Makin [22]
significantly improved this result by relaxing the continuity assumption.

In this paper we investigate completeness property for the problem (1.1)–(1.4)
_with non-regular boundary conditions_ , i.e. when $J_{32}J_{14}=0$. In this
case BVP (1.1)–(1.4) necessarily has _an incomplete system of root functions
whenever $Q(\cdot)\equiv 0$_.

In [32] L.L. Oridoroga and one of the authors (see also [31]) established the
first potential-dependent completeness result for the operator $L_{U}(Q)$ with
non-regular boundary conditions assuming that $Q\in
C^{1}([0,1];\mathbb{C}^{2\times 2})$. It reads as follows:

######  Theorem 1.1 (Theorem 5.1 in [32]).

Let $Q_{12},Q_{21}\in C^{1}[0,1]$. Then the system of root functions of the
operator $L_{U}(Q)$ is complete in $L^{2}([0,1];\mathbb{C}^{2})$ whenever the
following two conditions hold:

$\displaystyle|J_{32}|+|b_{1}J_{13}Q_{12}(0)+b_{2}J_{42}Q_{21}(1)|\neq 0,$
(1.13) $\displaystyle|J_{14}|+|b_{1}J_{13}Q_{12}(1)+b_{2}J_{42}Q_{21}(0)|\neq
0.$ (1.14)

The proof relies on formulas (1.11)–(1.12) for $n=1$. Namely, using triangular
transformation operators the authors obtained explicit form of the
coefficients $g_{1}(1)$ and $g_{2}(1)$ in (1.11)–(1.12) in terms of a
potential matrix $Q(\cdot)$ (see also [17, Remark 4.6] on the discussion about
smoothness assumption in this result). In [1], similar results using the same
method were obtained in the case of $B\neq B^{*}$ and analytic $Q(\cdot)$.

In [15] and [17], we generalized and clarified [32, Theorem 5.1] to establish
_potential-dependent completeness as well as spectral synthesis property_ for
the system of root functions of the $n\times n$ system with not necessarily
selfadjoint matrix $B$ and non-weakly-regular boundary conditions. In
particular, for $2\times 2$ Dirac-type systems [17, Proposition 4.5] improves
[32, Theorem 5.1] by relaxing $C^{1}$-smoothness of the potential $Q(\cdot)$
on $[0,1]$ to continuity of $Q(\cdot)$ at the endpoints $\\{0,1\\}$.

In a very recent paper [9] A.P. Kosarev and A.A. Shkalikov extended
completeness results from [32, 1, 17] to the case of $2\times 2$ Dirac-type
operators with non-constant matrix $B=\operatorname{diag}(b_{1}(x),b_{2}(x))$
and degenerate boundary conditions of a special form ($y_{1}(0)=y_{2}(1)=0$)
under the smoothness assumption $b_{1},b_{2},Q_{12},Q_{21}\in W_{1}^{1}[0,1]$.
In another recent preprint [26], A.S. Makin generalized and substantially
improved our result [17, Proposition 4.5] in the Dirac case by proposing a new
interesting approach (see Theorem 5.6). Namely, instead of continuity
condition on $Q(\cdot)$ he proposed a new more general integral limit
condition.

In this paper, we aim to further refine results from [32] and [17] when
potential matrix $Q(\cdot)$ has additional smoothness. However, finding
explicit form of the derivatives $g_{1}^{(k-1)}(1)$ and $g_{2}^{(k-1)}(1)$ in
asymptotic formulas (1.11)–(1.12) directly using transformation operators is
rather difficult. In this paper we propose another approach to compute these
coefficients by adapting Marchenko’s method from [33]. Namely, we establish
asymptotic expansion of the characteristic determinant $\Delta_{Q}(\lambda)$
of BVP (1.1)–(1.4) of the form (1.11)–(1.12) with coefficients explicitly
expressed as polynomials in boundary values of the functions $Q_{12},Q_{21}\in
W_{1}^{n}[0,1]$ and their derivatives as well as numbers $b_{1}$, $b_{2}$,
$J_{14}$, $J_{32}$, $J_{13}$, $J_{42}$ (see Theorem 4.1). In turn, we apply
this result to provide general explicit conditions of completeness in Theorem
5.1. For instance, in the case of $Q_{12},Q_{21}\in W_{1}^{2}[0,1]$, it reads
as follows.

######  Proposition 1.2.

Let $Q_{12},Q_{21}\in W_{1}^{2}[0,1]$. Assume that either condition (1.13)
holds or the following condition holds:

$b_{1}J_{13}Q_{12}^{\prime}(0)-b_{2}J_{42}Q_{21}^{\prime}(1)+ib_{1}b_{2}J_{14}Q_{12}(0)Q_{21}(1)\neq
0.$ (1.15)

Further, assume also that either condition (1.14) holds or the following
condition holds:

$b_{1}J_{13}Q_{12}^{\prime}(1)-b_{2}J_{42}Q_{21}^{\prime}(0)-ib_{1}b_{2}J_{32}Q_{12}(1)Q_{21}(0)\neq
0$ (1.16)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$.

It is evident that Proposition 1.2 (announced in [14]) even in the classical
Dirac case is not covered by previous completeness results from [32, 1, 17, 9,
26], if one of conditions (1.13)–(1.14) is violated (see Remark 5.8 for
detailed discussion).

Note, however, that general form of the coefficients in the expansions
(1.11)–(1.12) is somewhat cumbersome. If $J_{14}=J_{32}=0$, then we can
present more explicit and refined condition of completeness announced in [14,
Theorem 4].

The following function is highly involved in the following series of results,

$P(x):=J_{13}b_{1}Q_{12}(x)+J_{42}b_{2}Q_{21}(1-x).$ (1.17)

######  Theorem 1.3.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$ and let

$J_{32}=J_{14}=0,\qquad J_{13}J_{42}\neq 0.$ (1.18)

Further, let for some $n_{0},n_{1}\in\\{0,1,\ldots,n-1\\}$ the following
conditions hold

$\displaystyle P^{(k)}(0)$ $\displaystyle=0,\qquad
k\in\\{0,1,\ldots,n_{0}-1\\},$ (1.19) $\displaystyle P^{(k)}(1)$
$\displaystyle=0,\qquad k\in\\{0,1,\ldots,n_{1}-1\\}.$ (1.20)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$ provided that

$|n_{1}-n_{0}|\leqslant 1\qquad\text{and}\qquad P^{(n_{0})}(0)\neq 0,\qquad
P^{(n_{1})}(1)\neq 0.$ (1.21)

Extended version of Theorem 1.3, which also covers the case $|n_{1}-n_{0}|>1$,
is contained in Theorem 7.5.

Let us outline a few additional simple sufficient conditions guaranteeing
completeness property. In all results stated below we assume that
$Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\geqslant 3$. First, we mention
result that demonstrates substantial difference between Sturm-Liouville and
Dirac operators (see Corollary 7.6 and Remark 7.7). Namely, if
$J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$, then the following condition ensures
the desired completeness property,

$Q_{12}(1)\neq 0,\qquad P^{(n-3)}(0)\neq 0,\qquad P^{(k)}(1)=0,\quad
k\in\\{0,1,\ldots,n-1\\}.$ (1.22)

In particular, completeness is possible if $P(x)=0$, $x\in[a,1]$, for some
$a\in(0,1)$.

Further, under some algebraic assumption on the potential matrix $Q(\cdot)$,
we can eliminate any restrictions between $n_{0}$ and $n_{1}$ in condition
(1.21). Indeed, according to Corollary 7.6 (see also Proposition 7.8 for more
general result), if $J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$, then desired
completeness property holds whenever both of the following conditions hold

$\displaystyle Q^{(j)}(0)=Q^{(j)}(1)=0,\quad
j\in\\{0,1,\ldots,m-1\\},\quad\text{where}\quad
m=\text{\scalebox{0.85}{$\left\lceil{\frac{n-2}{3}}\right\rceil$}},$ (1.23)
$\displaystyle P^{(n_{0})}(0)P^{(n_{1})}(1)\neq 0\qquad\text{for some}\quad
n_{0},n_{1}\in\\{m,m+1,\ldots,n-1\\}.$ (1.24)

Similar result is valid if $J_{32}=0$, while $J_{14}\neq 0$. Namely, in this
case, Corollary 7.3 ensures the desired completeness property whenever

$P^{(n-1)}(0)\neq 0,\quad Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\quad 0\leqslant
j<m,\quad\text{where}\quad
m=\text{\scalebox{0.85}{$\left\lceil{\frac{n-1}{2}}\right\rceil$}}.$ (1.25)

See Proposition 7.1 and Remark 7.4 for more details.

We also find explicit completeness condition in the case $J_{32}=J_{13}=0$
(see Corollaries 7.13 and 7.15). Namely, if

$J_{32}=J_{13}=J_{42}=0,\qquad J_{14}\neq 0,\qquad\text{and}\qquad
Q_{12}^{(j_{0})}(0)Q_{21}^{(j_{1})}(1)\neq 0,$ (1.26)

for some $j_{0},j_{1}\in\\{0,1,\ldots,n-2\\}$ such that $j_{0}+j_{1}\leqslant
n-2$, then the system of root vectors of the BVP (1.1)–(1.4) is complete and
minimal in $L^{2}([0,1];\mathbb{C}^{2})$. Moreover, completeness property is
preserved whenever one of the following conditions holds,

$\displaystyle J_{32}=J_{13}=0,\quad J_{42}\neq 0,\qquad J_{14}\neq 0,\qquad
Q_{21}^{(j_{1})}(1)\neq 0,$ (1.27) $\displaystyle J_{32}=J_{13}=0,\quad
J_{42}\neq 0,\qquad J_{14}=0,\qquad Q_{21}^{(j_{0})}(0)Q_{21}^{(j_{1})}(1)\neq
0,$ (1.28)

for some $j_{0},j_{1}\in\\{0,1,\ldots,n-1\\}$.

Note also that an $m$-dissipative operator $L_{U}(0)$ automatically meets the
condition $J_{14}\neq 0$. Therefore, in this case, condition (1.27) shows that
under certain additional algebraic restrictions, completeness property for
$L_{U}(Q)$ depends only on behavior of $Q_{21}(\cdot)$ at the endpoint $1$.
This result and its applications to the spectral synthesis of dissipative
operators (previously investigated in [15]) will be discussed elsewhere.

It is surprising that the function $Q_{12}(\cdot)$ is not involved in
conditions (1.27)–(1.28). In turn, these conditions imply the following
peculiar criterion of completeness when the potential $Q(\cdot)$ is an
_analytic matrix function_ on $[0,1]$ and boundary conditions are of the form

$\alpha_{1}y_{1}(0)+\alpha_{2}y_{2}(0)+\alpha_{3}y_{1}(1)=0,\quad
y_{2}(1)=0,\quad\alpha_{2}\neq 0.$ (1.29)

Namely, in accordance with Corollary 7.17, the system of root vectors of the
BVP (1.1), (1.29) is complete in $L^{2}([0,1];\mathbb{C}^{2})$ _if and only if
$Q_{21}(\cdot)\not\equiv 0$_.

The paper is organized as follows. In Section 2 we recall key results from
[18] on transformation operators and define special functional spaces
$X_{\infty}$ and $X_{\infty}^{0}$. Here we also prove our first new
completeness result, Proposition 2.6, which easily follows from previous
results in [15, 17]. In Section 3 we establish asymptotic expansions for
solutions to system (1.1) (see Theorem 3.1) and “trace formulas” for certain
solution quotients (Proposition 3.7). In Section 4 we apply these trace
formulas to derive asymptotic expansion of the characteristic determinant
$\Delta_{Q}(\cdot)$ (Theorem 4.1). In Section 5 using this asymptotic
expansion we establish a general refined completeness result (Theorem 5.1). In
Sections 6 and 7 we “decipher” Theorem 5.1 to establish a series of explicit
completeness results outlined above.

The main results of this paper were announced in our short communication [14]
published in 2013. However, the recent publications [9, 26] (especially paper
[26] by A.S. Makin) influenced us to compare our results from [14] with the
new ones. Namely, we substantially extended the previous version of the paper
written along [14] by adding several new results on completeness (see Section
7).

##  2 Preliminaries

Following [27, 18] denote by $X_{\infty}:=X_{\infty}(\Omega)$ the linear space
composed of (equivalent classes of) measurable functions defined on the domain

$\Omega:=\\{(x,t):0\leqslant t\leqslant x\leqslant 1\\}$ (2.1)

satisfying

$\|f\|_{X_{\infty}}:=\operatorname*{ess\,sup}_{x\in[0,1]}\int_{0}^{x}|f(x,t)|dt<\infty.$
(2.2)

It can easily be shown that the space $X_{\infty}$ equipped with the norm
(2.2) forms a Banach space that is not separable. Denote by $X_{\infty}^{0}$
the (separable) subspace of $X_{\infty}$ obtained by taking the closure of the
set of continuous functions $C(\Omega)$. Evidently, the set $C^{1}(\Omega)$ of
smooth functions is also dense in the space $X_{\infty}^{0}$. In the sequel,
we need to following important property of the space $X_{\infty}^{0}$
established in [18].

######  Lemma 2.1.

Let $K\in X_{\infty}^{0}$ and $b\in\mathbb{R}\setminus\\{0\\}$. Then for each
$\delta>0$ there exists $R_{\delta}=R_{\delta}(K,b)>0$ such that the following
uniform estimate holds

$\left|{\int_{0}^{x}K(x,t)e^{ib\lambda
t}\,dt}\right|<\delta\cdot(e^{-b\operatorname{Im}\lambda}+1),\qquad|\lambda|>R_{\delta},\quad
x\in[0,1].$ (2.3)

######  Remark 2.2.

Note that [18, Lemma 3.2] is formulated with additional restriction
$|\operatorname{Im}\lambda|\leqslant h$, but the proof in more general case
remains the same. See also [20, Lemma 5.12] for stronger statement concerning
compact sets in $X_{\infty}^{0}(\Omega)$.

The space $X_{\infty}^{0}$ played crucial role in [18] to establish existence
of triangular transformation operators for solutions to system (1.1). In
particular, for each $a\in[0,1]$, the trace operator
$i_{a}:X_{\infty}^{0}(\Omega)\to L^{1}[0,a]$ (originally defined on
$C(\Omega)$ via $i_{a}\bigl{(}N(x,t)\bigr{)}:=N(a,t)$) is correctly extended
to $X_{\infty}^{0}(\Omega)$.

To formulate the next result ([18, Proposition 3.1]), denote by
$\Phi(x,\lambda)$ the fundamental matrix of the system (1.1) (uniquely)
determined by the initial condition $\Phi(0,\lambda)=I_{2}$, i.e.,

$\Phi(x,\lambda):=\begin{pmatrix}\Phi_{1}(x,\lambda)&\Phi_{2}(x,\lambda)\end{pmatrix},\quad\Phi_{k}(x,\lambda):=\begin{pmatrix}\varphi_{1k}(x,\lambda)\\\
\varphi_{2k}(x,\lambda)\end{pmatrix},\quad k\in\\{1,2\\},$ (2.4)

where $\displaystyle\Phi_{1}(0,\lambda):=\binom{1}{0},\
\Phi_{2}(0,\lambda)=\binom{0}{1}$.

######  Proposition 2.3.

Let $Q\in L^{1}([0,1];\mathbb{C}^{2\times 2})$. Then the functions
$\varphi_{jk}(\cdot,\lambda)$ admit the following representations

$\varphi_{jk}(x,\lambda)=\delta_{jk}e^{ib_{k}\lambda
x}+\int_{0}^{x}R_{1,j,k}(x,t)e^{ib_{1}\lambda
t}dt+\int_{0}^{x}R_{2,j,k}(x,t)e^{ib_{2}\lambda t}dt,$ (2.5)

for $x\in[0,1]$, $\lambda\in\mathbb{C}$, where $R_{l,j,k}\in
X_{\infty}^{0}(\Omega),$ $j,k,l\in\\{1,2\\}$.

Integral representations (2.5) for the entries of the fundemantal matrix
$\Phi(x,\lambda)$ played crucial role in our papers [16, 18]. Moreover, the
functions $g_{1},g_{2}(\in L^{1}[0,1])$ in formula (1.10) are linear
combinations of traces $i_{1}\bigl{(}R_{l,j,k}(x,t)\bigr{)}:=R_{l,j,k}(1,t)$
and due to inclusions $R_{l,j,k}\in X_{\infty}^{0}(\Omega)$ are well-defined.
Note, that for Sturm-Liouville operator the kernels of transformation
operators are always continuous, and similar difficulties in representation
(1.10) do not occur. Besides, for the case of Dirac operator with continuous
potential $Q(\cdot)$ treated in [33], the transformation operator matrix
kernel is continuous: $R_{l,j,k}\in C(\Omega)$, and therefore the traces
$R_{l,j,k}(1,t)$ and hence the functions $g_{1},g_{2}$ in (1.10) are also
continuous.

In the sequel we need alternative, equivalent to (2.5) representation of the
fundamental matrix $\Phi(x,\lambda)$.

######  Corollary 2.4.

Let $Q\in L^{1}([0,1];\mathbb{C}^{2\times 2})$. Then the fundamental matrix
$\Phi(x,\lambda)$ of the system (1.1) admits the following representation

$\Phi(x,\lambda)=\begin{pmatrix}e^{ib_{1}\lambda
x}v_{1}^{+}(x,\lambda)&e^{ib_{2}\lambda x}v_{2}^{-}(x,-\lambda)\\\
e^{ib_{1}\lambda x}v_{2}^{+}(x,\lambda)&e^{ib_{2}\lambda
x}v_{1}^{-}(x,-\lambda)\end{pmatrix},\quad
x\in[0,1],\quad\lambda\in\mathbb{C},$ (2.6)

where

$v_{k}^{\pm}(x,\lambda)=\delta_{1k}+\int_{0}^{x}R^{\pm}_{k}(x,t)e^{i(b_{2}-b_{1})\lambda
t}dt,\qquad k\in\\{1,2\\}.$ (2.7)

and $R^{\pm}_{1},R^{\pm}_{2}\in X_{\infty}^{0}(\Omega)$ are certain integrable
kernels.

###### Proof.

Let $j\in\\{1,2\\}$. Making a change of variable
$b_{1}t=(b_{2}-b_{1})s+b_{1}x$ we get for $x\in[0,1]$ and
$\lambda\in\mathbb{C}$,

$\int_{0}^{x}R_{1,j,1}(x,t)e^{ib_{1}\lambda t}\,dt=e^{i\lambda
b_{1}x}\int_{0}^{\frac{-b_{1}x}{b_{2}-b_{1}}}R_{1,j,1}\left(x,\text{\scalebox{0.75}{$\frac{(b_{2}-b_{1})s+b_{1}x}{b_{1}}$}}\right)e^{i(b_{2}-b_{1})\lambda
s}\,ds.$ (2.8)

Similarly, making a change of variable $b_{2}t=(b_{2}-b_{1})s+b_{1}x$ we get

$\int_{0}^{x}R_{2,j,1}(x,t)e^{ib_{2}\lambda t}\,dt=e^{i\lambda
b_{1}x}\int_{\frac{-b_{1}x}{b_{2}-b_{1}}}^{x}R_{2,j,1}\left(x,\text{\scalebox{0.75}{$\frac{(b_{2}-b_{1})s+b_{1}x}{b_{2}}$}}\right)e^{i(b_{2}-b_{1})\lambda
s}\,ds.$ (2.9)

Inserting these two relations into (2.5) we arrive at

$\varphi_{j1}(x,\lambda)=e^{ib_{1}\lambda
x}\left(\delta_{j1}+\int_{0}^{x}R_{j}^{+}(x,t)e^{i(b_{2}-b_{1})\lambda
t}\,dt\right),\qquad x\in[0,1],\quad\lambda\in\mathbb{C},$ (2.10)

where

$R_{j}^{+}(x,t)=\begin{cases}R_{1,j,1}\left(x,\frac{(b_{2}-b_{1})t+b_{1}x}{b_{1}}\right),&\quad
t\in\left[0,\frac{-b_{1}x}{b_{2}-b_{1}}\right],\vspace{5pt}\\\
R_{2,j,1}\left(x,\frac{(b_{2}-b_{1})t+b_{1}x}{b_{2}}\right),&\quad
t\in\left[\frac{-b_{1}x}{b_{2}-b_{1}},x\right].\end{cases}$ (2.11)

Starting from inclusions $R_{1,j,k}\in X_{\infty}^{0}(\Omega)$, it can be
shown that $R_{j}^{+}\in X_{\infty}^{0}(\Omega)$. This proves part of
relations (2.6)–(2.7) for $\Phi_{1}(\cdot,\lambda)$. Similar formula for
$\Phi_{2}(\cdot,\lambda)$ can be proved by performing change of variable
$b_{k}t=(b_{1}-b_{2})s+b_{2}x$ in the integral
$\int_{0}^{x}R_{k,j,2}(x,t)e^{ib_{k}\lambda t}\,dt$ for $k\in\\{1,2\\}$. ∎

Next, we recall the following abstract completeness theorem for operators
$L_{U}(Q)$ proved in [15, 17]. For BVP (1.1)–(1.4) it takes the following
form.

######  Theorem 2.5 (Theorem 2.3 in [15]).

Assume that there exist $C,R>0$ and $m\in\mathbb{Z}_{+}$ such that

$|\Delta_{Q}(it)|\geqslant\frac{Ce^{-b_{1}t}}{t^{m}},\qquad|\Delta_{Q}(-it)|\geqslant\frac{Ce^{b_{2}t}}{t^{m}},\qquad
t>R.$ (2.12)

Then the system of root functions of the BVP (1.1)–(1.4) $($of the operator
$L_{U}(Q))$ is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$.

As a first application of this result and formula for the asymptotic behavior
of solutions to system (1.1) obtained in [17], we establish a simple
completeness result in the case of degenerate boundary conditions of the form
$y_{1}(0)=y_{2}(1)=0$. We formulate it in more general case that covers both
Dirac-type system and the case when $b_{1}/b_{2}\notin\mathbb{R}$.

######  Proposition 2.6.

Let $\arg b_{1}\neq\arg b_{2}$, let $Q_{12}$, $Q_{21}$ be continuous at the
endpoints 0 and 1, and let the following condition hold

$J_{32}=J_{42}=J_{13}=0,\qquad J_{14}\neq 0,\qquad Q_{12}(0)Q_{21}(1)\neq 0.$
(2.13)

Then the system of root functions of the operator $L_{U}(Q)$ is complete in
$L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

Since $J_{32}=J_{42}=J_{13}=0$ then formula (1.8) for $\Delta_{Q}(\cdot)$
simplifies to

$\Delta_{Q}(\lambda)=J_{12}+J_{34}e^{i(b_{1}+b_{2})\lambda}+J_{14}\varphi_{22}(1,\lambda),\qquad\lambda\in\mathbb{C}.$
(2.14)

Since $\arg b_{1}\neq\arg b_{2}$, then there exists $z\in\mathbb{C}$ such that
$\operatorname{Re}(ib_{1}z)<0<\operatorname{Re}(ib_{2}z)$. Since $J_{14}\neq
0$, it easily follows (cf. [32, formula (3.38)] and [17, Proposition 3.4])
that

$|\Delta_{Q}(\lambda)|\geqslant
Ce^{\operatorname{Re}(ib_{2}\lambda)},\qquad|\arg\lambda-\arg
z|<\varepsilon,\quad|\lambda|\geqslant R,$ (2.15)

for sufficiently small $\varepsilon>0$ and sufficiently large $R$.

Next, we estimate $\Delta_{Q}(\lambda)$ on the ray $\arg\lambda=-\arg z$.
Condition $\operatorname{Re}(ib_{1}z)<0<\operatorname{Re}(ib_{2}z)$ implies
that

$\operatorname{Re}(ib_{2}\lambda)<0<\operatorname{Re}(ib_{1}\lambda),\qquad\arg\lambda=-\arg
z.$ (2.16)

According to [17, Proposition 3.2] system (1.1) has a matrix solution
$Y(x,\lambda)=\begin{pmatrix}y_{11}(x,\lambda)&y_{12}(x,\lambda)\\\
y_{21}(x,\lambda)&y_{22}(x,\lambda)\end{pmatrix}$ satisfying

$Y(0,\lambda)=\begin{pmatrix}1&\frac{b_{1}Q_{12}(0)+o(1)}{b_{1}-b_{2}}\cdot\frac{1}{\lambda}\\\
0&1\end{pmatrix}\quad\text{as}\quad\lambda\to\infty,\quad\arg\lambda=-\arg z,$
(2.17)

and

$Y(1,\lambda)=\begin{pmatrix}(1+o(1))\cdot e^{ib_{1}\lambda}&0\\\
\frac{b_{2}Q_{21}(1)+o(1)}{b_{2}-b_{1}}\cdot\frac{e^{ib_{1}\lambda}}{\lambda}&(1+o(1))\cdot
e^{ib_{2}\lambda}\end{pmatrix}\quad\text{as}\quad\lambda\to\infty,\ \
\arg\lambda=-\arg z.$ (2.18)

Evidently

$\Phi(x,\lambda)=Y(x,\lambda)[Y(0,\lambda)]^{-1},\qquad\arg\lambda=-\arg z.$
(2.19)

Therefore,

$\varphi_{22}(\lambda)=y_{22}(1,\lambda)-y_{12}(0,\lambda)y_{21}(1,\lambda)\\\
=(1+o(1))\cdot
e^{ib_{2}\lambda}-\frac{b_{1}Q_{12}(0)+o(1)}{b_{1}-b_{2}}\cdot\frac{b_{2}Q_{21}(1)+o(1)}{b_{2}-b_{1}}\cdot\frac{e^{ib_{1}\lambda}}{\lambda^{2}}$
(2.20)

as $\lambda\to\infty$ and $\arg\lambda=-\arg z$. Inserting (2.20) into (2.14)
with account of inequalities (2.16) and condition
$J_{14}Q_{12}(0)Q_{21}(1)\neq 0$ (see (2.13)) now yields

$|\Delta_{Q}(\lambda)|\geqslant
C\frac{e^{\operatorname{Re}(ib_{1}\lambda)}}{|\lambda|^{2}},\qquad\arg\lambda=-\arg
z,\quad|\lambda|\geqslant R,$ (2.21)

for some $C>0$ and sufficiently large $R>0$. Now completeness and minimality
of the system of root functions of the BVP (1.1)–(1.4) follows from Theorem
2.3 in [15] due of estimates (2.15) and (2.21). ∎

######  Remark 2.7.

Proposition 2.6 was implicitly contained in [17] but was not included due to
the lack of space. Note that it significantly relaxes smoothness assumptions
compared to [1, Theorems 1.5] where the same result was proved for analytic
$Q(\cdot)$ and non-Dirac-type case $(b_{1}/b_{2}\notin\mathbb{R})$.

In a very recent paper [9] A.P. Kosarev and A.A. Shkalikov considered the case
of $2\times 2$ Dirac-type operators with non-constant matrix
$B=\operatorname{diag}(b_{1}(x),b_{2}(x))$ and degenerate boundary conditions
of special form $(y_{1}(0)=y_{2}(1)=0)$. Namely, in [9, Theorem 4] they
established completeness property of the root vectors system of the
corresponding BVP under the smoothness assumption
$b_{1},b_{2},Q_{12},Q_{21}\in W_{1}^{1}[0,1]$ assuming the same algebraic
condition $Q_{12}(0)Q_{21}(1)\neq 0$. Note that in the case of a constant
matrix $B$, Proposition 2.6 ensures the completeness property under slightly
relaxed smoothness assumption on the potential matrix $Q(\cdot)$: continuity
at the endpoints 0 and 1 instead of absolute continuity on $[0,1]$.

##  3 Asymptotic expansion of solutions to the Dirac type system

Here we obtain announced in [14], refined asymptotic formulas for the
solutions to the system (1.1) following Marchenko’s method from [33], where
these expansions were obtained for the classical Dirac system under the
assumptions $Q\in C^{n}([0,1];\mathbb{C}^{2\times 2})$, $n\geqslant 0$, and
$Q\in W_{2}^{n}([0,1];\mathbb{C}^{2\times 2})$, $n\geqslant 1$ (see problems 1
and 3 in [33, §1.4], respectively).

We generalize these results to Dirac-type systems and relax smoothness
assumption to condition $Q\in W_{1}^{n}([0,1];\mathbb{C}^{2\times 2})$,
$n\geqslant 0$. Namely, condition $Q\in L^{1}([0,1];\mathbb{C}^{2\times 2})$
is covered by our Corollary 2.4 (the case $n=0$), while the case $n\geqslant
1$ is treated in Theorem 3.1 below, where the space $X_{\infty}^{0}$ again
plays significant role.

Recall that $B=\operatorname{diag}(b_{1},b_{2})$, where $b_{1}<0<b_{2}$. To
formulate the next result, we set

$b^{+}:=ib_{1},\quad b^{-}:=ib_{2},\quad q_{+}(x):=-ib_{1}Q_{12}(x),\quad
q_{-}(x):=-ib_{2}Q_{21}(x).$ (3.1)

Then, system (1.1) takes the form

$\begin{cases}y_{1}^{\prime}=b^{+}\lambda y_{1}+q_{+}(x)y_{2},\\\
y_{2}^{\prime}=b^{-}\lambda y_{2}+q_{-}(x)y_{1}.\end{cases}$ (3.2)

######  Theorem 3.1.

Let $n\in\mathbb{N}$ and $q_{+},q_{-}\in W_{1}^{n}[0,1]$. Then system (3.2)
has a matrix solution of the form

$Y(x,\lambda)=\begin{pmatrix}e^{b^{+}\lambda
x}u_{1}^{+}(x,\lambda)&e^{b^{-}\lambda x}u_{2}^{-}(x,-\lambda)\\\
e^{b^{+}\lambda x}u_{2}^{+}(x,\lambda)&e^{b^{-}\lambda
x}u_{1}^{-}(x,-\lambda)\end{pmatrix},\quad
x\in[0,1],\quad\lambda\in\mathbb{C}\setminus\\{0\\},$ (3.3)

where the functions $u_{j}^{\pm}(x,\lambda)$, $j\in\\{1,2\\}$, admit the
following representation

$\displaystyle
u_{1}^{\pm}(x,\lambda)=\sum_{k=0}^{n}\frac{b_{k}^{\pm}(x)}{(i(b_{1}-b_{2})\lambda)^{k}}+\frac{b_{n}^{\pm}(x,\lambda)}{(i(b_{1}-b_{2})\lambda)^{n}},$
(3.4) $\displaystyle
u_{2}^{\pm}(x,\lambda)=\sum_{k=0}^{n}\frac{a_{k}^{\pm}(x)}{(i(b_{1}-b_{2})\lambda)^{k}}+\frac{a_{n}^{\pm}(x,\lambda)}{(i(b_{1}-b_{2})\lambda)^{n}},$
(3.5)

where

$\displaystyle a_{0}^{\pm}(x)=0,\qquad b_{0}^{\pm}(x)=1,$ (3.6) $\displaystyle
a_{k}^{\pm}(x)=q_{\mp}(x)b_{k-1}^{\pm}(x)-\bigl{(}a_{k-1}^{\pm}(x)\bigr{)}^{\prime},\quad
k\in\\{1,\ldots,{n}\\},$ (3.7) $\displaystyle
b_{k}^{\pm}(x)=\int_{0}^{x}q_{\pm}(t)a_{k}^{\pm}(t)dt,\quad
k\in\\{1,\ldots,{n}\\},$ (3.8)

Besides, the functions $a_{n}^{\pm}(x,\lambda)$ and $b_{n}^{\pm}(x,\lambda)$
are given by

$\displaystyle a_{n}^{\pm}(x,\lambda)$ $\displaystyle=$
$\displaystyle\int_{0}^{x}A_{n}^{\pm}(x,t)e^{i(b_{2}-b_{1})\lambda t}dt,$
(3.9) $\displaystyle b_{n}^{\pm}(x,\lambda)$ $\displaystyle=$
$\displaystyle\int_{0}^{x}\left(\int_{t}^{x}A_{n}^{\pm}(s,t)q_{\pm}(s)ds\right)e^{i(b_{2}-b_{1})\lambda
t}dt,$ (3.10)

where $A_{n}^{\pm}\in X_{\infty}^{0}$ is the unique solution to the linear
integral equation

$A_{n}^{\pm}(x,\xi)=v_{n}^{\pm}(x-\xi)+\int_{0}^{\xi}\int_{t}^{t+x-\xi}q_{\mp}(t+x-\xi)q_{\pm}(s)A_{n}^{\pm}(s,t)\,ds\,dt,$
(3.11)

for $0\leqslant\xi\leqslant x\leqslant 1$, and
$v_{n}^{\pm}(\cdot):=-\left(a_{n}^{\pm}(\cdot)\right)^{\prime}+q_{\mp}(\cdot)b_{n}^{\pm}(\cdot)\in
L^{1}[0,1]$.

###### Proof.

Substituting the first column of (3.3) into (3.2) we obtain

$\displaystyle b^{+}\lambda\ e^{b^{+}\lambda x}u_{1}^{+}+e^{b^{+}\lambda
x}(u_{1}^{+})^{\prime}$ $\displaystyle=$ $\displaystyle b^{+}\lambda\
e^{b^{+}\lambda x}u_{1}^{+}+q_{+}(x)e^{b^{+}\lambda x}u_{2}^{+},$ (3.12)
$\displaystyle b^{+}\lambda\ e^{b^{+}\lambda x}u_{2}^{+}+e^{b^{+}\lambda
x}(u_{2}^{+})^{\prime}$ $\displaystyle=$ $\displaystyle b^{-}\lambda\
e^{b^{+}\lambda x}u_{2}^{+}+q_{-}(x)e^{b^{+}\lambda x}u_{1}^{+},$ (3.13)

or

$\displaystyle\bigl{(}u_{1}^{+}(x,\lambda)\bigr{)}^{\prime}$
$\displaystyle=q_{+}(x)u_{2}^{+}(x,\lambda),$ (3.14)
$\displaystyle\bigl{(}u_{2}^{+}(x,\lambda)\bigr{)}^{\prime}$
$\displaystyle=q_{-}(x)u_{1}^{+}(x,\lambda)-(b^{+}-b^{-})\lambda
u_{2}^{+}(x,\lambda).$ (3.15)

Similarly, inserting second column of (3.3) into (3.2) we arrive at

$\displaystyle\bigl{(}u_{1}^{-}(x,-\lambda)\bigr{)}^{\prime}$ $\displaystyle=$
$\displaystyle q_{-}(x)u_{2}^{-}(x,-\lambda),$ (3.16)
$\displaystyle\bigl{(}u_{2}^{-}(x,-\lambda)\bigr{)}^{\prime}$ $\displaystyle=$
$\displaystyle q_{+}(x)u_{1}^{-}(x,-\lambda)-(b^{-}-b^{+})\lambda
u_{2}^{-}(x,-\lambda).$ (3.17)

Changing $\lambda$ to $-\lambda$ in (3.16) and (3.17) we can combine (3.14),
(3.15), (3.16) and (3.17) into

$\displaystyle\bigl{(}u_{1}^{\pm}(x,\lambda)\bigr{)}^{\prime}$
$\displaystyle=$ $\displaystyle q_{\pm}(x)u_{2}^{\pm}(x,\lambda),$ (3.18)
$\displaystyle\bigl{(}u_{2}^{\pm}(x,\lambda)\bigr{)}^{\prime}$
$\displaystyle=$ $\displaystyle
q_{\mp}(x)u_{1}^{\pm}(x,\lambda)-(b^{+}-b^{-})\lambda u_{2}^{\pm}(x,\lambda).$
(3.19)

Next we show that functions

$a_{k}^{\pm}(\cdot),\ b_{k}^{\pm}(\cdot),\quad k\in\\{0,1,\ldots,n\\},$ (3.20)

are well defined by formulas (3.6)–(3.8). To this end we prove by induction
that

$a_{k}^{\pm}(\cdot)\in W_{1}^{n-k+1}[0,1],\quad b_{k}^{\pm}(\cdot)\in
W_{1}^{n-k+2}[0,1],\quad k\in\\{0,1,\ldots,n\\}.$ (3.21)

For $k=0$ this statement is clear in view of (3.6). Assume that (3.21) holds
for some $k\in\\{0,1,\ldots,n-1\\}$ and let us prove it for $k+1$. From (3.7)
we have

$a_{k+1}^{\pm}(\cdot)=q_{\mp}(\cdot)b_{k}^{\pm}(\cdot)-\bigl{(}a_{k}^{\pm}(\cdot)\bigr{)}^{\prime}\in
W_{1}^{n-k}[0,1],$ (3.22)

since $a_{k}^{\pm}\in W_{1}^{n-k+1}[0,1]$ (and hence
$(a_{k}^{\pm})^{\prime}\in W_{1}^{n-k}[0,1]$), $q_{\mp}\in W_{1}^{n}[0,1]$,
and $b_{k}^{\pm}\in W_{1}^{n-k+1}[0,1]$ for $0\leqslant k<n$. From (3.8) we
have

$b_{k+1}^{\pm}(x)=\int_{0}^{x}q_{\pm}(t)a_{k+1}^{\pm}(t)dt\in
W_{1}^{n-k+1}[0,1],$ (3.23)

because $q_{\pm}\in W_{1}^{n}[0,1]$ and $a_{k+1}^{\pm}\in W_{1}^{n-k}[0,1]$.
Thus (3.21) holds and we have as a consequence that

$a_{k}^{\pm}(\cdot),b_{k}^{\pm}(\cdot)\in W_{1}^{1}[0,1],\quad
k\in\\{0,1,\ldots,n\\}.$ (3.24)

Let’s rewrite equations (3.18) and (3.19) in terms of functions
$a_{n}^{\pm}(x,\lambda)$ and $b_{n}^{\pm}(x,\lambda)$. Inserting (3.4) and
(3.5) into (3.18) and setting

$c:=b^{+}-b^{-}=i(b_{1}-b_{2}),$ (3.25)

we get

$\sum_{k=0}^{n}\bigl{(}b_{k}^{\pm}(x)\bigr{)}^{\prime}(c\lambda)^{-k}+\bigl{(}b_{n}^{\pm}(x,\lambda)\bigr{)}^{\prime}(c\lambda)^{-n}\\\
=\sum_{k=0}^{n}q_{\pm}(x)a_{k}^{\pm}(x)(c\lambda)^{-k}+q_{\pm}(x)a_{n}^{\pm}(x,\lambda)(c\lambda)^{-n}.$
(3.26)

In view of (3.6) and (3.8) we have

$\bigl{(}b_{k}^{\pm}(x)\bigr{)}^{\prime}=q_{\pm}(x)a_{k}^{\pm}(x),\quad
k\in\\{0,1,\ldots,n\\}.$ (3.27)

Hence (3.26) is equivalent to

$\bigl{(}b_{n}^{\pm}(x,\lambda)\bigr{)}^{\prime}=q_{\pm}(x)a_{n}^{\pm}(x,\lambda).$
(3.28)

Inserting (3.4) and (3.5) into (3.19) we derive

$\sum_{k=0}^{n}\bigl{(}a_{k}^{\pm}(x)\bigr{)}^{\prime}(c\lambda)^{-k}+\bigl{(}a_{n}^{\pm}(x,\lambda)\bigr{)}^{\prime}(c\lambda)^{-n}\\\
=\sum_{k=0}^{n}q_{\mp}(x)b_{k}^{\pm}(x)(c\lambda)^{-k}+q_{\mp}(x)b_{n}^{\pm}(x,\lambda)(c\lambda)^{-n}\\\
-\sum_{k=0}^{n}a_{k}^{\pm}(x)(c\lambda)^{-k+1}-a_{n}^{\pm}(x,\lambda)(c\lambda)^{-n+1}.$
(3.29)

This identity with account of (3.6) can be rewritten as

$\sum_{k=0}^{n-1}\left(\left(a_{k}^{\pm}(x)\right)^{\prime}-q_{\mp}(x)b_{k}^{\pm}(x)+a_{k+1}^{\pm}(x)\right)(c\lambda)^{-k}=\Bigl{(}-\left(a_{n}^{\pm}(x)\right)^{\prime}-\left(a_{n}^{\pm}(x,\lambda)\right)^{\prime}\Bigr{.}\\\
\Bigl{.}+q_{\mp}(x)b_{n}^{\pm}(x,\lambda)+q_{\mp}(x)b_{n}^{\pm}(x)-c\lambda
a_{n}^{\pm}(x,\lambda)\Bigr{)}(c\lambda)^{-n}.$ (3.30)

Due to (3.7) each summand in the left hand side of (3.30) with $k\geqslant 1$
vanishes. Besides, combining (3.7) with (3.6) ensures that the first summand
(with $k=0$) in (3.30) also vanishes and the identity (3.30) turns into

$\bigl{(}a_{n}^{\pm}(x,\lambda)\bigr{)}^{\prime}=q_{\mp}(x)b_{n}^{\pm}(x,\lambda)-c\lambda
a_{n}^{\pm}(x,\lambda)+v_{n}^{\pm}(x).$ (3.31)

where

$v_{n}^{\pm}(x):=-\left(a_{n}^{\pm}(x)\right)^{\prime}+q_{\mp}(x)b_{n}^{\pm}(x),\qquad
n\geqslant 1.$ (3.32)

It follows from (3.24) that $v_{n}^{\pm}\in L^{1}[0,1]$.

Next we show that combining (3.9) with (3.10) implies (3.28) for any function
$A_{n}^{\pm}(\cdot,\cdot)\in L^{1}(\Omega)$ where $\Omega=\\{(x,t):0\leqslant
t\leqslant x\leqslant 1\\}$ is given by (2.1). Indeed, changing order of
integration and taking definition (3.9) into account we get

$b_{n}^{\pm}(x,\lambda)=\int_{0}^{x}q_{\pm}(s)\left(\int_{0}^{s}A_{n}^{\pm}(s,t)e^{-c\lambda
t}dt\right)ds=\int_{0}^{x}q_{\pm}(s)a_{n}^{\pm}(s,\lambda)ds.$ (3.33)

Differentiating this identity leads to (3.28).

Making change of variables in (3.9) we get

$a_{n}^{\pm}(x,\lambda)=\int_{0}^{x}A_{n}^{\pm}(x,x-u)e^{-c\lambda(x-u)}du.$
(3.34)

Assume for the moment that (we prove this fact later)

$A_{n}^{\pm}(\cdot,\cdot-u)\in W_{1}^{1}[u,1],\quad\text{for a.e.}\ \
u\in[0,1].$ (3.35)

Assuming this, we can differentiate in (3.34) for almost every $x\in[0,1]$ and
with account of (3.9) obtain

$\left(a_{n}^{\pm}(x,\lambda)\right)^{\prime}=A_{n}^{\pm}(x,0)\\\
+\int_{0}^{x}\left(\frac{d}{dx}\bigl{(}A_{n}^{\pm}(x,x-u)\bigr{)}e^{-c\lambda(x-u)}-c\lambda
A_{n}^{\pm}(x,x-u)e^{-c\lambda(x-u)}\right)du\\\ =A_{n}^{\pm}(x,0)-c\lambda
a_{n}^{\pm}(x,\lambda)+\int_{0}^{x}\frac{d}{dx}\bigl{(}A_{n}^{\pm}(x,x-u)\bigr{)}e^{-c\lambda(x-u)}du.$
(3.36)

Inserting (3.10) and (3.36) into (3.31) and making change of variable in the
integral at the right-hand side we arrive at the following equation

$A_{n}^{\pm}(x,0)+\int_{0}^{x}\frac{d}{dx}\bigl{(}A_{n}^{\pm}(x,x-u)\bigr{)}e^{-c\lambda(x-u)}du\\\
=v_{n}^{\pm}(x)+q_{\mp}(x)\int_{0}^{x}\left(\int_{x-u}^{x}A_{n}^{\pm}(s,x-u)q_{\pm}(s)ds\right)e^{-c\lambda(x-u)}du$
(3.37)

Evidently, this equation will turn into identity if we set

$\displaystyle A_{n}^{\pm}(x,0)=v_{n}^{\pm}(x),\quad x\in[0,1],$ (3.38)
$\displaystyle\frac{d}{dx}\bigl{(}A_{n}^{\pm}(x,x-u)\bigr{)}=q_{\mp}(x)\int_{x-u}^{x}A_{n}^{\pm}(s,x-u)q_{\pm}(s)ds,\quad
0\leqslant u\leqslant x\leqslant 1.$ (3.39)

Thus, it suffices to show that integro-differential equation (3.39) with
initial condition (3.38) has a solution $A_{n}^{\pm}(\cdot,\cdot)\in
X_{\infty}^{0}(\Omega)$.

Changing $x$ to $t$ in (3.39) and integrating thus obtained equality over
$t\in[u,x]$ we get

$\int_{u}^{x}\frac{d}{dt}\bigl{(}A_{n}^{\pm}(t,t-u)\bigr{)}dt=\int_{u}^{x}q_{\mp}(t)\int_{t-u}^{t}A_{n}^{\pm}(s,t-u)q_{\pm}(s)dsdt$
(3.40)

or in view of (3.35) and (3.38)

$A_{n}^{\pm}(x,x-u)-v_{n}^{\pm}(u)=\int_{0}^{x-u}\int_{t}^{t+u}q_{\mp}(t+u)q_{\pm}(s)A_{n}^{\pm}(s,t)dsdt,\\\
0\leqslant u\leqslant x\leqslant 1.$ (3.41)

Setting $u=x-\xi$ here we easily derive

$A_{n}^{\pm}(x,\xi)=v_{n}^{\pm}(x-\xi)+\int_{0}^{\xi}\int_{t}^{t+x-\xi}q_{\mp}(t+x-\xi)q_{\pm}(s)A_{n}^{\pm}(s,t)dsdt,\\\
0\leqslant\xi\leqslant x\leqslant 1.$ (3.42)

It is clear that (3.42) is equivalent to the initial value problem
(3.38)–(3.39). Hence we need to prove that the integral equation (3.42) has a
solution in the class $X_{\infty}^{0}$.

Note that for $x\in[0,1]$ one has

$\|v_{n}^{\pm}(x-\cdot)\|_{L^{1}[0,x]}=\int_{0}^{x}\left|{v_{n}^{\pm}(x-t)}\right|dt=\int_{0}^{x}\left|{v_{n}^{\pm}(t)}\right|dt\leqslant\|v_{n}^{\pm}\|_{L^{1}[0,1]}.$
(3.43)

Hence the function $V_{n}^{\pm}(x,\xi):=v_{n}^{\pm}(x-\xi)$ belongs to the
space $X_{\infty}$. It is also straightforward to show that $V_{n}^{\pm}\in
X_{\infty}^{0}$ by approximating $v_{n}^{\pm}(\cdot)$ in $L^{1}$-norm with
continuous functions.

Consider the following operator $T^{\pm}$ from $X_{\infty}$ into $X_{\infty}$,

$(T^{\pm}F)(x,\xi):=\int_{0}^{\xi}\int_{t}^{t+x-\xi}q_{\mp}(t+x-\xi)q_{\pm}(s)F(s,t)dsdt,\quad
0\leqslant\xi\leqslant x\leqslant 1.$ (3.44)

Integral equation (3.42) can be rewritten in the following abstract form

$(I_{H}-T^{\pm})A_{n}^{\pm}=V_{n}^{\pm},$ (3.45)

where $V_{n}^{\pm}$ is a fixed element of $X_{\infty}^{0}$. Let us prove
existence of $A_{n}^{\pm}\in X_{\infty}^{0}$ satisfying (3.45).

At first let us rewrite the definition of $T^{\pm}$ in more convenient way

$(T^{\pm}F)(x_{1},x_{2})=\iint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant
t_{2}\leqslant x_{2}}{0\leqslant t_{1}-t_{2}\leqslant
x_{1}-x_{2}}}q_{\mp}(t_{2}+x_{1}-x_{2})q_{\pm}(t_{1})F(t_{1},t_{2})dt_{1}dt_{2},\\\
0\leqslant x_{2}\leqslant x_{1}\leqslant 1.$ (3.46)

Recall that $q_{\pm}\in W_{1}^{n}[0,1]\subset C[0,1]$. Let us prove by
induction in $k$ that

$|((T^{\pm})^{k+1}F)(x_{1},x_{2})|\\\ \leqslant
C^{k+1}\iint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant t_{2}\leqslant
x_{2}}{0\leqslant t_{1}-t_{2}\leqslant
x_{1}-x_{2}}}\frac{(x_{2}-t_{2})^{k}}{k!}\cdot\frac{((x_{1}-x_{2})-(t_{1}-t_{2}))^{k}}{k!}\cdot|F(t_{1},t_{2})|dt_{1}dt_{2},\\\
0\leqslant x_{2}\leqslant x_{1}\leqslant 1.$ (3.47)

where

$C=\|q_{+}\|_{C[0,1]}\cdot\|q_{-}\|_{C[0,1]}.$ (3.48)

The base case $k=0$ directly follows from (3.46) and (3.48). Now assuming that
(3.47) is proved for $k=l-1$ we prove it for $k=l$, where $l\in\mathbb{N}$. We
have for $0\leqslant x_{2}\leqslant x_{1}\leqslant 1$

$\left|{((T^{\pm})^{l+1}F)(x_{1},x_{2})}\right|\leqslant
C\iint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant t_{2}\leqslant
x_{2}}{0\leqslant t_{1}-t_{2}\leqslant
x_{1}-x_{2}}}|((T^{\pm})^{l}F)(t_{1},t_{2})|dt_{1}dt_{2}\\\ \leqslant
C^{l+1}\iiiint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant s_{2}\leqslant
t_{2}\leqslant x_{2}}{0\leqslant s_{1}-s_{2}\leqslant t_{1}-t_{2}\leqslant
x_{1}-x_{2}}}\frac{(t_{2}-s_{2})^{l-1}}{(l-1)!}\times\frac{((t_{1}-t_{2})-(s_{1}-s_{2}))^{l-1}}{(l-1)!}\\\
\times|F(s_{1},s_{2})|ds_{1}ds_{2}dt_{1}dt_{2}=C^{l+1}\iint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant
s_{2}\leqslant x_{2}}{0\leqslant s_{1}-s_{2}\leqslant
x_{1}-x_{2}}}|F(s_{1},s_{2})|ds_{1}ds_{2}\\\
\iint\limits_{\genfrac{}{}{0.0pt}{2}{s_{2}\leqslant t_{2}\leqslant
x_{2}}{s_{1}-s_{2}\leqslant t_{1}-t_{2}\leqslant
x_{1}-x_{2}}}\frac{(t_{2}-s_{2})^{l-1}}{(l-1)!}\times\frac{((t_{1}-t_{2})-(s_{1}-s_{2}))^{l-1}}{(l-1)!}dt_{1}dt_{2}\\\
=C^{l+1}\iint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant s_{2}\leqslant
x_{2}}{0\leqslant s_{1}-s_{2}\leqslant
x_{1}-x_{2}}}\frac{(x_{2}-s_{2})^{l}}{l!}\times\frac{((x_{1}-x_{2})-(s_{1}-s_{2}))^{l}}{l!}\times|F(s_{1},s_{2})|ds_{1}ds_{2}\\\
\leqslant\frac{C^{l+1}}{l!^{2}}\iint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant
s_{2}\leqslant x_{2}}{0\leqslant s_{1}-s_{2}\leqslant
x_{1}-x_{2}}}|F(s_{1},s_{2})|ds_{1}ds_{2}.$ (3.49)

So (3.47) is proved. Now by (3.47) we have for $k\in\mathbb{Z}_{+}$ and
$x_{1}\in[0,1]$

$\|((T^{\pm})^{k+1}F)(x_{1},\cdot)\|_{L^{1}[0,x_{1}]}=\int_{0}^{x_{1}}\left|{((T^{\pm})^{k+1}F)(x_{1},x_{2})}\right|dx_{2}\\\
\leqslant\frac{C^{k+1}}{k!^{2}}\iiint\limits_{\genfrac{}{}{0.0pt}{2}{0\leqslant
t_{2}\leqslant x_{2}\leqslant x_{1}}{0\leqslant t_{1}-t_{2}\leqslant
x_{1}-x_{2}}}|F(t_{1},t_{2})|dt_{1}dt_{2}dx_{2}\\\
\leqslant\frac{C^{k+1}}{k!^{2}}\int_{0}^{x_{1}}\int_{0}^{x_{1}}\left(\int_{0}^{t_{1}}\left|{F(t_{1},t_{2})}\right|dt_{2}\right)dt_{1}dx_{2}\\\
\leqslant\frac{C^{k+1}}{k!^{2}}\int_{0}^{x_{1}}\int_{0}^{x_{1}}\|F\|_{X_{\infty}}\,dt_{1}dx_{2}\leqslant\frac{C^{k+1}}{k!^{2}}\|F\|_{X_{\infty}}.$
(3.50)

Hence

$\|(T^{\pm})^{k+1}F\|_{X_{\infty}}\leqslant\frac{C^{k+1}}{k!^{2}}\|F\|_{X_{\infty}},\quad
F\in X_{\infty},\quad k\in\mathbb{Z}_{+},$ (3.51)

or

$\|(T^{\pm})^{k}\|\leqslant\frac{C^{k}}{((k-1)!)^{2}},\quad k\in\mathbb{N}.$
(3.52)

This implies that the operator $T^{\pm}$ has zero spectral radius. Hence the
operator $I_{H}-T^{\pm}$ has bounded inverse and
$A_{n}^{\pm}=(I_{H}-T^{\pm})^{-1}V_{n}^{\pm}\in X_{\infty}$.

Let us prove that, in fact, the operator $T^{\pm}$ maps $X_{\infty}^{0}$ into
itself. Let $F\in X_{\infty}^{0}$ and let $F_{m}\to F$ in $X_{\infty}$ as
$m\to\infty$ where $F_{m}\in C(\Omega)$, $m\in\mathbb{N}$. Since $q_{\pm}\in
C[0,1]$, it is evident from (3.44) that $T^{\pm}F_{m}\in C(\Omega)$,
$m\in\mathbb{N}$. Combining (3.44) with (3.48) we get for
$0\leqslant\xi\leqslant x\leqslant 1$,

$\left|{(T^{\pm}F)(x,\xi)-(T^{\pm}F_{m})(x,\xi)}\right|\leqslant
C\int_{0}^{\xi}\int_{t}^{t+x-\xi}\left|{F(s,t)-F_{m}(s,t)}\right|\,ds\,dt\\\
=C\int_{0}^{x}\int_{\max\\{0,s-x+\xi\\}}^{\min\\{s,\xi\\}}\left|{F(s,t)-F_{m}(s,t)}\right|\,dt\,ds\\\
\leqslant C\int_{0}^{x}\|F(s,\cdot)-F_{m}(s,\cdot)\|_{L^{1}[0,s]}\,ds\leqslant
C\|F-F_{m}\|_{X_{\infty}}.$ (3.53)

This estimate trivially implies that
$\|T^{\pm}F_{m}-T^{\pm}F\|_{X_{\infty}}\to 0$ as $m\to\infty$. Hence
$T^{\pm}F\in X_{\infty}^{0}$. This concludes the proof of the desired
inclusion $A_{n}^{\pm}\in X_{\infty}^{0}$.

Finally let’s prove (3.35). In view of (3.41) it is equivalent to

$B^{\pm}(\cdot,u)\in L^{1}[0,1-u],\quad u\in[0,1],$ (3.54)

where

$B^{\pm}(t,u):=\int_{t}^{t+u}q_{\mp}(t+u)q_{\pm}(s)A_{n}^{\pm}(s,t)\,ds,\quad
t,u\geqslant 0,\ \ t+u\leqslant 1.$ (3.55)

Combining (3.48) with inclusion $A_{n}^{\pm}\in X_{\infty}$ and doing
estimation similarly to (3.53) implies

$\int_{0}^{1-u}|B^{\pm}(t,u)|dt\leqslant
C\int_{0}^{1-u}\int_{t}^{t+u}|A_{n}^{\pm}(s,t)|\,ds\,dt\leqslant
C\|A_{n}^{\pm}\|_{X_{\infty}}.$ (3.56)

Hence (3.54) holds and the proof is complete. ∎

Lemma 2.1 allows us to estimate growth of remainder terms
$a_{n}^{\pm}(x,\lambda)$ and $b_{n}^{\pm}(x,\lambda)$ in expansions
(3.4)–(3.5) in the upper half-plane.

######  Lemma 3.2.

Let $h\in\mathbb{R}$ be fixed. Then under the assumption of Theorem 3.1 the
functions $a_{n}^{\pm}(x,\lambda)$ and $b_{n}^{\pm}(x,\lambda)$ given by
(3.9)–(3.10) satisfy the following asymptotic relations uniformly in
$x\in[0,1]$,

$a_{n}^{\pm}(x,\lambda)=o(1),\qquad
b_{n}^{\pm}(x,\lambda)=o(1),\qquad\lambda\to\infty,\ \
\operatorname{Im}\lambda\geqslant h.$ (3.57)

###### Proof.

Let $\delta>0$. Combining (3.9) with the inclusion $A_{n}^{\pm}\in
X_{\infty}^{0}$, Lemma 2.1 and inequality $b_{1}-b_{2}<0$, we have

$\left|{a_{n}^{\pm}(x,\lambda)}\right|=\left|{\int_{0}^{x}A_{n}^{\pm}(x,t)e^{i(b_{2}-b_{1})\lambda
t}dt}\right|\leqslant\delta\cdot(e^{(b_{1}-b_{2})\operatorname{Im}\lambda}+1)\\\
\leqslant\delta\cdot(e^{(b_{1}-b_{2})h}+1),\qquad\operatorname{Im}\lambda\geqslant
h,\quad|\lambda|>R_{\delta},\quad x\in[0,1].$ (3.58)

This trivially implies the first relation in (3.57).

Recall that $q_{\pm}\in W_{1}^{n}[0,1]\subset C[0,1]$. Hence formula (3.33)
implies estimate

$\left|{b_{n}^{\pm}(x,\lambda)}\right|\leqslant\|q_{\pm}\|_{C[0,1]}\cdot\sup_{t\in[0,1]}|a_{n}^{\pm}(t,\lambda)|,\quad\lambda\in\mathbb{C}\setminus\\{0\\}.$
(3.59)

Combining the first relation in (3.57) with estimate (3.59) implies the second
relation in (3.57). ∎

######  Remark 3.3.

Note that in the recent papers [9, 10] A.P. Kosarev and A.A. Shkalikov
established asymptotic expansions similar to (3.4)–(3.5) with estimate of the
remainder term as in (3.57) for $2\times 2$ Dirac-type operators with non-
constant matrix $B(\cdot)$ using Birkhoff-Tamarkin method (see also [38]).
Unique feature of Marchenko method, that we used in the proof of formulas
(3.4)–(3.5), is the integral representation of the remainder term as a Fourier
transform of the solution to certain explicit integral equation.

Formulas (3.7) and (3.8) contain undesirable integration operations, which are
avoidable by considering the quotient
$u_{2}^{\pm}(x,\lambda)[u_{1}^{\pm}(x,\lambda)]^{-1}$.

First we need the following abstract result of folklore nature on the
asymptotic behavior of rational functions.

######  Lemma 3.4.

Let holomorphic in unbounded domain $D\subset\mathbb{C}\setminus\\{0\\}$
functions $u_{1}(\lambda)$ and $u_{2}(\lambda)$ be given by

$\displaystyle
u_{1}(\lambda)=\sum_{k=0}^{n}b_{k}\cdot(c\lambda)^{-k}+b_{n}(\lambda)\cdot(c\lambda)^{-n},\qquad\lambda\in
D,$ (3.60) $\displaystyle
u_{2}(\lambda)=\sum_{k=0}^{n}a_{k}\cdot(c\lambda)^{-k}+a_{n}(\lambda)\cdot(c\lambda)^{-n},\qquad\lambda\in
D,$ (3.61)

where $b_{0}=1$, $c\in\mathbb{C}\setminus\\{0\\}$ is fixed, and

$a_{n}(\lambda)=o(1),\quad
b_{n}(\lambda)=o(1)\quad\text{as}\quad\lambda\to\infty,\quad\lambda\in D.$
(3.62)

Assume that $u_{1}(\lambda)\neq 0$ for $\lambda\in D$. Then the following
identity holds

$\frac{u_{2}(\lambda)}{u_{1}(\lambda)}=\sum_{k=0}^{n}\sigma_{k}\cdot(c\lambda)^{-k}+\sigma_{n}(\lambda)\cdot(c\lambda)^{-n},\qquad\lambda\in
D,$ (3.63)

where

$\sigma_{0}=a_{0},\qquad\sigma_{k}=a_{k}-\sum_{j=1}^{k}b_{j}\sigma_{k-j},\qquad
k\in\\{1,\ldots,{n}\\},$ (3.64)

and

$\sigma_{n}(\lambda)=\frac{1}{u_{1}(\lambda)}\left(a_{n}(\lambda)-b_{n}(\lambda)\sum_{k=0}^{n}\sigma_{k}\cdot(c\lambda)^{-k}\right.\\\
\left.-\sum_{k=n+1}^{2n}\sum_{j=n-k}^{n}b_{j}\sigma_{k-j}\cdot(c\lambda)^{k-n}\right),\qquad\lambda\in
D.$ (3.65)

In particular,

$\sigma_{n}(\lambda)=o(1)\quad\text{as}\quad\lambda\to\infty,\quad\lambda\in
D.$ (3.66)

###### Proof.

Multiplying (3.63) by $u_{1}(\lambda)$, expanding the product of two sums and
comparing coefficients at powers of $\lambda$ one can arrive at formulas
(3.64)–(3.65) after straightforward calculations. Since $b_{0}=1$, it is
evident that $u_{1}(\lambda)=1+o(1)$ as $\lambda\to\infty$, $\lambda\in D$.
Inserting this asymptotic formula and formulas (3.62) into (3.65) one readily
arrives at (3.66). ∎

First we apply Lemma 3.4 to establish asymptotic expansion of the quotient
$\frac{u_{2}^{\pm}(x,\lambda)}{u_{1}^{\pm}(x,\lambda)}$ with coefficients
$\sigma^{\pm}_{k}(x)$ that are expressed via $a_{k}^{\pm}(x),b_{k}^{\pm}(x)$.

######  Proposition 3.5.

Under the assumptions of Theorem 3.1 consider the functions
$u_{2}^{\pm}(x,\lambda)$ and $u_{1}^{\pm}(x,\lambda)$ $($see (3.4) and
(3.5)$)$ from the representation of the fundamental solution $Y(x,\lambda)$ of
the form (3.3) to system (3.2). Let $h\in\mathbb{R}$ be fixed. Then the
following representation holds for the quotient of these functions

$\sigma^{\pm}(x,\lambda):=\frac{u_{2}^{\pm}(x,\lambda)}{u_{1}^{\pm}(x,\lambda)}=\sum_{k=1}^{n}\frac{\sigma^{\pm}_{k}(x)}{(c\lambda)^{k}}+\frac{\sigma^{\pm}_{n}(x,\lambda)}{(c\lambda)^{n}},\quad\lambda\in\mathbb{C}\setminus\\{0\\},$
(3.67)

where $c=b^{+}-b^{-}=i(b_{1}-b_{2})$ $($see (3.25)$)$ and

$\displaystyle\sigma_{k}^{\pm}(x)=a_{k}^{\pm}(x)-\sum_{j=1}^{k-1}b_{j}^{\pm}(x)\sigma_{k-j}^{\pm}(x),\quad
x\in[0,1],\quad k\in\\{1,\ldots,{n}\\},$ (3.68)
$\displaystyle\sigma^{\pm}_{n}(0,\lambda)=0,\qquad\lambda\in\mathbb{C}\setminus\\{0\\},$
(3.69) $\displaystyle\sigma^{\pm}_{n}(x,\lambda)=o(1),\quad\lambda\to\infty,\
\operatorname{Im}\lambda\geqslant h,\ \ \text{uniformly in}\ \ x\in[0,1].$
(3.70)

###### Proof.

With account notation (3.25) formulas (3.4)–(3.5) take the form

$\displaystyle
u_{1}^{\pm}(x,\lambda)=\sum_{k=0}^{n}b_{k}^{\pm}(x)(c\lambda)^{-k}+b_{n}^{\pm}(x,\lambda)(c\lambda)^{-n},\qquad\lambda\neq
0,$ (3.71) $\displaystyle
u_{2}^{\pm}(x,\lambda)=\sum_{k=0}^{n}a_{k}^{\pm}(x)(c\lambda)^{-k}+a_{n}^{\pm}(x,\lambda)(c\lambda)^{-n},\qquad\lambda\neq
0.$ (3.72)

Lemma 3.2 implies asymptotic relations (3.57) for $a_{n}^{\pm}(x,\lambda)$ and
$b_{n}^{\pm}(x,\lambda)$. Relations (3.71)–(3.72), (3.24), (3.57) and
equalities $a_{0}^{\pm}(x)=0$, $b_{0}^{\pm}(x)=1$ (see (3.6)) imply that

$u_{1}^{\pm}(x,\lambda)=1+o(1),\quad
u_{2}^{\pm}(x,\lambda)=o(1)\quad\text{as}\quad\lambda\to\infty,\ \
\operatorname{Im}\lambda\geqslant h,$ (3.73)

uniformly in $x\in[0,1]$. Set

$\Theta_{h,R}:=\\{\lambda\in\mathbb{C}:\operatorname{Im}\lambda\geqslant h,\
|\lambda|>R\\}.$ (3.74)

The first relation in (3.73) yields

$u_{1}^{\pm}(x,\lambda)\neq 0,\qquad\lambda\in\Theta_{h,R},\quad x\in[0,1],$
(3.75)

for some $R>0$. Hence the function $\sigma^{\pm}(x,\lambda)$ is correctly
define on $[0,1]\times\Theta_{h,R}$.

Therefore, for each $x\in[0,1]$ Lemma 3.4 is applicable to functions
$u_{1}(\lambda)=u_{1}^{\pm}(x,\lambda)$ and
$u_{2}(\lambda)=u_{2}^{\pm}(x,\lambda)$ in the domain $D=\Theta_{h,R}$. In
turn, it implies formulas (3.68) for the coefficients in the expansion (3.67).

Next we establish uniform asymptotic relation
$\sigma^{\pm}_{n}(x,\lambda)=o(1)$ as $\lambda\to\infty$ and
$\operatorname{Im}\lambda\geqslant h$. Combining identity (3.68) with
inclusions (3.21) one can prove the following inclusions via induction:

$\sigma_{k}^{\pm}\in W_{1}^{n-k+1}[0,1]\subset C[0,1],\quad
k\in\\{1,\ldots,{n}\\}.$ (3.76)

Combining these inclusions with inclusions (3.24), uniform asymptotic
relations (3.57) and formula (3.65) for $\sigma_{n}^{\pm}(x,\lambda)$ implies
uniform estimate (3.70). Further, combining (3.8) with (3.9) and (3.10) yields

$\displaystyle b_{1}^{\pm}(0)=\ldots=b_{n}^{\pm}(0)=0,\qquad
a_{n}^{\pm}(0,\lambda)=b_{n}^{\pm}(0,\lambda)=0.$ (3.77)

Inserting these equalities in formula (3.65) for $\sigma_{n}^{\pm}(0,\lambda)$
trivially implies that $\sigma_{n}^{\pm}(0,\lambda)=0$. The proof is now
complete. ∎

######  Remark 3.6.

Let us calculate several first values of $\sigma_{k}^{\pm}(\cdot)$ using
formulas (3.68) and (3.6)–(3.8). For brevity we omit the argument:

$\sigma_{1}^{\pm}=a_{1}^{\pm}=q_{\mp}.$ (3.78)
$\sigma_{2}^{\pm}=a_{2}^{\pm}-b_{1}^{\pm}\sigma_{1}^{\pm}=q_{\mp}b_{1}^{\pm}-(a_{1}^{\pm})^{\prime}-b_{1}^{\pm}q_{\mp}=-q_{\mp}^{\prime}.$
(3.79)
$\sigma_{3}^{\pm}=a_{3}^{\pm}-b_{1}^{\pm}\sigma_{2}^{\pm}-b_{2}^{\pm}\sigma_{1}^{\pm}=q_{\mp}b_{2}^{\pm}-(a_{2}^{\pm})^{\prime}+b_{1}^{\pm}q_{\mp}^{\prime}-b_{2}^{\pm}q_{\mp}\\\
=-(q_{\mp}b_{1}^{\pm}-(a_{1}^{\pm})^{\prime})^{\prime}+b_{1}^{\pm}q_{\mp}^{\prime}=q_{\mp}^{\prime\prime}-q_{\mp}(b_{1}^{\pm})^{\prime}=q_{\mp}^{\prime\prime}-q_{\mp}q_{\pm}a_{1}^{\pm}=q_{\mp}^{\prime\prime}-q_{\mp}^{2}q_{\pm}.$
(3.80)

Surprisingly, despite the fact that formulas for $b_{k}^{\pm}(\cdot)$ contain
integration operations, the final formula for $\sigma_{k}^{\pm}(\cdot)$ is
only expressed via functions $q_{\pm}$ and their derivatives. In fact, this is
valid for all $k\in\\{1,\ldots,{n}\\}$. Direct proof of this fact via formulas
(3.68) and (3.6)–(3.8) is rather tedious and cumbersome. We will overcome this
difficulty in the next result which was announced in [14, Proposition 1], by
using Marchenko’s idea to establish equation of Ricatti type for the quotient
$\sigma^{\pm}(x,\lambda)$.

######  Proposition 3.7.

Under the assumptions of Proposition 3.5 the following explicit recurrent
formulas hold for the functions $\sigma_{k}^{\pm}(\cdot)$,

$\displaystyle\sigma^{\pm}_{1}(x)$ $\displaystyle=q_{\mp}(x),$ (3.81)
$\displaystyle\sigma^{\pm}_{k+1}(x)$
$\displaystyle=-(\sigma^{\pm}_{k}(x))^{\prime}-q_{\pm}(x)\cdot\sum_{j=1}^{k-1}\sigma^{\pm}_{j}(x)\sigma^{\pm}_{k-j}(x),$
(3.82)

for $k\in\\{1,\ldots,{n-1}\\}$ and $x\in[0,1]$.

###### Proof.

First we derive a differential equation on the function
$\sigma^{\pm}(x,\lambda)$ given by (3.67). From its definition we have

$u_{2}^{\pm}(x,\lambda)=u_{1}^{\pm}(x,\lambda)\sigma^{\pm}(x,\lambda).$ (3.83)

Substituting this into (3.18) we get

$(u_{1}^{\pm}(x,\lambda))^{\prime}=q_{\pm}(x)u_{1}^{\pm}(x,\lambda)\sigma^{\pm}(x,\lambda).$
(3.84)

Inserting (3.84) into (3.85) we arrive at the second equation

$(u_{1}^{\pm})^{\prime}\sigma^{\pm}+u_{1}^{\pm}(\sigma^{\pm})^{\prime}=q_{\mp}(x)u_{1}^{\pm}-c\lambda
u_{1}^{\pm}\sigma^{\pm}.$ (3.85)

Excluding the derivative $(u_{1}^{\pm})^{\prime}$ from system (3.84)–(3.85) we
arrive at the following relation

$q_{\pm}(x)u_{1}^{\pm}(\sigma^{\pm})^{2}+u_{1}^{\pm}(\sigma^{\pm})^{\prime}=q_{\mp}(x)u_{1}^{\pm}-c\lambda
u_{1}^{\pm}\sigma^{\pm}.$ (3.86)

Taking into account that $u_{1}^{\pm}(x,\lambda)\neq 0$ on
$[0,1]\times\Theta_{h,R}$ (see (3.75)) and dividing equality (3.86) by
$u_{1}^{\pm}(x,\lambda)$ we arrive at the following non-linear differential
equation (of Ricatti type) on $\sigma^{\pm}(t,\lambda)$,

$(\sigma^{\pm}(x,\lambda))^{\prime}+c\lambda\sigma^{\pm}(x,\lambda)+q_{\pm}(x)(\sigma^{\pm}(x,\lambda))^{2}=q_{\mp}(x),\quad
x\in[0,1],\quad\lambda\in\Theta_{h,R}.$ (3.87)

To avoid cumbersome expression for derivate
$(\sigma_{n}^{\pm}(x,\lambda))^{\prime}$ when inserting expression (3.67) for
$\sigma^{\pm}(x,\lambda)$ into equation (3.87) we integrate this equation
first (after replacing $x$ with $t$) for $t\in[0,x]$:

$\sigma^{\pm}(x,\lambda)-\sigma^{\pm}(0,\lambda)+c\lambda\int_{0}^{x}\sigma^{\pm}(t,\lambda)dt+\int_{0}^{x}q_{\pm}(t)(\sigma^{\pm}(t,\lambda))^{2}dt\\\
=\int_{0}^{x}q_{\mp}(t)dt,\quad x\in[0,1],\quad\lambda\in\Theta_{h,R}.$ (3.88)

Now substituting r.h.s. of expansion (3.67) into (3.88) implies the following
equation

$c^{\pm}_{0}(x)+\frac{c^{\pm}_{1}(x)}{c\lambda}+\ldots+\frac{c^{\pm}_{n-1}(x)}{(c\lambda)^{n-1}}+\frac{c^{\pm}_{n}(x,\lambda)}{(c\lambda)^{n}}=0,\quad
x\in[0,1],\quad\lambda\in\Theta_{h,R},$ (3.89)

where

$\displaystyle
c_{0}^{\pm}(x):=\int_{0}^{x}(\sigma_{1}^{\pm}(t)-q_{\mp}(t))dt,$ (3.90)
$\displaystyle
c_{k}^{\pm}(x):=\int_{0}^{x}\sigma_{k+1}^{\pm}(t)dt+\sigma_{k}^{\pm}(x)-\sigma_{k}^{\pm}(0)+\sum_{j=1}^{k-1}\int_{0}^{x}q_{\pm}(t)\sigma_{j}^{\pm}(t)\sigma_{k-j}^{\pm}(t)dt,$
(3.91)

for $k\in\\{1,\ldots,{n-1}\\}$ and

$c_{n}^{\pm}(x,\lambda):=c\lambda\int_{0}^{x}\sigma_{n}^{\pm}(t,\lambda)dt+\sigma_{n}^{\pm}(x,\lambda)-\sigma_{n}^{\pm}(0,\lambda)\\\
+\sigma_{n}^{\pm}(x)-\sigma_{n}^{\pm}(0)+\sum_{j=1}^{n-1}\int_{0}^{x}q_{\pm}(t)\sigma_{j}^{\pm}(t)\sigma_{n-j}^{\pm}(t)dt\\\
+\sum_{k=1}^{n}(c\lambda)^{-k}\sum_{j=k}^{n}\int_{0}^{x}q_{\pm}(t)\sigma_{j}^{\pm}(t)\sigma_{n+k-j}^{\pm}(t)dt\\\
+\sum_{k=1}^{n}2(c\lambda)^{-k}\int_{0}^{x}q_{\pm}(t)\sigma_{k}^{\pm}(t)\sigma_{n}^{\pm}(t,\lambda)dt+(c\lambda)^{-n}\int_{0}^{x}q_{\pm}(t)(\sigma_{n}^{\pm}(t,\lambda))^{2}dt.$
(3.92)

To estimate the remainder term in (3.89) we establish the following uniform
asymptotic relation:

$c_{n}^{\pm}(x,\lambda)=o(\lambda),\quad\lambda\to\infty,\ \
\operatorname{Im}\lambda\geqslant h,\quad\text{uniformly in}\ \ x\in[0,1].$
(3.93)

With account of boundedness of coefficients $\sigma_{k}^{\pm}$ and $q_{\pm}$
(see (3.76)), asymptotic relation (3.70) yields the following uniform in
$x\in[0,1]$ relations as $\lambda\to\infty$ and
$\operatorname{Im}\lambda\geqslant h$:

$\displaystyle c\lambda\int_{0}^{x}\sigma_{n}^{\pm}(t,\lambda)dt$
$\displaystyle=$ $\displaystyle o(\lambda),$ (3.94)
$\displaystyle\sigma_{n}^{\pm}(x,\lambda)-\sigma_{n}^{\pm}(0,\lambda)$
$\displaystyle=$ $\displaystyle o(1),$ (3.95)
$\displaystyle\sigma_{n}^{\pm}(x)-\sigma_{n}^{\pm}(0)+\sum_{j=1}^{n-1}\int_{0}^{x}q_{\pm}(t)\sigma_{j}^{\pm}(t)\sigma_{n-j}^{\pm}(t)dt$
$\displaystyle=$ $\displaystyle O(1),$ (3.96)
$\displaystyle\sum_{k=1}^{n}(c\lambda)^{-k}\sum_{j=k}^{n}\int_{0}^{x}q_{\pm}(t)\sigma_{j}^{\pm}(t)\sigma_{n+k-j}^{\pm}(t)dt$
$\displaystyle=$ $\displaystyle O(\lambda^{-1}),$ (3.97)
$\displaystyle\sum_{k=1}^{n}2(c\lambda)^{-k}\int_{0}^{x}q_{\pm}(t)\sigma_{k}^{\pm}(t)\sigma_{n}^{\pm}(t,\lambda)dt$
$\displaystyle=$ $\displaystyle o(\lambda^{-1}),$ (3.98)
$\displaystyle(c\lambda)^{-n}\int_{0}^{x}q_{\pm}(t)(\sigma_{n}^{\pm}(t,\lambda))^{2}dt$
$\displaystyle=$ $\displaystyle o(\lambda^{-n}).$ (3.99)

Inserting asymptotic estimates (3.94)–(3.99) into equality (3.92) we arrive at
the desired asymptotic relation (3.93).

Now fixing $x=x_{0}\in[0,1]$ in (3.89), multiplying obtained equality by
$(c\lambda)^{n-1}$ and tending $\lambda$ to infinity with account of
asymptotic relation (3.93) implies the following equalities,

$c_{k}^{\pm}(x)=0,\quad x\in[0,1],\quad k\in\\{1,\ldots,{n}\\}.$ (3.100)

Differentiating equalities (3.100) with account of definitions (3.90) and
(3.91) and smoothness of coefficients $\sigma_{k}^{\pm}$, we arrive at the
desired relations (3.81) and (3.82), which completes the proof. ∎

######  Remark 3.8.

(i) Note that recurrent formulas (3.82) for coefficients $\sigma_{k}^{\pm}$
provide an explicit form of local polynomial integrals of motion for non-
linear Shrödinger equation $($see [34]$)$.

(ii) Formula (3.82) allows to easily and explicitly calculate values of
$\sigma_{k}^{\pm}(\cdot)$ via entries $q_{\pm}(\cdot)$ of the potential matrix
$Q(\cdot)$, which would be quite tedious to do via formulas (3.68) and
(3.6)–(3.8). For instance,

$\displaystyle\sigma_{4}^{\pm}=-(\sigma_{3}^{\pm})^{\prime}-2q_{\mp}\sigma_{1}^{\pm}\sigma_{2}^{\pm}=-q_{\mp}^{\prime\prime\prime}+q_{\mp}^{2}q_{\pm}^{\prime}+4q_{\mp}^{\prime}q_{\mp}q_{\pm},$
(3.101)
$\displaystyle\sigma_{5}^{\pm}=q_{\mp}^{({4})}-6q_{\mp}^{\prime\prime}q_{\mp}q_{\pm}-5q_{\mp}^{\prime}q_{\mp}^{\prime}q_{\pm}-6q_{\mp}^{\prime}q_{\mp}q_{\pm}^{\prime}-q_{\mp}^{2}q_{\pm}^{\prime\prime}+2q_{\mp}^{3}q_{\pm}^{2},$
(3.102)
$\displaystyle\sigma_{6}^{\pm}=-q_{\mp}^{(5)}+8q_{\mp}^{\prime\prime\prime}q_{\mp}q_{\pm}+18q_{\mp}^{\prime\prime}q_{\mp}^{\prime}q_{\pm}+12q_{\mp}^{\prime\prime}q_{\mp}q_{\pm}^{\prime}+11q_{\mp}^{\prime}q_{\mp}^{\prime}q_{\pm}^{\prime}$
$\displaystyle\qquad\qquad+8q_{\mp}^{\prime}q_{\mp}q_{\pm}^{\prime\prime}+q_{\mp}^{2}q_{\pm}^{\prime\prime\prime}-16q_{\mp}^{\prime}q_{\mp}^{2}q_{\pm}^{2}-6q_{\mp}^{3}q_{\pm}^{\prime}q_{\pm}.$
(3.103)

Moreover, one can obtain explicit expression for $\sigma_{k}^{\pm}$ as a
linear combination of products of the functions $q_{\pm}$ and their
derivatives in general case $($see Lemma 6.1 below$)$.

######  Remark 3.9.

Asymptotic expansions of solutions to the system (1.1) obtained in this
section play important role in establishing refined asymptotic formulas for
eigenvalues and eigenfunctions of the corresponding BVP. In this connection,
we mention that recently L. Rzepnicki [35] obtained sharp asymptotic formulas
for deviations $\lambda_{m}-\lambda_{m}^{0}=\delta_{m}+\rho_{m}$ in the case
of Dirichlet BVP for $2\times 2$ Dirac system with $Q\in
L^{p}([0,1];\mathbb{C}^{2\times 2})$, $1\leqslant p<2$. Namely, $\delta_{m}$
is explicitly expressed via Fourier coefficients and Fourier transforms of
$Q_{12}$ and $Q_{21}$, while
$\\{\rho_{m}\\}_{m\in\mathbb{Z}}\in\ell^{p^{\prime}/2}(\mathbb{Z})$ where
$p^{\prime}$ is the conjugate exponent. Similar result was obtained for
eigenvectors. For Sturm-Liouville operators with singular potentials, A.
Gomilko and L. Rzepnicki obtained similar results in another recent paper [7].

##  4 Asymptotic expansion of the characteristic determinant

In this section we apply Proposition 3.7 to obtain asymptotic expansion of the
characteristic determinant $\Delta_{Q}(\cdot)$ given by (1.8) with explicit,
easy to calculate coefficients.

Recall, that $\Phi(x,\lambda)=(\varphi_{jk}(x,\lambda))_{j,k=1}^{2}$ is the
fundamental matrix solution to system (1.1) (uniquely) determined by the
initial condition $\Phi(0,\lambda)=I_{2}$ (see (2.4)). It is evident, that the
eigenvalues of the problem (1.1)–(1.4) are the roots of the characteristic
equation $\Delta_{Q}(\lambda):=\det U(\lambda)=0$, where

$U(\lambda):=\begin{pmatrix}U_{1}(\Phi_{1}(x,\lambda))&U_{1}(\Phi_{2}(x,\lambda))\\\
U_{2}(\Phi_{1}(x,\lambda))&U_{2}(\Phi_{2}(x,\lambda))\end{pmatrix}=:\begin{pmatrix}u_{11}(\lambda)&u_{12}(\lambda)\\\
u_{21}(\lambda)&u_{22}(\lambda)\end{pmatrix}.$ (4.1)

With account of notations (1.5) we arrive at the following expression for the
characteristic determinant:

$\Delta_{Q}(\lambda)=J_{12}+J_{34}e^{i(b_{1}+b_{2})\lambda}+J_{32}\varphi_{11}(\lambda)+J_{13}\varphi_{12}(\lambda)+J_{42}\varphi_{21}(\lambda)+J_{14}\varphi_{22}(\lambda),$
(4.2)

where $\varphi_{jk}(\lambda):=\varphi_{jk}(1,\lambda)$.

######  Theorem 4.1.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ and functions
$\sigma^{\pm}_{k}(x),k\in\\{1,\ldots,{n}\\}$, be defined by the formulas
(3.1), (3.81), (3.82). Then the characteristic determinant
$\Delta_{Q}(\lambda)$ of the system (1.1) admits the following representations

$\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=e^{ib_{1}\lambda}\cdot\Bigl{(}J_{32}+\sum_{k=1}^{n}\frac{c_{k}^{+}}{(c\lambda)^{k}}+o(\lambda^{-n})\Bigr{)}\cdot(1+o(1)),\qquad\operatorname{Im}\lambda\to+\infty,$
(4.3) $\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=e^{ib_{2}\lambda}\cdot\Bigl{(}J_{14}-\sum_{k=1}^{n}\frac{c_{k}^{-}}{(c\lambda)^{k}}+o(\lambda^{-n})\Bigr{)}\cdot(1+o(1)),\qquad\operatorname{Im}\lambda\to-\infty,$
(4.4)

where $c=i(b_{1}-b_{2})$, and $c_{k}^{\pm}$ for $k\in\\{1,\ldots,{n}\\}$ are
given by

$\displaystyle c_{k}^{+}$
$\displaystyle:=J_{13}(-1)^{k-1}\sigma_{k}^{-}(0)+J_{42}\sigma_{k}^{+}(1)-J_{14}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(0)\sigma_{k-j}^{+}(1),$
(4.5) $\displaystyle c_{k}^{-}$
$\displaystyle:=J_{13}(-1)^{k-1}\sigma_{k}^{-}(1)+J_{42}\sigma_{k}^{+}(0)+J_{32}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(1)\sigma_{k-j}^{+}(0).$
(4.6)

###### Proof.

According to Theorem 3.1 there exists a matrix solution to system (1.1) of the
form (3.3), in which functions $u_{1}^{\pm}(x,\pm\lambda)$ and
$u_{2}^{\pm}(x,\pm\lambda)$ admit expansions (3.4) and (3.5), respectively.
Combining conditions (3.6), (3.77) on one hand and conditions (3.83), (3.67),
(3.69) on the other hand we obtain

$u_{1}^{\pm}(0,\pm\lambda)=1,\qquad
u_{2}^{\pm}(0,\pm\lambda)=\sigma^{\pm}(0,\pm\lambda)=o(1),\quad\lambda\to\infty.$
(4.7)

Inserting these expressions into (3.3) we get

$Y(0,\lambda)=\begin{pmatrix}1&\sigma^{-}(0,-\lambda)\\\
\sigma^{+}(0,\lambda)&1\end{pmatrix}=I_{2}+o_{2}(1),\quad\lambda\to\infty,$
(4.8)

where $o_{2}(1)$ is a $2\times 2$ with $o(1)$ entries. Hence

$\omega(\lambda):=\det(Y(0,\lambda))=1-\sigma^{-}(0,-\lambda)\sigma^{+}(0,\lambda)=1+o(1),\quad\lambda\to\infty.$
(4.9)

Therefore, for some $R>0$ we have $\det(Y(0,\lambda))\neq 0,\ |\lambda|>R$ and
hence $Y(x,\lambda)$ is a fundamental system of solutions to the system (3.2)
for all $|\lambda|>R$. Hence $Y(x,\lambda)$ and $\Phi(x,\lambda)$ related by
the identity

$\Phi(x,\lambda)=Y(x,\lambda)[Y(0,\lambda)]^{-1},\quad|\lambda|>R.$ (4.10)

Putting (2.4), (3.3) and (4.8) into (4.10) we get for $|\lambda|>R$

$\displaystyle\varphi_{11}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{\omega(\lambda)}\left(e^{b^{+}\lambda}u_{1}^{+}(1,\lambda)-e^{b^{-}\lambda}u_{2}^{-}(1,-\lambda)\sigma^{+}(0,\lambda)\right),$
(4.11) $\displaystyle\varphi_{12}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{\omega(\lambda)}\left(-e^{b^{+}\lambda}u_{1}^{+}(1,\lambda)\sigma^{-}(0,-\lambda)+e^{b^{-}\lambda}u_{2}^{-}(1,-\lambda)\right),$
(4.12) $\displaystyle\varphi_{21}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{\omega(\lambda)}\left(e^{b^{+}\lambda}u_{2}^{+}(1,\lambda)-e^{b^{-}\lambda}u_{1}^{-}(1,-\lambda)\sigma^{+}(0,\lambda)\right),$
(4.13) $\displaystyle\varphi_{22}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{\omega(\lambda)}\left(-e^{b^{+}\lambda}u_{2}^{+}(1,\lambda)\sigma^{-}(0,-\lambda)+e^{b^{-}\lambda}u_{1}^{-}(1,-\lambda)\right).$
(4.14)

Inserting (4.11)–(4.14) into (4.2) we get in view of (3.1)

$\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=J_{12}+J_{34}e^{i(b_{1}+b_{2})\lambda}$
$\displaystyle+\frac{e^{ib_{1}\lambda}}{\omega(\lambda)}\left(J_{32}u_{1}^{+}(1,\lambda)-J_{13}u_{1}^{+}(1,\lambda)\sigma^{-}(0,-\lambda)\right.$
$\displaystyle\qquad\left.+J_{42}u_{2}^{+}(1,\lambda)-J_{14}u_{2}^{+}(1,\lambda)\sigma^{-}(0,-\lambda)\right)$
$\displaystyle+\frac{e^{ib_{2}\lambda}}{\omega(\lambda)}\left(-J_{32}u_{2}^{-}(1,-\lambda)\sigma^{+}(0,\lambda)+J_{13}u_{2}^{-}(1,-\lambda)\right.$
$\displaystyle\qquad\left.-J_{42}u_{1}^{-}(1,-\lambda)\sigma^{+}(0,\lambda)+J_{14}u_{1}^{-}(1,-\lambda)\right).$
(4.15)

First, consider the case $\operatorname{Im}\lambda\geqslant R$. In view of
(3.70) we may assume that for these values of $\lambda$ the function
$\sigma^{+}(1,\lambda)$ is correctly defined. Using (3.83), (3.67), (3.69),
(3.73) and (3.70) and taking into account notation (4.5) we rewrite the factor
in parentheses of the third summand in (4) as follows

$\displaystyle
J_{32}u_{1}^{+}(1,\lambda)-J_{13}u_{1}^{+}(1,\lambda)\sigma^{-}(0,-\lambda)+J_{42}u_{2}^{+}(1,\lambda)-J_{14}u_{2}^{+}(1,\lambda)\sigma^{-}(0,-\lambda)$
$\displaystyle\quad=u_{1}^{+}(1,\lambda)(J_{32}-J_{13}\sigma^{-}(0,-\lambda)+J_{42}\sigma^{+}(1,\lambda)-J_{14}\sigma^{+}(1,\lambda)\sigma^{-}(0,-\lambda))$
$\displaystyle\quad=(1+o(1))\cdot\Bigl{(}J_{32}-J_{13}\sum_{k=1}^{n}\frac{\sigma_{k}^{-}(0)}{(-c\lambda)^{k}}+J_{42}\Bigl{(}\sum_{k=1}^{n}\frac{\sigma_{k}^{+}(1)}{(c\lambda)^{k}}+\frac{\sigma_{n}^{+}(1,\lambda)}{(c\lambda)^{n}}\Bigr{)}\Bigr{.}$
$\displaystyle\qquad\qquad\Bigl{.}-J_{14}\Bigl{(}\sum_{k=1}^{n}\frac{\sigma_{k}^{-}(0)}{(-c\lambda)^{k}}\Bigr{)}\Bigl{(}\sum_{k=1}^{n}\frac{\sigma_{k}^{+}(1)}{(c\lambda)^{k}}+\frac{\sigma_{n}^{+}(1,\lambda)}{(c\lambda)^{n}}\Bigr{)}\Bigr{)}$
$\displaystyle\quad=(1+o(1))\cdot\Bigl{(}J_{32}+\sum_{k=1}^{n}(c\lambda)^{-k}\Bigl{(}-J_{13}(-1)^{k}\sigma_{k}^{-}(0)+J_{42}\sigma_{k}^{+}(1)\Bigr{.}\Bigr{.}$
$\displaystyle\qquad\qquad\Bigl{.}\Bigl{.}-J_{14}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(0)\sigma_{k-j}^{+}(1)\Bigr{)}+o(\lambda^{-n})\Bigr{)}$
$\displaystyle\quad=(1+o(1))\cdot\Bigl{(}J_{32}+\sum_{k=1}^{n}\frac{c_{k}^{+}}{(c\lambda)^{k}}+o(\lambda^{-n})\Bigr{)},\qquad\operatorname{Im}\lambda\to+\infty,\quad\operatorname{Im}\lambda\geqslant
R.$ (4.16)

Similar to (3.58) by combining (3.9) with inclusion $A_{n}^{\pm}\in
X_{\infty}^{0}$, Lemma 2.1 and inequality $b_{2}-b_{1}>0$ we derive

$|a_{n}^{\pm}(x,-\lambda)|=o(e^{(b_{2}-b_{1})\operatorname{Im}\lambda}),\qquad\operatorname{Im}\lambda\to+\infty,\quad\operatorname{Im}\lambda\geqslant
R,$ (4.17)

uniformly in $x\in[0,1]$. Combining (3.59) with (4.17) we have

$|b_{n}^{-}(1,-\lambda)|=o(e^{(b_{2}-b_{1})\operatorname{Im}\lambda}),\qquad\operatorname{Im}\lambda\to+\infty,\quad\operatorname{Im}\lambda\geqslant
R.$ (4.18)

Now using (3.4), (3.5), (4.7), (4.9), (4.17) and (4.18) we can estimate the
last summand at (4)

$\displaystyle\left|{\frac{e^{ib_{2}\lambda}}{\omega(\lambda)}\left(u_{2}^{-}(1,-\lambda)(J_{32}\sigma^{+}(0,\lambda)-J_{13})+u_{1}^{-}(1,-\lambda)(J_{42}\sigma^{+}(0,\lambda)-J_{14})\right)}\right|$
$\displaystyle\quad=e^{-b_{2}\operatorname{Im}\lambda}\cdot(1+o(1))\cdot\left|\left(o(1)+\frac{e^{(b_{2}-b_{1})\operatorname{Im}\lambda}}{\lambda^{n}}o(1)\right)\cdot(o(1)-J_{13})\right.$
$\displaystyle\quad\quad\left.+\left(1+o(1)+\frac{e^{(b_{2}-b_{1})\operatorname{Im}\lambda}}{\lambda^{n}}o(1)\right)\cdot(o(1)-J_{14})\right|=\frac{e^{-b_{1}\operatorname{Im}\lambda}}{|\lambda|^{n}}o(1),$
(4.19)

as $\operatorname{Im}\lambda\to+\infty$ and $\operatorname{Im}\lambda\geqslant
R$. Since $b_{1}<0<b_{2}$ then

$\left|{J_{12}+J_{34}e^{i(b_{1}+b_{2})\lambda}}\right|=\frac{e^{-b_{1}\operatorname{Im}\lambda}}{|\lambda|^{n}}o(1),\qquad\operatorname{Im}\lambda\to+\infty,\quad\operatorname{Im}\lambda\geqslant
R.$ (4.20)

Putting (4), (4) and (4.20) into (4) we arrive at (4.3).

Now consider the case when $\operatorname{Im}\lambda\leqslant-R$. Relation
(4.4) can be obtained similarly. We reproduce only the step corresponding to
the equation (4). For such $\lambda$ the function $\sigma^{-}(1,-\lambda)$ is
correctly defined and so we have

$-J_{32}u_{2}^{-}(1,-\lambda)\sigma^{+}(0,\lambda)+J_{13}u_{2}^{-}(1,-\lambda)\\\
-J_{42}u_{1}^{-}(1,-\lambda)\sigma^{+}(0,\lambda)+J_{14}u_{1}^{-}(1,-\lambda)\\\
=u_{1}^{-}(1,-\lambda)(J_{14}+J_{13}\sigma^{-}(1,-\lambda)-J_{42}\sigma^{+}(0,\lambda)-J_{32}\sigma^{-}(1,-\lambda)\sigma^{+}(0,\lambda))\\\
=(1+o(1))\cdot\Bigl{(}J_{14}+J_{13}\Bigl{(}\sum_{k=1}^{n}\frac{\sigma_{k}^{-}(1)}{(-c\lambda)^{k}}+\frac{\sigma_{n}^{-}(1,-\lambda)}{(-c\lambda)^{n}}\Bigr{)}-J_{42}\sum_{k=1}^{n}\frac{\sigma_{k}^{+}(0)}{(c\lambda)^{k}}\Bigr{.}\\\
\Bigl{.}-J_{32}\Bigl{(}\sum_{k=1}^{n}\frac{\sigma_{k}^{-}(1)}{(-c\lambda)^{k}}+\frac{\sigma_{n}^{-}(1,-\lambda)}{(-c\lambda)^{n}}\Bigr{)}\Bigl{(}\sum_{k=1}^{n}\frac{\sigma_{k}^{+}(0)}{(c\lambda)^{k}}\Bigr{)}\Bigr{)}\\\
=(1+o(1))\cdot\Bigl{(}J_{14}+\sum_{k=1}^{n}(c\lambda)^{-k}\Bigl{(}J_{13}(-1)^{k}\sigma_{k}^{-}(1)-J_{42}\sigma_{k}^{+}(0)\Bigr{.}\Bigr{.}\\\
\Bigl{.}\Bigl{.}-J_{32}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(1)\sigma_{k-j}^{+}(0)\Bigr{)}+o(\lambda^{-n})\Bigr{)}\\\
=(1+o(1))\cdot\Bigl{(}J_{14}-\sum_{k=1}^{n}\frac{c_{k}^{-}}{(c\lambda)^{k}}+o(\lambda^{-n})\Bigr{)}$
(4.21)

as $\operatorname{Im}\lambda\to-\infty$ and
$\operatorname{Im}\lambda\leqslant-R$. The proof is now complete. ∎

######  Remark 4.2.

Let us demonstrate alternative method of obtaining formulas (4.3)–(4.6)
originally proposed in [29] for Sturm-Liouville operators. It is based on
integral representation of the characteristic determinant obtained for
$2\times 2$ Dirac type system with integrable potential matrix in [18] $($see
(1.10)$)$:

$\displaystyle\Delta_{Q}(\lambda)$
$\displaystyle=\Delta_{0}(\lambda)+\int^{1}_{0}g_{1}(t)e^{ib_{1}\lambda
t}dt+\int^{1}_{0}g_{2}(t)e^{ib_{2}\lambda t}dt,$ (4.22)

where $g_{1},g_{2}\in L^{1}[0,1]$ are expressed via the kernel of the
transformation operator.

Assuming that $g_{1},g_{2}\in W_{1}^{n}[0,1]$ and integrating by parts, one
derives

$\int^{1}_{0}g_{1}(t)e^{ib_{1}\lambda
t}dt=\left[g_{1}(t)\frac{e^{ib_{1}\lambda
t}}{ib_{1}\lambda}\right]_{0}^{1}-\frac{1}{ib_{1}\lambda}\int^{1}_{0}g_{1}^{\prime}(t)e^{ib_{1}\lambda
t}dt\\\
=\sum_{k=1}^{n}\frac{g_{1}^{(k-1)}(0)-g_{1}^{(k-1)}(1)e^{ib_{1}\lambda}}{(-ib_{1}\lambda)^{k}}+\frac{1}{(-ib_{1}\lambda)^{n}}\int^{1}_{0}g_{1}^{(n)}(t)e^{ib_{1}\lambda
t}dt.$ (4.23)

Inserting this formula into representation (4.22) yields

$\Delta_{Q}(\lambda)=\Delta_{0}(\lambda)+\sum_{k=1}^{n}\frac{g_{1}^{(k-1)}(0)-g_{1}^{(k-1)}(1)e^{ib_{1}\lambda}}{(-ib_{1}\lambda)^{k}}+\frac{1}{(-ib_{1}\lambda)^{n}}\int^{1}_{0}g_{1}^{(n)}(t)e^{ib_{1}\lambda
t}dt\\\
+\sum_{k=1}^{n}\frac{g_{2}^{(k-1)}(0)-g_{2}^{(k-1)}(1)e^{ib_{2}\lambda}}{(-ib_{2}\lambda)^{k}}+\frac{1}{(-ib_{2}\lambda)^{n}}\int^{1}_{0}g_{2}^{(n)}(t)e^{ib_{2}\lambda
t}dt,\quad\lambda\neq 0.$ (4.24)

In turn, combining this formula with (1.9) for $\Delta_{0}(\lambda)$, noting
that $b_{1}<0<b_{2}$ and applying the Riemann-Lebesgue Lemma implies desired
asymptotic expansion (1.11)–(1.12). Comparing (4.3)–(4.4) and (1.11)–(1.12) we
see that

$c_{k}^{+}=-\left(\frac{b_{2}-b_{1}}{b_{1}}\right)^{k}g_{1}^{(k-1)}(1),\qquad
c_{k}^{-}=\left(\frac{b_{2}-b_{1}}{b_{2}}\right)^{k}g_{2}^{(k-1)}(1),$ (4.25)

for $k\in\\{1,\ldots,{n}\\}$. Note, however, that finding explicit form of the
values $g_{1}^{(k-1)}(1)$ and $g_{2}^{(k-1)}(1)$ directly using transformation
operators is rather difficult. This is the reason why in this paper $($as well
as in [14]$)$ we use approach above to compute coefficients $c_{k}^{\pm}$ by
evolving Marchenko’s method from [33].

Let us compute values $c_{k}^{\pm}$ (defined in (4.5)–(4.6)) for small values
of $k$ explicitly via functions $q_{\pm}$. Recall that

$q_{+}(x)=-ib_{1}Q_{12}(x),\qquad q_{-}(x)=-ib_{2}Q_{21}(x),\qquad x\in[0,1].$
(4.26)

######  Lemma 4.3.

(i) Let $Q_{12},Q_{21}\in W_{1}^{1}[0,1]$. Then

$\displaystyle
c_{1}^{+}=J_{13}q_{+}(0)+J_{42}q_{-}(1)=-i\bigr{(}J_{13}b_{1}Q_{12}(0)+J_{42}b_{2}Q_{21}(1)\bigl{)},$
(4.27) $\displaystyle
c_{1}^{-}=J_{13}q_{+}(1)+J_{42}q_{-}(0)=-i\bigr{(}J_{13}b_{1}Q_{12}(1)+J_{42}b_{2}Q_{21}(0)\bigl{)}.$
(4.28)

(ii) Let $Q_{12},Q_{21}\in W_{1}^{2}[0,1]$. Then

$\displaystyle
c_{2}^{+}=J_{13}q_{+}^{\prime}(0)-J_{42}q_{-}^{\prime}(1)-J_{14}q_{+}(0)q_{-}(1),$
(4.29) $\displaystyle
c_{2}^{-}=J_{13}q_{+}^{\prime}(1)-J_{42}q_{-}^{\prime}(0)+J_{32}q_{+}(1)q_{-}(0).$
(4.30)

(iii) Let $Q_{12},Q_{21}\in W_{1}^{3}[0,1]$. Then

$c_{3}^{+}=J_{13}\bigl{(}q_{+}^{\prime\prime}(0)-q_{+}^{2}(0)q_{-}(0)\bigr{)}+J_{42}\bigl{(}q_{-}^{\prime\prime}(1)-q_{-}^{2}(1)q_{+}(1)\bigr{)}\\\
+J_{14}\bigl{(}q_{+}^{\prime}(0)q_{-}(1)-q_{+}(0)q_{-}^{\prime}(1)\bigr{)},$
(4.31)
$c_{3}^{-}=J_{13}\bigl{(}q_{+}^{\prime\prime}(1)-q_{+}^{2}(1)q_{-}(1)\bigr{)}+J_{42}\bigl{(}q_{-}^{\prime\prime}(0)-q_{-}^{2}(0)q_{+}(0)\bigr{)}\\\
-J_{32}\bigl{(}q_{+}^{\prime}(1)q_{-}(0)-q_{+}(1)q_{-}^{\prime}(0)\bigr{)}.$
(4.32)

###### Proof.

(i) Since $q_{\pm}\in W_{1}^{1}[0,1]$ then $\sigma_{1}^{\pm}=q_{\mp}$ (see
(3.81)). Hence it follows from (4.5)–(4.6) that

$\displaystyle
c_{1}^{+}=J_{13}\sigma_{1}^{-}(0)+J_{42}\sigma_{1}^{+}(1)=J_{13}q_{+}(0)+J_{42}q_{-}(1),$
(4.33) $\displaystyle
c_{1}^{-}=J_{13}\sigma_{1}^{-}(1)+J_{42}\sigma_{1}^{+}(0)=J_{13}q_{+}(1)+J_{42}q_{-}(0).$
(4.34)

(ii) Since $q_{\pm}\in W_{1}^{2}[0,1]$ then
$\sigma_{2}^{\pm}=-q_{\mp}^{\prime}$ (see (3.79)). Formulas (4.5)–(4.6) for
$k=2$ have the form

$\displaystyle
c_{2}^{+}=-J_{13}\sigma_{2}^{-}(0)+J_{42}\sigma_{2}^{+}(1)-J_{14}\sigma_{1}^{-}(0)\sigma_{1}^{+}(1),$
(4.35) $\displaystyle
c_{2}^{-}=-J_{13}\sigma_{2}^{-}(1)+J_{42}\sigma_{2}^{+}(0)+J_{32}\sigma_{1}^{-}(1)\sigma_{1}^{+}(0).$
(4.36)

Inserting formulas $\sigma_{1}^{\pm}=q_{\mp}$ and
$\sigma_{2}^{\pm}=-q_{\mp}^{\prime}$ into (4.35)–(4.36) we arrive at
(4.29)–(4.30).

(iii) Since $q_{\pm}\in W_{1}^{3}[0,1]$ then formulas (3.78)–(3.80) imply that

$\sigma_{1}^{\pm}=q_{\mp},\qquad\sigma_{2}^{\pm}=-q_{\mp}^{\prime},\qquad\sigma_{3}^{\pm}=q_{\mp}^{\prime\prime}-q_{\mp}^{2}q_{\pm}.$
(4.37)

Further, formulas (4.5)–(4.6) for $k=3$ have the form

$\displaystyle
c_{3}^{+}=J_{13}\sigma_{3}^{-}(0)+J_{42}\sigma_{3}^{+}(1)+J_{14}\sigma_{1}^{-}(0)\sigma_{2}^{+}(1)-J_{14}\sigma_{2}^{-}(0)\sigma_{1}^{+}(1),$
(4.38) $\displaystyle
c_{3}^{-}=J_{13}\sigma_{3}^{-}(1)+J_{42}\sigma_{3}^{+}(0)-J_{32}\sigma_{1}^{-}(1)\sigma_{2}^{+}(0)+J_{14}\sigma_{2}^{-}(1)\sigma_{1}^{+}(0).$
(4.39)

Inserting (4.37) into (4.38)–(4.39) we arrive at (4.31)–(4.32). ∎

##  5 General completeness results for non-regular BVP

In this section we apply asymptotic expansion of the characteristic
determinant from Theorem 4.1 to obtain a new general explicit completeness
result (Theorem 5.1) in the case of non-regular boundary conditions that
substantially supplement [32, Theorem 5.1] and [17, Proposition 4.5]. We then
apply Theorem 5.1 to obtain more explicit completeness results when some of
the determinants $J_{jk}$ vanish. We conclude this section with an explicit
criterion of completeness in the case of analytic potential $Q$ and degenerate
boundary conditions of a special form.

The following statement is our first result on completeness.

######  Theorem 5.1 (Theorem 3 in [14]).

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$, and let the
numbers $c_{1}^{\pm},\ldots,c_{n}^{\pm}$ be defined via (4.5)–(4.6). Assume
that

$|J_{32}|+|c_{1}^{+}|+\ldots+|c_{n}^{+}|\neq
0,\qquad|J_{14}|+|c_{1}^{-}|+\ldots+|c_{n}^{-}|\neq 0.$ (5.1)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

Set $c_{0}^{+}:=J_{32}$, and $c_{0}^{-}:=J_{14}$. Condition (5.1) implies
existence of $k^{+},k^{-}\in\\{0,1,\ldots,n\\}$ such that

$\displaystyle c_{k^{+}}^{+}\neq 0,\qquad c_{k}^{+}=0,\quad
k\in\\{0,\ldots,k^{+}-1\\},$ (5.2) $\displaystyle c_{k^{-}}^{-}\neq 0,\qquad
c_{k}^{-}=0,\quad k\in\\{0,\ldots,k^{-}-1\\}.$ (5.3)

Combining relations (5.2)–(5.3) with Theorem 4.1 (see formulas (4.3)–(4.6))
implies that

$|\Delta_{Q}(it)|\geqslant\frac{Ce^{-b_{1}t}}{t^{k^{+}}},\qquad|\Delta_{Q}(-it)|\geqslant\frac{Ce^{b_{2}t}}{t^{k^{-}}},\qquad
t>R,$ (5.4)

for some $C,R>0$. Hence due to Theorem 2.5 the system of root vectors of the
BVP (1.1)–(1.4) is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$. ∎

######  Remark 5.2.

Note that formulas (3.81)–(3.82) and (4.5)–(4.6) imply that coefficients
$c^{\pm}_{k}$ are polynomials in $b_{1}$, $b_{2}$, $J_{32}$, $J_{14}$,
$J_{42}$, $J_{13}$, $Q_{12}^{(j)}(0)$, $Q_{21}^{(j)}(0)$, $Q_{12}^{(j)}(1)$,
$Q_{21}^{(j)}(1)$, $j\in\\{0,1,\ldots,k-1\\}$ $($see also Remark 3.8 and Lemma
4.3$)$.

Combining Theorem 5.1 with Lemma 4.3 leads to a series of more explicit
completeness results.

######  Corollary 5.3.

Let $Q_{12},Q_{21}\in W_{1}^{1}[0,1]$. Then the system of root vectors of the
BVP (1.1)–(1.4) is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$
whenever both of the following conditions hold:

$\displaystyle|J_{32}|+|J_{13}b_{1}Q_{12}(0)+J_{42}b_{2}Q_{21}(1)|\neq 0,$
(5.5) $\displaystyle|J_{14}|+|J_{13}b_{1}Q_{12}(1)+J_{42}b_{2}Q_{21}(0)|\neq
0.$ (5.6)

###### Proof.

Combining formulas (4.27)–(4.28) from Lemma 4.3 with conditions (5.5)–(5.6)
implies that

$|J_{32}|+|c_{1}^{+}|\neq 0,\qquad|J_{14}|+|c_{1}^{-}|\neq 0.$ (5.7)

Therefore, Theorem 5.1 implies the desired completeness property. ∎

Recall that the functions $q_{\pm}$ are given by (4.26).

######  Corollary 5.4.

Let $Q_{12},Q_{21}\in W_{1}^{2}[0,1]$. Then the system of root vectors of the
BVP (1.1)–(1.4) is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$
whenever both of the following conditions hold:

$\displaystyle|J_{32}|+|J_{13}q_{+}(0)+J_{42}q_{-}(1)|+|J_{13}q_{+}^{\prime}(0)-J_{42}q_{-}^{\prime}(1)-J_{14}q_{+}(0)q_{-}(1)|\neq
0,$ (5.8)
$\displaystyle|J_{14}|+|J_{13}q_{+}(1)+J_{42}q_{-}(0)|+|J_{13}q_{+}^{\prime}(1)-J_{42}q_{-}^{\prime}(0)+J_{32}q_{+}(1)q_{-}(0)|\neq
0.$ (5.9)

###### Proof.

Combining formulas (4.27)–(4.30) from with conditions (5.8)–(5.9) implies that

$|J_{32}|+|c_{1}^{+}|+|c_{2}^{+}|\neq
0,\qquad|J_{14}|+|c_{1}^{-}|+|c_{2}^{-}|\neq 0.$ (5.10)

As in the previous result, Theorem 5.1 completes the proof. ∎

Inserting notations (4.26) into formulas (5.8)–(5.9) one readily transforms
the formulation of Corollary 5.4 into formulation of Proposition 1.2 from the
introduction.

######  Corollary 5.5.

Let $Q_{12},Q_{21}\in W_{1}^{3}[0,1]$. Assume that either condition (5.8)
holds or

$J_{13}\bigl{(}q_{+}^{\prime\prime}(0)-q_{+}^{2}(0)q_{-}(0)\bigr{)}+J_{42}\bigl{(}q_{-}^{\prime\prime}(1)-q_{-}^{2}(1)q_{+}(1)\bigr{)}\\\
+J_{14}\bigl{(}q_{+}^{\prime}(0)q_{-}(1)-q_{+}(0)q_{-}^{\prime}(1)\bigr{)}\neq
0.$ (5.11)

Further, assume that either condition (5.9) holds or

$J_{13}\bigl{(}q_{+}^{\prime\prime}(1)-q_{+}^{2}(1)q_{-}(1)\bigr{)}+J_{42}\bigl{(}q_{-}^{\prime\prime}(0)-q_{-}^{2}(0)q_{+}(0)\bigr{)}\\\
-J_{32}\bigl{(}q_{+}^{\prime}(1)q_{-}(0)-q_{+}(1)q_{-}^{\prime}(0)\bigr{)}\neq
0.$ (5.12)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

Combining formulas (4.27)–(4.32) with conditions (5.11)–(5.12) implies that

$|J_{32}|+|c_{1}^{+}|+|c_{2}^{+}|+|c_{3}^{+}|\neq
0,\qquad|J_{14}|+|c_{1}^{-}|+|c_{2}^{-}|+|c_{3}^{-}|\neq 0,$ (5.13)

which coincides with condition (5.1) from Theorem 5.1. ∎

Next we compare Corollaries 5.3, 5.4 and 5.5 with a very interesting recent
completeness result from [26] by A.S. Makin, which reads as follows.

######  Theorem 5.6 (Theorem 1 in [26]).

Let $B=\operatorname{diag}(-1,1)$ and $Q_{12},Q_{21}\in L^{1}[0,1]$. Assume
that either $J_{32}\neq 0$ or the following conditions hold with some
$\rho_{1},\rho_{2}>0$:

$\displaystyle\lim_{h\to
0}\frac{1}{h^{\rho_{1}}}\int_{0}^{h}Q_{12}(x)\,dx=\widetilde{Q}_{12}^{0}\neq
0,\qquad\lim_{h\to
0}\frac{1}{h^{\rho_{2}}}\int_{1-h}^{1}Q_{21}(x)\,dx=\widetilde{Q}_{21}^{1}\neq
0,$ (5.14)
$\displaystyle\qquad\qquad(|J_{13}|+|J_{42}|)|\rho_{1}-\rho_{2}|+|J_{13}\widetilde{Q}_{12}^{0}-J_{42}\widetilde{Q}_{21}^{1}|\neq
0.$ (5.15)

Further, assume either $J_{14}\neq 0$ or the following conditions hold with
some $\rho_{3},\rho_{4}>0$:

$\displaystyle\lim_{h\to
0}\frac{1}{h^{\rho_{3}}}\int_{1-h}^{1}Q_{12}(x)\,dx=\widetilde{Q}_{12}^{1}\neq
0,\qquad\lim_{h\to
0}\frac{1}{h^{\rho_{4}}}\int_{0}^{h}Q_{21}(x)\,dx=\widetilde{Q}_{21}^{0}\neq
0,$ (5.16)
$\displaystyle\qquad\qquad(|J_{13}|+|J_{42}|)|\rho_{3}-\rho_{4}|+|J_{13}\widetilde{Q}_{12}^{1}-J_{42}\widetilde{Q}_{21}^{0}|\neq
0.$ (5.17)

Then the system of root functions of the operator $L_{U}(Q)$ is complete in
$L^{2}([0,1];\mathbb{C}^{2})$.

Restricting ourselves to the case $J_{32}=0$ and $J_{14}\neq 0$, we note, that
when $Q(\cdot)$ is smooth and several first derivatives $Q_{12}^{(j)}(0)$ and
$Q_{21}^{(j)}(1)$ vanish simultaneously, then Theorem 5.6 can be reformulated
as follows (see also [26, Remark 2]).

######  Corollary 5.7 ([26]).

Let $-b_{1}=b_{2}=1$ and let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some
$n\in\mathbb{Z}_{+}$. Assume that $J_{32}=0$, $J_{14}\neq 0$, and let the
following conditions hold,

$Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\quad j\in\\{0,1,\ldots,n-2\\},\qquad
Q_{12}^{(n-1)}(0)Q_{21}^{(n-1)}(1)\neq 0.$ (5.18)

Then the system of root functions of the operator $L_{U}(Q)$ is complete in
$L^{2}([0,1];\mathbb{C}^{2})$ whenever

$J_{13}Q_{12}^{(n-1)}(0)\neq J_{42}(-1)^{n-1}Q_{21}^{(n-1)}(1).$ (5.19)

###### Proof.

Since $J_{32}=0$ and $J_{14}\neq 0$, then Theorem 5.6 applies if and only if
condition (5.15) holds. Conditions (5.18) imply

$\displaystyle\lim_{h\to
0}\frac{1}{h^{n}}\int_{0}^{h}Q_{12}(x)\,dx=\frac{Q_{12}^{(n-1)}(0)}{n!}\neq
0,$ (5.20) $\displaystyle\lim_{h\to
0}\frac{1}{h^{n}}\int_{1-h}^{1}Q_{21}(x)\,dx=\frac{(-1)^{n-1}Q_{21}^{(n-1)}(1)}{n!}\neq
0.$ (5.21)

Hence relations (5.14) hold with

$\rho_{1}=\rho_{2}=n,\qquad\widetilde{Q}_{12}^{0}=\frac{Q_{12}^{(n-1)}(0)}{n!},\qquad\widetilde{Q}_{21}^{1}=\frac{(-1)^{n-1}Q_{21}^{(n-1)}(1)}{n!}.$
(5.22)

In turn, condition (5.15) turns into

$(|J_{13}|+|J_{42}|)|\rho_{1}-\rho_{2}|+\bigl{|}{J_{13}\widetilde{Q}_{12}^{0}-J_{42}\widetilde{Q}_{21}^{1}}\bigr{|}\\\
=\frac{\bigl{|}{J_{13}Q_{12}^{(n-1)}(0)-J_{42}(-1)^{n-1}Q_{21}^{(n-1)}(1)}\bigr{|}}{n!}\neq
0,$ (5.23)

which is equivalent to (5.19). ∎

######  Remark 5.8.

Theorem 5.1 was announced in 2013 in [14]. After that, there appeared several
papers [17, 9, 26] concerned with the completeness property for BVP
(1.1)–(1.4) subject to non-regular boundary conditions. Let us compare these
recent results with Corollaries 5.3 and 5.4 in the classical Dirac case.
First, note that Corollary 5.3 coincides with [17, Proposition 4.5], but it
requires more restrictive smoothness assumption $Q\in W_{1}^{1}$ compared to
continuity of $Q(\cdot)$ at the endpoints 0 and 1 imposed in [17].

Further, let $-b_{1}=b_{2}=1$, $Q_{12},Q_{21}\in W_{1}^{2}[0,1]$, and let the
following conditions hold,

$J_{32}=0,\quad J_{14}\neq 0,\qquad Q_{12}(0)Q_{21}(1)\neq 0,\quad
J_{13}Q_{12}(0)=J_{42}Q_{21}(1).$ (5.24)

Then condition (5.6) holds, but condition (5.5) fails. Therefore, [17,
Proposition 4.5] is not applicable. Moreover, [26, Theorem 1] $($see Theorem
5.6 above for its formulation$)$ also cannot be applied. Indeed, under
assumptions (5.24), Theorem 5.6 is equivalent to Corollary 5.7 with $n=1$, but
the last condition in (5.24) ensures that condition (5.19) does not hold. On
the other hand, if

$J_{13}Q_{12}^{\prime}(0)+J_{42}Q_{21}^{\prime}(1)+iJ_{14}Q_{12}(0)Q_{21}(1)\neq
0,$ (5.25)

then condition (5.8) holds, and Corollary 5.4 ensures the desired completeness
property.

##  6 Properties of the functions $\sigma_{j}^{\pm}(x)$

Explicit expressions for values $c^{\pm}_{k}$ (given by (4.5)–(4.6)) for large
$k$ are somewhat cumbersome in general case. However, if either some of the
values $Q_{12}^{(j)}(0)$, $Q_{12}^{(j)}(1)$, $Q_{21}^{(j)}(0)$,
$Q_{21}^{(j)}(1)$ are zero, or two of the determinants
$J_{14},J_{32},J_{13},J_{42}$ are zero, then conditions of Theorem 5.1 can be
expressed explicitly via values $Q_{12}^{(j)}(0)$, $Q_{12}^{(j)}(1)$,
$Q_{21}^{(j)}(0)$, $Q_{21}^{(j)}(1)$ in a very simple fashion. To establish
these explicit completeness results, we need to “decipher” formulas
(3.81)–(3.82) for the functions $\sigma_{j}^{\pm}(\cdot)$. Let us introduce
some notations,

$\displaystyle\mathbb{Z}_{+}^{k}:=\\{\alpha=(\alpha_{1},\ldots,\alpha_{k}):\alpha_{1},\ldots,\alpha_{k}\in\mathbb{Z}_{+}\\},\quad\mathbb{Z}_{+}=\\{0,1,2,3,\ldots\\},$
(6.1) $\displaystyle\Sigma_{\alpha}:=\alpha_{1}+\ldots+\alpha_{k},\qquad
m_{\alpha}:=\max\\{\alpha_{1},\ldots,\alpha_{k}\\},$ (6.2) $\displaystyle
q^{(\alpha)}:=q^{(\alpha_{1})}\cdot\ldots\cdot q^{(\alpha_{k})},\qquad q\in
W_{1}^{m_{\alpha}}[0,1],$ (6.3)

for $\alpha=(\alpha_{1},\ldots,\alpha_{k})\in\mathbb{Z}_{+}^{k}$.

######  Lemma 6.1.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ and functions
$\sigma^{\pm}_{k}(x),k\in\\{1,\ldots,{n}\\}$, be defined via formulas (3.1),
(3.81), (3.82). Then the following identities hold,

$\sigma^{\pm}_{j}=(-1)^{j-1}q_{\mp}^{(j-1)}+\sum_{\genfrac{}{}{0.0pt}{2}{k\in\mathbb{N},\
2k+1\leqslant j,}{\genfrac{}{}{0.0pt}{2}{\alpha\in\mathbb{Z}_{+}^{k+1},\
\beta\in\mathbb{Z}_{+}^{k},}{\Sigma_{\alpha}+\Sigma_{\beta}=j-1-2k}}}C_{j,k,\alpha,\beta}\cdot
q_{\mp}^{(\alpha)}\cdot q_{\pm}^{(\beta)},\qquad j\in\\{1,\ldots,{n}\\},$
(6.4)

with some $C_{j,k,\alpha,\beta}$ that depend only on sequences
$\alpha=(\alpha_{1},\ldots,\alpha_{k+1})$,
$\beta=(\beta_{1},\ldots,\beta_{k})$ and indexes $j,k$ $($in particular, these
coefficients are the same in formulas for $\sigma_{j}^{+}$ and
$\sigma_{j}^{-})$. Moreover,

$C_{j,1,\alpha,\beta}=(-1)^{j},\qquad j\geqslant
3,\quad\alpha=(0,0)\in\mathbb{Z}_{+}^{2},\quad\beta=(j-3)\in\mathbb{Z}_{+}^{1},$
(6.5)

i.e. the coefficient at $q_{\mp}\cdot q_{\mp}\cdot q_{\pm}^{(j-3)}$ in the sum
in the r.h.s. of (6.4) equals to $(-1)^{j}$.

###### Proof.

Identity (6.4) for $j=1$ coincides with (3.81), while for $j=2$ it is proved
in (3.79). Let $j\geqslant 3$ and assume that identity (6.4) is valid for
$\sigma_{1}^{\pm}$, $\ldots$, $\sigma_{j-1}^{\pm}$. To simplify the exposition
of the proof let us introduce some notations. First, we set

$\gamma_{h}:=(h)\in\mathbb{Z}_{+}^{1},\quad
h\in\mathbb{Z}_{+},\qquad\beta_{\varnothing}:=()\in\mathbb{Z}_{+}^{0},\qquad\alpha_{0}:=(0,0)\in\mathbb{Z}_{+}^{2},$
(6.6)

i.e. $\beta_{\varnothing}$ is an empty sequence. Next, we note that

$(-1)^{h-1}q_{\pm}^{(h-1)}=C_{h,0,\alpha,\beta}\cdot q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\beta)},\qquad\alpha=\gamma_{h-1}\in\mathbb{Z}_{+}^{1},\quad\beta=\beta_{\varnothing}\in\mathbb{Z}_{+}^{0},$
(6.7)

for $h\in\\{1,\ldots,{n}\\}$. Here
$q_{\pm}^{(\beta)}=q_{\pm}^{(\beta_{\varnothing})}:=1$ and
$C_{h,0,\alpha,\beta}=(-1)^{h-1}$. Let us also set

$\displaystyle\Psi_{h}$ $\displaystyle:=\\{(k,\alpha,\beta):k\in\mathbb{N},\
2k+1\leqslant h,\ \alpha\in\mathbb{Z}_{+}^{k+1},\ \beta\in\mathbb{Z}_{+}^{k},\
\Sigma_{\alpha}+\Sigma_{\beta}=h-1-2k\\},$ (6.8) $\displaystyle\Psi_{h}^{0}$
$\displaystyle:=\\{(0,\gamma_{h-1},\beta_{\varnothing})\\}\cup\Psi_{h}$
$\displaystyle=\\{(k,\alpha,\beta):k\in\mathbb{Z}_{+},\ 2k+1\leqslant h,\
\alpha\in\mathbb{Z}_{+}^{k+1},\ \beta\in\mathbb{Z}_{+}^{k},\
\Sigma_{\alpha}+\Sigma_{\beta}=h-1-2k\\},$ (6.9)

for $h\in\mathbb{N}$. With account of these notations, induction hypothesis
reads as follows:

$\sigma^{\pm}_{h}=(-1)^{h-1}q_{\pm}^{(h-1)}+\sum_{(k,\alpha,\beta)\in\Psi_{h}}C_{h,k,\alpha,\beta}\cdot
q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\beta)}=\sum_{(k,\alpha,\beta)\in\Psi_{h}^{0}}C_{h,k,\alpha,\beta}\cdot
q_{\mp}^{(\alpha)}\cdot q_{\pm}^{(\beta)},$ (6.10)

for $h\in\\{1,\ldots,{j-1}\\}$ where

$\displaystyle C_{h,0,\alpha,\beta}=(-1)^{h-1},$
$\displaystyle\alpha=\gamma_{h-1}\in\mathbb{Z}_{+}^{1},\quad\beta=\beta_{\varnothing}\in\mathbb{Z}_{+}^{0},\quad
1\leqslant h<j,$ (6.11) $\displaystyle C_{h,1,\alpha,\beta}=(-1)^{h},$
$\displaystyle\alpha=\alpha_{0}\in\mathbb{Z}_{+}^{2},\quad\beta=\gamma_{h-3}\in\mathbb{Z}_{+}^{1},\quad
3\leqslant h<j.$ (6.12)

Combining induction hypothesis (6.10) with formula (3.82) yields,

$\sigma_{j}^{\pm}=-(\sigma_{j-1}^{\pm})^{\prime}-q_{\pm}\cdot\sum_{h=1}^{j-2}\sigma_{h}^{\pm}\sigma_{j-1-h}^{\pm}\\\
=(-1)^{j-1}q_{\mp}^{(j-1)}-\sum_{(k,\alpha,\beta)\in\Psi_{j-1}}C_{j-1,k,\alpha,\beta}\cdot\left(q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\beta)}\right)^{\prime}\\\ -\sum_{h=1}^{j-2}\
\sum_{(k_{1},\widetilde{\alpha}_{1},\widetilde{\beta}_{1})\in\Psi_{h}^{0}}\
\sum_{(k_{2},\widetilde{\alpha}_{2},\widetilde{\beta}_{2})\in\Psi_{j-1-h}^{0}}C_{h,k_{1},\widetilde{\alpha}_{1},\widetilde{\beta}_{1}}C_{j-1-h,k_{2},\widetilde{\alpha}_{2},\widetilde{\beta}_{2}}\times\\\
\times
q_{\mp}^{(\widetilde{\alpha}_{1})}q_{\mp}^{(\widetilde{\alpha}_{2})}\times
q_{\pm}^{(\widetilde{\beta}_{1})}q_{\pm}^{(\widetilde{\beta}_{2})}q_{\pm}.$
(6.13)

For $u\in(u_{1},\ldots,u_{h})\in\mathbb{Z}_{+}^{h}$ and
$u\in(v_{1},\ldots,v_{k})\in\mathbb{Z}_{+}^{k}$ we denote

$u^{\frown}v:=(u_{1},\ldots,u_{h},v_{1},\ldots,v_{k}),$ (6.14)

the concatenation of the sequences $u$ and $v$. With account of this notation,
for each summand in the triple sum above we have

$q_{\mp}^{(\widetilde{\alpha}_{1})}q_{\mp}^{(\widetilde{\alpha}_{2})}\times
q_{\pm}^{(\widetilde{\beta}_{1})}q_{\pm}^{(\widetilde{\beta}_{2})}q_{\pm}=q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\beta)},$ (6.15)

where

$\displaystyle\alpha=\widetilde{\alpha}_{1}^{\frown}\widetilde{\alpha}_{2}\in\mathbb{Z}_{+}^{k_{1}+k_{2}+2},\quad\beta=\widetilde{\beta}_{1}^{\frown}\widetilde{\beta}_{2}^{\frown}\gamma_{0}\in\mathbb{Z}_{+}^{k_{1}+k_{2}+1},\quad
k=k_{1}+k_{2}+1,$ (6.16)
$\displaystyle\Sigma_{\alpha}+\Sigma_{\beta}=\Sigma_{\widetilde{\alpha}_{1}}+\Sigma_{\widetilde{\beta}_{1}}+\Sigma_{\widetilde{\alpha}_{1}}+\Sigma_{\widetilde{\beta}_{1}}+0=j-3-2k_{1}-2k_{2}=j-1-2k.$
(6.17)

It follows from inequalities $2k_{1}+1\leqslant h$ and $2k_{2}+1\leqslant
j-1-h$ that $2k+1\leqslant j$. Therefore, the triple sum in (6.13) has the
desired form

$\sum_{(k,\alpha,\beta)\in\Psi_{j}}\widetilde{C}_{j,k,\alpha,\beta}\cdot
q_{\mp}^{(\alpha)}\cdot q_{\pm}^{(\beta)},$ (6.18)

with certain coefficients $\widetilde{C}_{j,k,\alpha,\beta}$.

Further we note that for
$\alpha=(\alpha_{1},\ldots,\alpha_{k+1})\in\mathbb{Z}_{+}^{k+1}$ and
$\beta=(\beta_{1},\ldots,\beta_{k})\in\mathbb{Z}_{+}^{k}$ we have

$\left(q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\beta)}\right)^{\prime}=\left(q_{\mp}^{(\alpha)}\right)^{\prime}\cdot
q_{\pm}^{(\beta)}+q_{\mp}^{(\alpha)}\cdot\left(q_{\pm}^{(\beta)}\right)^{\prime}=\sum_{h=1}^{k+1}q_{\mp}^{(\widetilde{\alpha}_{h})}\cdot
q_{\pm}^{(\beta)}+\sum_{h=1}^{k}q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\widetilde{\beta}_{h})},$ (6.19)

where

$\displaystyle\widetilde{\alpha}_{h}=(\alpha_{1},\ldots,\alpha_{h-1},\alpha_{h}+1,\alpha_{h+1},\ldots,\alpha_{k+1}),\qquad
h\in\\{1,\ldots,{k+1}\\},$ (6.20)
$\displaystyle\widetilde{\beta}_{h}=(\beta_{1},\ldots,\beta_{h-1},\beta_{h}+1,\beta_{h+1},\ldots,\beta_{k}),\qquad
h\in\\{1,\ldots,{k}\\}.$ (6.21)

Evidently,

$\displaystyle\Sigma_{\widetilde{\alpha}_{h}}+\Sigma_{\beta}=\Sigma_{\alpha}+\Sigma_{\beta}+1,\qquad
h\in\\{1,\ldots,{k+1}\\},$ (6.22)
$\displaystyle\Sigma_{\alpha}+\Sigma_{\widetilde{\beta}_{h}}=\Sigma_{\alpha}+\Sigma_{\beta}+1,\qquad
h\in\\{1,\ldots,{k}\\}.$ (6.23)

Therefore for the second summand in (6.13) we have

$\sum_{(k,\alpha,\beta)\in\Psi_{j-1}}C_{j-1,k,\alpha,\beta}\cdot\left(q_{\mp}^{(\alpha)}\cdot
q_{\pm}^{(\beta)}\right)^{\prime}=\sum_{(k,\alpha,\beta)\in\Psi_{j}}\widehat{C}_{j,k,\alpha,\beta}\cdot
q_{\mp}^{(\alpha)}\cdot q_{\pm}^{(\beta)},$ (6.24)

with certain coefficients $\widehat{C}_{j,k,\alpha,\beta}$. Inserting (6.24)
and expression (6.18) for the triple sum from (6.13) into (6.13) we arrive at
the desired formula (6.4).

Finally, we prove relation (6.5) via induction. For $j=3$ it is evident from
(3.80). Let $j\geqslant 4$ and assume that relation (6.5) is valid for $j-1$.
With account of notations (6.6) it takes the form

$C_{j-1,1,\alpha_{0},\gamma_{j-4}}=(-1)^{j-1}.$ (6.25)

Let us prove that $C_{j,1,\alpha_{0},\gamma_{j-3}}=(-1)^{j}$. Let us analyze
how monomial $q_{\mp}\cdot q_{\mp}\cdot q_{\pm}^{(j-3)}$ can appear in the
expression (6.13) for $\sigma_{j}^{\pm}$. It is evident that each monomial in
the triple sum from (6.13) has a factor $q_{\pm}$ not present in $q_{\mp}\cdot
q_{\mp}\cdot q_{\pm}^{(j-3)}$. Hence the monomial $q_{\mp}\cdot q_{\mp}\cdot
q_{\pm}^{(j-3)}$ can only appear in the second sum. Since this monomial has
only one derivative factor $q_{\pm}^{(j-3)}$ it follows from (6.19) that it is
only contained in this derivative term

$C_{j-1,1,\alpha_{0},\gamma_{j-4}}\cdot\left(q_{\mp}\cdot q_{\mp}\cdot
q_{\pm}^{(j-4)}\right)^{\prime}\\\ =2C_{j-1,1,\alpha_{0},\gamma_{j-4}}\cdot
q_{\mp}^{\prime}\cdot q_{\mp}\cdot
q_{\pm}^{(j-4)}+C_{j-1,1,\alpha_{0},\gamma_{j-4}}\cdot q_{\mp}\cdot
q_{\mp}\cdot q_{\pm}^{(j-3)}.$ (6.26)

Taking into account this observaion and induction hypothesis (6.25), and
comparing (6.4) and (6.13) yields

$C_{j,1,\alpha_{0},\gamma_{j-3}}=-C_{j-1,1,\alpha_{0},\gamma_{j-4}}=-(-1)^{j-1}=(-1)^{j}.$
(6.27)

The proof is now complete. ∎

We need the following simple corollaries of formula (6.4).

######  Lemma 6.2.

Under the assumptions of Lemma 6.1, let

$a\in[0,1],\qquad m\in\\{0,1,\ldots,n-1\\},\qquad m_{0}:=\min\\{2m+2,n\\}.$
(6.28)

Then the following implications hold

$\displaystyle q_{-}^{(j)}(a)=0,\ \ 0\leqslant
j<m\quad\Longrightarrow\quad\sigma^{+}_{j}(a)=(-1)^{j-1}q_{-}^{(j-1)}(a),\ \
1\leqslant j\leqslant m_{0},$ (6.29) $\displaystyle q_{+}^{(j)}(a)=0,\ \
0\leqslant
j<m\quad\Longrightarrow\quad\sigma^{-}_{j}(a)=(-1)^{j-1}q_{+}^{(j-1)}(a),\ \
1\leqslant j\leqslant m_{0}.$ (6.30)

###### Proof.

Let us prove (6.29) (implication (6.30) can be proved similarly). Assume that
$q_{-}^{(j)}(a)=0,\ \ 0\leqslant j<m$. In particular, if $m=0$, then no
assumptions on $q_{-}$ are needed. Let $j\in\\{1,\ldots,{m_{0}}\\}$. Since
$j\leqslant n$, then Lemma 6.1 ensures formula (6.4) for
$\sigma_{j}^{+}(\cdot)$. Consider a factor $q_{-}^{(\alpha)}(a)$ of any
individual summand in the sum in (6.4) for $\sigma_{j}^{+}(a)$, where

$\alpha=(\alpha_{1},\ldots,\alpha_{k+1})\in\mathbb{Z}_{+}^{k+1},\qquad\alpha_{1}+\ldots+\alpha_{k+1}\leqslant
j-1-2k,$ (6.31)

and $k\in\mathbb{N}$, $2k+1\leqslant j$. Since $j\leqslant m_{0}\leqslant
2m+2$, it follows from the second relation in (6.31) that

$\min\\{\alpha_{1},\ldots,\alpha_{k+1}\\}\leqslant\frac{j-1-2k}{k+1}\leqslant\frac{j-3}{2}\leqslant\frac{2m-1}{2}<m.$
(6.32)

Hence, it follows from our assumption that $q_{-}^{(\alpha_{h})}(a)=0$ for
some $h\in\\{1,\ldots,{k+1}\\}$. Therefore, $q_{-}^{(\alpha)}(a)=0$. Thus, it
follows from (6.4) that

$\sigma^{+}_{j}(a)=(-1)^{j-1}q_{-}^{(j-1)}(a),$ (6.33)

which finishes the proof. ∎

######  Corollary 6.3.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$, $a\in[0,1]$, and
$m\in\\{0,1,\ldots,n-1\\}$.

(i) Let the following conditions hold,

$q_{-}^{(j)}(a)=0,\quad j\in\\{0,\ldots,m-1\\},\qquad q_{-}^{(m)}(a)\neq 0.$
(6.34)

Then

$\sigma^{+}_{j}(a)=0,\quad
j\in\\{1,\ldots,{m}\\},\qquad\sigma^{+}_{m+1}(a)\neq 0.$ (6.35)

(ii) Let the following conditions hold,

$q_{+}^{(j)}(a)=0,\quad j\in\\{0,\ldots,m-1\\},\qquad q_{+}^{(m)}(a)\neq 0.$
(6.36)

Then

$\sigma^{-}_{j}(a)=0,\quad
j\in\\{1,\ldots,{m}\\},\qquad\sigma^{-}_{m+1}(a)\neq 0.$ (6.37)

###### Proof.

(i) Since $m<n$, it is evident that $m_{0}\geqslant m+1$ where $m_{0}$ is
given by (6.28). Combining this observation with condition (6.34) and
implication (6.29) from Lemma 6.2 yields that

$\sigma^{+}_{j}(a)=(-1)^{j-1}q_{-}^{(j-1)}(a),\qquad
j\in\\{1,\ldots,{m+1}\\}.$ (6.38)

Combining these relations with condition (6.34) trivially implies (6.35).

(ii) The proof is similar. ∎

##  7 Explicit refined completeness results for non-regular BVP

In this final section, under additional algebraic assumptions either on the
potential matrix $Q(\cdot)$ or on the boundary conditions, we establish a
series of more explicit refined completeness results that clarify somewhat
cumbersome conditions of Theorem 5.1.

The following function is highly involved in most of the results of this
section,

$P(x):=J_{13}b_{1}Q_{12}(x)+J_{42}b_{2}Q_{21}(1-x),\qquad x\in[0,1].$ (7.1)

The next result covers the general case of non-regular boundary conditions,
assuming that some of the values $Q_{12}^{(j)}(0)$, $Q_{21}^{(j)}(1)$ vanish.

######  Proposition 7.1.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Assume also
that the following conditions hold,

$J_{32}=0,\qquad J_{14}\neq 0,\qquad
Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0\quad\text{for}\quad 0\leqslant j<m,$ (7.2)

with some $m\in\\{1,\ldots,{n-1}\\}$. Then the system of root vectors of the
BVP (1.1)–(1.4) is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$
whenever at least one of the following conditions holds

$\displaystyle\text{(i)}\quad P^{(n_{0})}(0)\neq 0\quad\text{for some}\quad
n_{0}\in\\{m,m+1,\ldots,\widetilde{m}\\},\quad\widetilde{m}:=\min\\{2m,n-1\\},$
(7.3) $\displaystyle\text{(ii)}\quad 2m+2\leqslant n\quad\text{and}\quad
P^{(2m+1)}(0)+J_{14}b_{1}b_{2}Q_{12}^{(m)}(0)Q_{21}^{(m)}(1)\neq 0.$ (7.4)

###### Proof.

In notations (4.26) condition (7.2) takes the form

$q_{+}^{(j)}(0)=q_{-}^{(j)}(1)=0\quad\text{for}\quad 0\leqslant j<m.$ (7.5)

In turn, Lemma 6.2 implies that

$\sigma^{-}_{j}(0)=(-1)^{j-1}q_{+}^{(j-1)}(0),\qquad\sigma^{+}_{j}(1)=(-1)^{j-1}q_{-}^{(j-1)}(1),\qquad
j\in\\{1,\ldots,{m_{0}}\\},$ (7.6)

where $m_{0}=\min\\{2m+2,n\\}$. Since $J_{14}\neq 0$, to apply Theorem 5.1, we
just need to verify that $c_{k}^{+}\neq 0$ for some $k\in\\{1,\ldots,{n}\\}$.
First let $k\leqslant 2m+1$. Then

$\min\\{j-1,k-j-1\\}\leqslant\frac{k-2}{2}\leqslant\frac{2m-1}{2}<m,\qquad
j\in\\{1,\ldots,{k-1}\\}.$ (7.7)

Hence condition (7.5) implies that

$q_{+}^{(j-1)}(0)q_{-}^{(k-j-1)}(1)=0,\qquad j\in\\{1,\ldots,{k-1}\\}.$ (7.8)

Inserting relations (7.6) and (7.8) into formula (4.5) and taking into account
definition (7.1) we arrive at

$c_{k}^{+}=J_{13}(-1)^{k-1}\sigma_{k}^{-}(0)+J_{42}\sigma_{k}^{+}(1)-J_{14}\sum_{j=1}^{k-1}(-1)^{j}\sigma_{j}^{-}(0)\sigma_{k-j}^{+}(1)\\\
=J_{13}q_{+}^{(k-1)}(0)+J_{42}(-1)^{k-1}q_{-}^{(k-1)}(1)-J_{14}\sum_{j=1}^{k-1}\underset{=0}{\underbrace{q_{+}^{(j-1)}(0)q_{-}^{(k-j-1)}(1)}}=P^{(k-1)}(0).$
(7.9)

In turn, if condition (7.3) holds, then $c_{n_{0}+1}^{+}=P^{(n_{0})}(0)\neq 0$
and the desired completeness property is ensured by Theorem 5.1.

Finally we assume condition (7.4). Similarly as above we have

$c_{2m+2}^{+}=P^{(2m+1)}(0)-J_{14}q_{+}^{(m)}(0)q_{-}^{(m)}(1)-J_{14}\sum_{\genfrac{}{}{0.0pt}{2}{j=1}{j\neq
m+1}}^{2m+1}\underset{=0}{\underbrace{q_{+}^{(j-1)}(0)q_{-}^{(2m+1-j)}(1)}}\\\
=P^{(2m+1)}(0)+J_{14}b_{1}b_{2}Q_{12}^{(m)}(0)Q_{21}^{(m)}(1)\neq 0.$ (7.10)

Again, Theorem 5.1 completes the proof. ∎

######  Remark 7.2.

Note, that when $m=0$, then Proposition 7.1 is still valid. In this case no
assumption on $Q(\cdot)$ is posed, and this result coincides with Corollary
5.4 above. Thus, Proposition 7.1 can be viewed as a non-trivial generalization
of Corollary 5.4.

We can now establish significant enhancement of Corollary 5.7 by Makin. Recall
that $\left\lceil{x}\right\rceil$, $x\in\mathbb{R}$, denotes the smallest
integer $k$ such that $k\geqslant x$.

######  Corollary 7.3.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Assume that
$J_{32}=0$, $J_{14}\neq 0$, and let the following conditions hold,

$P^{(n-1)}(0)\neq 0,\quad Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\quad 0\leqslant
j<m,\quad\text{where}\quad
m=\text{\scalebox{0.9}{$\left\lceil{\frac{n-1}{2}}\right\rceil$}},$ (7.11)

and $P(\cdot)$ is given by (7.1). Then the system of root functions of the
operator $L_{U}(Q)$ is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

The proof is immediate from Propositon 7.1(i) applied with
$m=\left\lceil{\frac{n-1}{2}}\right\rceil$ and $n_{0}=n-1$. Indeed, by
definition of $\left\lceil{\cdot}\right\rceil$ we have $n_{0}=n-1\leqslant
2m$. Hence the first condition in (7.11) ensures condition (7.3). ∎

######  Remark 7.4.

Let us compare Corollary 7.3 with Corollary 5.7 by Makin from [26]. Namely,
let $-b_{1}=b_{2}=1$ and $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some
$n\geqslant 3$, and assume that $P^{(n-1)}(0)\neq 0$ $($which coincides with
condition (5.19) in the Dirac case$)$. Then Corollary 5.7 ensures completeness
property under more restrictive assumption (5.18) compared to our relaxed
assumption (7.11) from Corollary 7.3. Namely, Corollary 5.7 requires
additional $2\left\lfloor{\frac{n-1}{2}}\right\rfloor\geqslant 2$ zero
derivative values compared to (7.11),

$Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\quad m\leqslant j\leqslant
n-2,\quad\text{where}\quad
m=\text{\scalebox{0.85}{$\left\lceil{\frac{n-1}{2}}\right\rceil$}}.$ (7.12)

Therefore, Proposition 7.1 demonstrates that completeness property is valid
under less restrictive condition than the one in [26, Theorem 1], when
$Q(\cdot)$ is rather smooth.

Next result substantially complements Theorem 5.1 from [32]. Part (i) of the
next theorem is announced in [14, Theorem 4].

######  Theorem 7.5.

Let $J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$, and $Q_{12},Q_{21}\in
W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Let for some
$n_{0},n_{1}\in\\{0,1,\ldots,n-1\\}$ the following conditions hold

$\displaystyle P^{(k)}(0)$ $\displaystyle=0,\qquad
k\in\\{0,1,\ldots,n_{0}-1\\},$ (7.13) $\displaystyle P^{(k)}(1)$
$\displaystyle=0,\qquad k\in\\{0,1,\ldots,n_{1}-1\\},$ (7.14)

where $P(\cdot)$ is given by (7.1). Then the system of root vectors of the BVP
(1.1)–(1.4) is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$ whenever
one of the following conditions holds

$\displaystyle\text{(i)}\quad|n_{1}-n_{0}|\leqslant 1,\qquad
P^{(n_{0})}(0)\neq 0,\quad P^{(n_{1})}(1)\neq 0.$ (7.15)
$\displaystyle\text{(ii)}\quad n_{0}\geqslant n_{1}+2,\qquad
P^{(n_{1})}(1)\neq 0,$ $\displaystyle\qquad\qquad\qquad\qquad
J_{13}P^{(n_{1}+2)}(0)+(-1)^{n_{1}}J_{42}b_{1}^{2}Q_{12}^{2}(0)P^{(n_{1})}(1)\neq
0,$ (7.16) $\displaystyle\text{(iii)}\quad n_{1}\geqslant n_{0}+2,\qquad
P^{(n_{0})}(0)\neq 0,$ $\displaystyle\qquad\qquad\qquad\qquad
J_{13}P^{(n_{0}+2)}(1)+(-1)^{n_{0}}J_{42}b_{1}^{2}Q_{12}^{2}(1)P^{(n_{0})}(0)\neq
0.$ (7.17)

###### Proof.

With account of notations (4.26) and (7.1) we have

$\displaystyle
J_{13}q_{+}^{(j)}(0)+J_{42}(-1)^{j}q_{-}^{(j)}(1)=-iP^{(j)}(0),\qquad
0\leqslant j<n,$ (7.18) $\displaystyle
J_{13}q_{+}^{(j)}(1)+J_{42}(-1)^{j}q_{-}^{(j)}(0)=-iP^{(j)}(1),\qquad
0\leqslant j<n.$ (7.19)

Combining these relations with conditions (7.13)–(7.14) yields

$\displaystyle J_{13}q_{+}^{(j)}(0)=J_{42}(-1)^{j-1}q_{-}^{(j)}(1),\qquad
0\leqslant j<n_{0},$ (7.20) $\displaystyle
J_{13}q_{+}^{(j)}(1)=J_{42}(-1)^{j-1}q_{-}^{(j)}(0),\qquad 0\leqslant
j<n_{1}.$ (7.21)

Since $J_{13}J_{42}\neq 0$, it follows from these identities that

$q_{-}^{(j_{0})}(1)q_{+}^{(j_{1})}(1)=(-1)^{j_{0}+j_{1}}q_{+}^{(j_{0})}(0)q_{-}^{(j_{1})}(0),\qquad
0\leqslant j_{0}<n_{0},\quad 0\leqslant j_{1}<n_{1}.$ (7.22)

Further, since $J_{14}=J_{32}=0$, then the coefficients $c_{j}^{\pm}$ given by
(4.5)–(4.6) are simplified to,

$c_{j}^{+}=J_{13}(-1)^{j-1}\sigma_{j}^{-}(0)+J_{42}\sigma_{j}^{+}(1),\qquad
c_{j}^{-}=J_{13}(-1)^{j-1}\sigma_{j}^{-}(1)+J_{42}\sigma_{j}^{+}(0),$ (7.23)

for $j\in\\{1,\ldots,{n}\\}$. Inserting identity (6.4) (see (6.8)–(6.10)) and
relation (7.19) into each of the equalities in (7.23) yields

$\displaystyle c_{j}^{+}$
$\displaystyle=\underset{=-iP^{(j-1)}(0)}{\underbrace{J_{13}q_{+}^{(j-1)}(0)+J_{42}(-1)^{j-1}q_{-}^{(j-1)}(1)}}$
$\displaystyle+\sum_{(k,\alpha,\beta)\in\Psi_{j}}C_{j,k,\alpha,\beta}\cdot\left(J_{13}(-1)^{j-1}q_{+}^{(\alpha)}(0)\cdot
q_{-}^{(\beta)}(0)+J_{42}q_{-}^{(\alpha)}(1)\cdot q_{+}^{(\beta)}(1)\right),$
(7.24) $\displaystyle c_{j}^{-}$
$\displaystyle=\underset{=-iP^{(j-1)}(1)}{\underbrace{J_{13}q_{+}^{(j-1)}(1)+J_{42}(-1)^{j-1}q_{-}^{(j-1)}(0)}}$
$\displaystyle+\sum_{(k,\alpha,\beta)\in\Psi_{j}}C_{j,k,\alpha,\beta}\cdot\left(J_{13}(-1)^{j-1}q_{+}^{(\alpha)}(1)\cdot
q_{-}^{(\beta)}(1)+J_{42}q_{-}^{(\alpha)}(0)\cdot q_{+}^{(\beta)}(0)\right).$
(7.25)

Let $(k,\alpha,\beta)\in\Psi_{j}$ with
$\alpha=(\alpha_{1},\ldots,\alpha_{k+1})\in\mathbb{Z}_{+}^{k+1}$ and
$\beta=(\beta_{1},\ldots,\beta_{k})\in\mathbb{Z}_{+}^{k}$. First assume that

$\alpha_{h}<n_{0},\quad h\in\\{1,\ldots,{k+1}\\},\qquad\beta_{h}<n_{1},\quad
h\in\\{1,\ldots,{k}\\}.$ (7.26)

Combining relations (7.21) and (7.22) with equality
$\Sigma_{\alpha}+\Sigma_{\beta}=j-1-2k$ implies

$J_{42}q_{-}^{(\alpha)}(1)\cdot
q_{+}^{(\beta)}(1)=J_{42}q_{-}^{(\alpha_{k+1})}(1)\times\prod_{h=1}^{k}q_{-}^{(\alpha_{h})}(1)q_{+}^{(\beta_{h})}(1)\\\
=J_{13}(-1)^{\alpha_{k+1}-1}q_{+}^{(\alpha_{k+1})}(0)\times\prod_{h=1}^{k}(-1)^{\alpha_{h}+\beta_{h}}q_{+}^{(\alpha_{h})}(0)q_{-}^{(\beta_{h})}(0)\\\
=J_{13}(-1)^{\Sigma_{\alpha}+\Sigma_{\beta}-1}\cdot q_{+}^{(\alpha)}(0)\cdot
q_{-}^{(\beta)}(0)=J_{13}(-1)^{j}q_{+}^{(\alpha)}(0)\cdot q_{-}^{(\beta)}(0).$
(7.27)

Similarly, combining relations (7.20) and (7.22) yields

$J_{42}q_{-}^{(\alpha)}(0)\cdot
q_{+}^{(\beta)}(0)=J_{13}(-1)^{j}q_{+}^{(\alpha)}(1)\cdot q_{-}^{(\beta)}(1),$
(7.28)

whenever

$\alpha_{h}<n_{1},\quad h\in\\{1,\ldots,{k+1}\\},\qquad\beta_{j}<n_{0},\quad
h\in\\{1,\ldots,{k}\\}.$ (7.29)

Now let $j\leqslant\min\\{n_{0},n_{1}\\}+2$. For any triplet
$(k,\alpha,\beta)\in\Psi_{j}$ we have

$\displaystyle\alpha_{h}\leqslant j-1-2k\leqslant
j-3<\min\\{n_{0},n_{1}\\},\qquad h\in\\{1,\ldots,{k+1}\\},$ (7.30)
$\displaystyle\beta_{h}\leqslant j-1-2k\leqslant
j-3<\min\\{n_{0},n_{1}\\},\qquad h\in\\{1,\ldots,{k}\\}.$ (7.31)

Therefore, both relations (7.26) and (7.29) hold. In turn, combining
identities (7.27) and (7.28) with equalities (7.24) and (7.25) implies

$c_{j}^{+}=-iP^{(j-1)}(0),\qquad c_{j}^{-}=-iP^{(j-1)}(1),\qquad
j\leqslant\min\\{n_{0},n_{1}\\}+2.$ (7.32)

(i) First assume that condition (7.15) holds. Then

$n_{0}+1\leqslant\min\\{n_{0},n_{1}\\}+2,\qquad
n_{1}+1\leqslant\min\\{n_{0},n_{1}\\}+2.$ (7.33)

Hence combining relations (7.32) with condition
$P^{(n_{0})}(0)P^{(n_{1})}(1)\neq 0$ yields

$c_{n_{0}+1}^{+}=-iP^{(n_{0})}(0)\neq 0,\qquad
c_{n_{1}+1}^{-}=-iP^{(n_{1})}(1)\neq 0.$ (7.34)

Therefore, Theorem 5.1 implies the desired completeness property, which
finishes the proof in this case.

(ii) Now assume that conditions (7.16) holds. Since $n_{1}\leqslant
n_{0}-2\leqslant n_{0}+1$, then as in the previous step

$c_{n_{1}+1}^{-}=-iP^{(n_{1})}(1)\neq 0.$ (7.35)

However, since $n_{0}\geqslant n_{1}+2$ and condition (7.13) holds, then

$c_{j}^{+}=-iP^{(j-1)}(0)=0,\qquad j\in\\{1,\ldots,{n_{1}+2}\\}.$ (7.36)

To this end, we evaluate $c_{n_{1}+3}^{+}$ and show that the second condition
in (7.16) yields $c_{n_{1}+3}^{+}\neq 0$. Set $j=n_{1}+3$ and let
$(k,\alpha,\beta)\in\Psi_{j}$. Similarly to inequalities (7.30)–(7.31) we
conclude that

$\displaystyle\alpha_{h}\leqslant j-1-2k\leqslant j-3=n_{1}<n_{0},\qquad
h\in\\{1,\ldots,{k+1}\\},$ (7.37) $\displaystyle\beta_{h}\leqslant
j-1-2k\leqslant j-3=n_{1},\qquad h\in\\{1,\ldots,{k}\\}.$ (7.38)

Hence condition (7.26) holds for all triples $(k,\alpha,\beta)$ except the
case when $\beta_{h}=j-3=n_{1}$ for some $h$. This is possible only when

$k=1,\quad\alpha=\alpha_{0}=(0,0)\in\mathbb{Z}_{+}^{2}\quad\text{and}\quad\beta=\gamma_{h-3}=(h-3)\in\mathbb{Z}_{+}^{1}.$
(7.39)

Therefore, identity (7.27) holds for each summand in the sum in the r.h.s. in
(7.24) except this one exceptional triple. In turn, with account of formula
(6.5), relation (7.24) simplifies too

$c_{j}^{+}=-iP^{(j-1)}(0)+(-1)^{j}\left(J_{13}(-1)^{j-1}q_{+}^{2}(0)\cdot
q_{-}^{(j-3)}(0)+J_{42}q_{-}^{2}(1)\cdot q_{+}^{(j-3)}(1)\right)\\\
=-iP^{(n_{1}+2)}(0)-J_{13}q_{+}^{2}(0)\cdot
q_{-}^{(n_{1})}(0)-J_{42}q_{-}^{2}(1)\cdot(-1)^{n_{1}}q_{+}^{(n_{1})}(1).$
(7.40)

Since $n_{0}>0$, it follows from (7.20) that $J_{13}q_{+}(0)=-J_{42}q_{-}(1)$,
which implies that

$J_{42}q_{-}^{2}(1)=\frac{J_{13}}{J_{42}}J_{13}q_{+}^{2}(0)=-\frac{J_{13}}{J_{42}}J_{13}b_{1}^{2}Q_{12}^{2}(0).$
(7.41)

Inserting (7.41) and (7.19) into (7.40) and taking into account condition
(7.16), we arrive at

$c_{n_{1}+3}^{+}=-iP^{(n_{1}+2)}(0)-J_{13}q_{+}^{2}(0)\left(q_{-}^{(n_{1})}(0)+\frac{J_{13}}{J_{42}}\cdot(-1)^{n_{1}}q_{+}^{(n_{1})}(1)\right)\\\
=-i\left(P^{(n_{1}+2)}(0)-\frac{J_{13}}{J_{42}}q_{+}^{2}(0)\cdot(-1)^{n_{1}}P^{(n_{1})}(1)\right)\\\
=\frac{-i}{J_{42}}\left(J_{42}P^{(n_{1}+2)}(0)+(-1)^{n_{1}}J_{13}b_{1}^{2}Q_{12}^{2}(0)P^{(n_{1})}(1)\right)\neq
0.$ (7.42)

As above, Theorem 5.1 finishes the proof in this case.

(iii) The case of condition (7.17) is treated similarly, which completes the
proof. ∎

######  Corollary 7.6.

Let $J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$, $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$
for some $n\geqslant 3$, and let $P(\cdot)$ be given by (7.1). Assume that the
following two conditions hold

$\displaystyle\qquad P^{(n_{0})}(0)\neq 0\quad\text{for some}\quad
n_{0}\in\\{0,1,\ldots,n-3\\},$ (7.43) $\displaystyle Q_{12}(1)\neq
0,\quad\text{and}\quad P^{(k)}(1)=0\quad\text{for each}\quad
k\in\\{0,1,\ldots,n-1\\}.$ (7.44)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

Replacing $n_{0}$ with a smaller value if needed, we can assume condition
(7.13). Condition (7.44) implies condition (7.14) with $n_{1}=n_{0}+2<n$. It
is also evident that $P^{(n_{0}+2)}(1)=0$. Since $Q_{12}(1)\neq 0$ and
$P^{(n_{0})}(0)\neq 0$, then condition (7.17) holds. Theorem 7.5(iii) now
finishes the proof. ∎

######  Remark 7.7.

Note that condition (7.44) holds, in particular, if $P(x)=0$ for $x\in[a,1]$
with some $a\in(0,1)$. Thus, Corollary 7.6 shows substantial difference
between Sturm-Liouville equation and $2\times 2$ Dirac system in the case of
non-regular boundary conditions.

Namely, consider Sturm-Liouville equation
$-y^{\prime\prime}+q(x)y=\lambda^{2}y$, $x\in[0,1]$, subject to degenerate
boundary conditions $y(0)-\alpha y(1)=y^{\prime}(0)+\alpha y^{\prime}(1)=0$,
$\alpha\neq 0$. Let $\widetilde{P}(x):=q(x)-q(1-x)$, $x\in[0,1]$. Based on
[29, 22], if $\widetilde{P}^{(k)}(0)\neq 0$ for some $k\in\mathbb{Z}_{+}$,
then the system of root vectors of such BVP is complete in $L^{2}[0,1]$. On
the other hand, if $\widetilde{P}(x)=0$ for $x\in[a,1]$ with some $a\in(0,1)$,
then according to [29] this system of root vectors is incomplete and has
infinite defect.

On the other hand, as Corollary 7.6 shows, in the case of Dirac operator,
completeness is possible if $P(x)=0$ for $x\in[a,1]$ as long as
$P^{(k)}(0)\neq 0$ for some $k\in\mathbb{Z}_{+}$ and $Q_{12}(1)\neq 0$.
Moreover, if also $P(x)=0$ for $x\in[0,\delta]$ with some $\delta\in(0,1)$,
then according to [32, Proposition 5.12], [17, Proposition 4.13] the system of
root vectors of the BVP for the $2\times 2$ Dirac equation with
$J_{14}=J_{32}=0$ is also incomplete.

Theorem 7.5(i) has inconvenient restriction $|n_{0}-n_{1}|\leqslant 1$. We can
relax it if some of derivatives $Q^{(j)}(0)$, $Q^{(j)}(1)$ vanish, as the
following result shows.

######  Proposition 7.8.

Let $J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$, and $Q_{12},Q_{21}\in
W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Let for some
$m_{0},m_{1}\in\\{0,1,\ldots,n-1\\}$ the following conditions hold

$\displaystyle Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\qquad 0\leqslant j<m_{0},$
(7.45) $\displaystyle Q_{12}^{(j)}(1)=Q_{21}^{(j)}(0)=0,\qquad 0\leqslant
j<m_{1}.$ (7.46)

Further, let for some $n_{0},n_{1}\in\\{0,1,\ldots,n-1\\}$ the following
conditions hold

$\displaystyle P^{(j)}(0)$ $\displaystyle=0,\quad
j\in\\{0,1,\ldots,n_{0}-1\\},\qquad P^{(n_{0})}(0)\neq 0,$ (7.47)
$\displaystyle P^{(j)}(1)$ $\displaystyle=0,\quad
j\in\\{0,1,\ldots,n_{1}-1\\},\qquad P^{(n_{1})}(1)\neq 0,$ (7.48)

where $P(\cdot)$ is given by (7.1). Then the system of root vectors of the BVP
(1.1)–(1.4) is complete and minimal in $L^{2}([0,1];\mathbb{C}^{2})$ whenever
the following condition holds

$-2m_{0}-1\leqslant n_{0}-n_{1}\leqslant 2m_{1}+1.$ (7.49)

In particular, this holds if $n_{0}\leqslant 3m_{1}+1$ and $n_{1}\leqslant
3m_{0}+1$.

###### Proof.

As in the proof of Theorem 7.5 conditions (7.47)–(7.48) imply identities
(7.18)–(7.22). With account of notation (4.26), conditions (7.45)–(7.46) take
the form

$\displaystyle q_{+}^{(j)}(0)=q_{-}^{(j)}(1)=0,\qquad 0\leqslant j<m_{0},$
(7.50) $\displaystyle q_{+}^{(j)}(1)=q_{-}^{(j)}(0)=0,\qquad 0\leqslant
j<m_{1}.$ (7.51)

First let $j=n_{0}+1$. We will prove that $c_{j}^{+}=c_{n_{0}+1}^{+}\neq 0$.
Let us analyze individual summands

$Q^{+}_{j,k,\alpha,\beta}:=J_{13}(-1)^{j-1}q_{+}^{(\alpha)}(1)\cdot
q_{-}^{(\beta)}(1)+J_{42}q_{-}^{(\alpha)}(0)\cdot q_{+}^{(\beta)}(0)$ (7.52)

in the sum in the r.h.s. of formula (7.24) for $c_{j}^{+}$. To this end, let
$(k,\alpha,\beta)\in\Psi_{j}$, i.e.

$\displaystyle\alpha=(\alpha_{1},\ldots,\alpha_{k+1})\in\mathbb{Z}_{+}^{k+1},\qquad\beta=(\beta_{1},\ldots,\beta_{k})\in\mathbb{Z}_{+}^{k},$
(7.53)
$\displaystyle\alpha_{1}+\ldots+\alpha_{k+1}+\beta_{1}+\ldots+\beta_{k}=j-1-2k,$
(7.54)

where $k\in\mathbb{N}$, $2k+1\leqslant j$. Note that if $\alpha_{h}<m_{1}$ for
some $h\in\\{1,\ldots,{k+1}\\}$, then condition (7.51) implies that
$q_{+}^{(\alpha)}(1)=q_{-}^{(\alpha)}(0)=0$. Similarly, if $\beta_{h}<m_{0}$
for some $h\in\\{1,\ldots,{k}\\}$, then condition (7.50) implies that
$q_{-}^{(\beta)}(1)=q_{+}^{(\beta)}(0)=0$. Therefore,
$Q^{+}_{j,k,\alpha,\beta}=0$ if either $\alpha_{h}<m_{1}$ for some $h$ or
$\beta_{h}<m_{0}$ for some $h$.

Next consider the opposite case, i.e.

$\alpha_{h}\geqslant m_{1},\quad
h\in\\{1,\ldots,{k+1}\\},\qquad\beta_{h}\geqslant m_{0},\quad
h\in\\{1,\ldots,{k}\\}.$ (7.55)

Combining this inequalities with relation (7.54), definition $j=n_{0}+1$ and
the second inequality in (7.49) implies

$\displaystyle\alpha_{h}\leqslant j-1-2k-km_{1}-km_{0}\leqslant
n_{0}-2-m_{1}-m_{0}<n_{0},\quad 1\leqslant h\leqslant k+1,$ (7.56)
$\displaystyle\beta_{h}\leqslant j-1-2k-(k+1)m_{1}-(k-1)m_{0}\leqslant
n_{0}-2-2m_{1}<n_{1},\quad 1\leqslant h\leqslant k.$ (7.57)

Hence, condition (7.26) holds and just as in the proof of Theorem 7.5,
combining identities (7.20)–(7.22) with condition (7.26) implies identity
(7.27), which again means that $Q^{+}_{j,k,\alpha,\beta}=0$. Hence, formula
(7.24) implies that $c_{j}^{+}=-iP^{(j-1)}(0)=-iP^{(n_{0})}(0)\neq 0$ due to
the second condition in (7.47).

Similarly, using inequality $n_{1}-2-2m_{0}<n_{0}$ we can prove that

$Q^{-}_{j,k,\alpha,\beta}:=J_{13}(-1)^{j-1}q_{+}^{(\alpha)}(0)\cdot
q_{-}^{(\beta)}(0)+J_{42}q_{-}^{(\alpha)}(1)\cdot
q_{+}^{(\beta)}(1)=0,\quad(k,\alpha,\beta)\in\Psi_{j},$ (7.58)

where $j=n_{1}+1$. In turn, this implies that
$c_{j}^{-}=c_{n_{1}+1}^{-}=-iP^{(n_{1})}(1)\neq 0$ due to the second condition
in (7.48). Theorem 5.1 implies the desired completeness property, which
finishes the proof of the main statement.

It remains to consider the case when $n_{0}\leqslant 3m_{1}+1$ and
$n_{1}\leqslant 3m_{0}+1$. Note that the last condition in (7.47) means that

$P^{(n_{0})}(0)=J_{13}b_{1}Q_{12}^{(n_{0})}(0)+J_{42}(-1)^{n_{0}}b_{2}Q_{21}^{(n_{0})}(1)\neq
0.$ (7.59)

Which implies that either $Q_{12}^{(n_{0})}(0)\neq 0$ or
$Q_{21}^{(n_{0})}(1)\neq 0$. Combining this fact with (7.45) implies that
$n_{0}\geqslant m_{0}$. Hence

$n_{0}-n_{1}\geqslant m_{0}-n_{1}\geqslant m_{0}-3m_{0}-1=-2m_{0}-1.$ (7.60)

Similarly, we derive that $n_{1}\geqslant m_{1}$ and that
$n_{0}-n_{1}\leqslant 2m_{1}+1$. Therefore, inequalities (7.49) hold whenever
$n_{0}\leqslant 3m_{1}+1$ and $n_{1}\leqslant 3m_{0}+1$. ∎

######  Remark 7.9.

(i) If $m_{0}=0$, then condition (7.45) holds automatically since there are no
$j$ satisfying inequality $0\leqslant j<m_{0}$. Similarly, if $m_{1}=0$, then
condition (7.46) holds automatically. In particular, if $m_{0}=m_{1}=0$, then
condition (7.49) turns into $|n_{0}-n_{1}|\leqslant 1$, and so in this case
Proposition 7.8 is equivalent to Theorem 7.5(i).

(ii) Emphasize that in condition (7.45) we do not assume that either
$Q_{12}^{(m_{0})}(0)\neq 0$ or $Q_{21}^{(m_{0})}(1)\neq 0$. Similar can be
said on condition (7.46).

In turn, with sufficient number of zero derivatives $Q^{(j)}(0)$, $Q^{(j)}(1)$
we can eliminate any restrictions between $n_{0}$ and $n_{1}$ in Theorem
7.5(i).

######  Corollary 7.10.

Let $J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$, and let $Q_{12},Q_{21}\in
W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Let $P(\cdot)$ be given by (7.1).
Let the following conditions hold

$\displaystyle Q^{(j)}(0)=Q^{(j)}(1)=0,\quad
j\in\\{0,1,\ldots,m-1\\},\quad\text{where}\quad
m=\text{\scalebox{0.9}{$\left\lceil{\frac{n-2}{3}}\right\rceil$}},$ (7.61)
$\displaystyle P^{(n_{0})}(0)P^{(n_{1})}(1)\neq 0\qquad\text{for some}\quad
n_{0},n_{1}\in\\{m,m+1,\ldots,n-1\\}.$ (7.62)

Then the system of root vectors of the BVP (1.1)–(1.4) is complete and minimal
in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

It is clear that conditions (7.45)–(7.46) hold with $m_{0}=m_{1}=m$. It
follows from the definition of $\left\lceil{x}\right\rceil$ that $3m\geqslant
n-2$. Hence $n_{0}\leqslant n-1\leqslant 3m+1=3m_{1}+1$ and $n_{1}\leqslant
n-1\leqslant 3m+1=3m_{0}+1$. The second part of Proposition 7.8 completes the
proof. ∎

Let us compare Corollary 7.10 with Theorem 5.6 by Makin. First we reformulate
Theorem 5.6 in the case of smooth potential when $J_{32}=J_{14}=0$ and
$J_{13}J_{42}\neq 0$.

######  Corollary 7.11 ([26]).

Let $B=\operatorname{diag}(-1,1)$, $J_{32}=J_{14}=0$, $J_{13}J_{42}\neq 0$,
and let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\in\mathbb{N}$. Assume
that

$Q_{12}^{(\rho_{1})}(0)\cdot Q_{12}^{(\rho_{2})}(1)\cdot
Q_{21}^{(\rho_{3})}(0)\cdot Q_{21}^{(\rho_{4})}(1)\neq 0,$ (7.63)

for some $\rho_{1},\rho_{2},\rho_{3},\rho_{4}\in\\{0,1,\ldots,n-1\\}$. Let for
some $n_{0},n_{1}\in\\{0,1,\ldots,n-1\\}$ the following conditions hold

$\displaystyle Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\quad
j\in\\{0,1,\ldots,n_{0}-1\\},\qquad P^{(n_{0})}(0)\neq 0,$ (7.64)
$\displaystyle Q_{12}^{(j)}(1)=Q_{21}^{(j)}(0)=0,\quad
j\in\\{0,1,\ldots,n_{1}-1\\},\qquad P^{(n_{1})}(1)\neq 0,$ (7.65)

where $P(\cdot)$ is given by (7.1). Then the system of root vectors of the BVP
(1.1)–(1.4) is complete in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

Condition guarantees existence of
$\rho_{1},\rho_{2},\rho_{3},\rho_{4}\leqslant n$ for which conditions (5.14)
and (5.16) hold. Let us verify condition (5.15). Condition (7.64) guarantees
that $\rho_{1},\rho_{2}\geqslant n_{0}+1$. In turn since

$0\neq
P^{(n_{0})}(0)=J_{13}Q_{12}^{(n_{0})}(0)-J_{42}(-1)^{n_{0}}Q_{21}^{(n_{0})}(1),$
(7.66)

it follows that either $Q_{12}^{(n_{0})}(0)\neq 0$ or $Q_{21}^{(n_{0})}(1)\neq
0$. Assume without loss of generality that $Q_{12}^{(n_{0})}(0)\neq 0$, then
$\rho_{1}=n_{0}+1$ (see (5.20)). If $Q_{21}^{(n_{0})}(1)=0$ then
$\rho_{2}>\rho_{1}$. Since by assumption $J_{13}J_{42}\neq 0$, then
$(|J_{13}|+|J_{42}|)|\rho_{1}-\rho_{2}|\neq 0$ and hence condition (5.15)
holds. If $Q_{21}^{(n_{0})}(1)\neq 0$ then $\rho_{1}=\rho_{2}=n_{0}+1$ and
condition (5.15) is verified as in the proof of Corollary 5.7. Condition
(5.17) can be verified similarly. Theorem 5.6 completes the proof. ∎

######  Remark 7.12.

Let us compare Corollary 7.10 with Corollary 7.11 by Makin [26]. Namely, let
$-b_{1}=b_{2}=1$ and let assumptions of Corollary 7.10 be satisfied $($recall
that $m=\left\lceil{\frac{n-2}{3}}\right\rceil)$, which ensures the desired
completeness property. On the other hand, for Corollary 7.11 to guarantee the
completeness property, condition (7.61) should be satisfied as well as
additional $2(n_{0}+n_{1}-2m)$ derivative values at the endpoints should
vanish,

$\displaystyle Q_{12}^{(j)}(0)=Q_{21}^{(j)}(1)=0,\qquad
j\in\\{m,m+1,\ldots,n_{0}-1\\},$ (7.67) $\displaystyle
Q_{12}^{(j)}(1)=Q_{21}^{(j)}(0)=0,\qquad j\in\\{m,m+1,\ldots,n_{1}-1\\}.$
(7.68)

Thus, Proposition 7.8 requires much less than [26, Theorem 1] in the discussed
case.

Finally we study the case $J_{32}=J_{13}=0$. As it turns out, completeness
conditions are the most simple and explicit in this case.

######  Corollary 7.13.

Let $Q_{12},Q_{21}\in W_{1}^{n}[0,1]$ for some $n\geqslant 2$. Let the
following condition hold

$J_{32}=J_{13}=J_{42}=0,\qquad J_{14}\neq 0,\qquad
Q_{12}^{(j_{0})}(0)Q_{21}^{(j_{1})}(1)\neq 0,$ (7.69)

for some $j_{0},j_{1}\in\\{0,1,\ldots,n-2\\}$ such that $j_{0}+j_{1}\leqslant
n-2$. Then the system of root vectors of the BVP (1.1)–(1.4) is complete and
minimal in $L^{2}([0,1];\mathbb{C}^{2})$.

###### Proof.

As in the previous results, we will verify condition (5.1) from Theorem 5.1.
Since $J_{14}\neq 0$, it follows that $c_{0}^{-}=J_{14}\neq 0$. Due to the