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Structure of the spectrum of a nonselfadjoint Dirac operator A. S. Makin MIREA — Russian Technological University, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 1, DOI: https://doi.org/10.4213/sm9709 
Bibliographic databases:
    § 1. IntroductionAn important class of inverse spectral problems is the problem of recovering a system of differential equations from its spectral data. Problems of this kind for the Dirac and Dirac-type operators are best studied. In particular, these problems for the canonical Dirac system on a finite interval where $\mathbf{y}=\operatorname{col}(y_1(x),y_2(x))$,
$$
\begin{equation*}
B=\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix}\quad\text{and} \quad V(x)=\begin{pmatrix} p(x)&q(x) \\ q(x)&-p(x) \end{pmatrix},
\end{equation*}
\notag
$$
are well studied when the operator is selfadjoint. In the case of Dirichlet or Neumann boundary conditions, the continuous potential was recovered from two spectra in [1]. Similar results for the Dirac operator with an integrable potential were deduced in [2]. The above problems for nonseparable boundary conditions, including periodic, antiperiodic and quasi-periodic ones, were solved in [3] and [4]. In the nonselfadjoint case the problem of the recovery of the potential $V(x)$ from the spectral data is much more complicated, since many methods used successfully for selfadjoint operators are inapplicable. For example, the spectra of periodic (antiperiodic) boundary-value problems for the operator (1.1) were characterized in [3] in terms of special conformal mappings, which do not exist for complex-valued potentials, while the alternation property of eigenvalues of the corresponding Dirichlet and Neumann problems, which is often used to prove the solvability of the basic equation, has no sense in the complex case. Inverse problems for nonselfadjoint differential systems of the first order were considered in [5] and [6]. Among the recent publications, we note [7]. This paper studies the Dirac system (1.1) for complex-valued functions $p$ and $q$ in the space $L_2(0,\pi)$ $(V\in L_2(0,\pi))$, with the two-point boundary conditions
$$
\begin{equation}
\begin{gathered} \, U_1(\mathbf{y})= a_{11}y_1(0)+a_{12}y_2(0)+a_{13}y_1(\pi)+ a_{14}y_2(\pi)=0, \\ U_2(\mathbf{y})= a_{21}y_1(0)+a_{22}y_2(0)+a_{23}y_1(\pi)+ a_{24}y_2(\pi)=0, \end{gathered}
\end{equation}
\tag{1.2}
$$
where the coefficients $a_{ij}$ are arbitrary complex numbers and the matrix
$$
\begin{equation*}
A=\begin{pmatrix} a_{11}&a_{12}&a_{13} &a_{14} \\ a_{21}&a_{22}&a_{23} &a_{24} \end{pmatrix}
\end{equation*}
\notag
$$
has linearly independent rows. Our main goal is to investigate the structure of the spectrum of the eigenvalue problem for (1.1) with boundary conditions like (1.2) and nonsmooth complex-valued potential $V(x)$. We let $\|\mathbf{f}\|=(|f_1|^2+|f_2|^2)^{1/2}$ denote the norm of an arbitrary vector $\mathbf{f}=\operatorname{col}(f_1,f_2)\in\mathbb{C}^2$ and set $\langle \mathbf{f},\mathbf{g}\rangle=f_1\overline g_1+f_2\overline g_2$; we let $\|W\|= \sup_{\|\mathbf{f}\|=1}\|W\mathbf{f}\|$ denote the norm of an arbitrary $2\times2$ matrix $W$. We let $L_{2,2}(a,b)$ denote the space of two-dimensional vector functions $\mathbf{f}(t)=\operatorname{col}(f_1(t),f_2(t))$ with norm $\displaystyle\|\mathbf{f}\|_{L_{2,2}(a,b)}=\biggl(\int_a^b\|\mathbf{f}(t)\|\,dt\biggr)^{1/2}$ and $L_{2,2}^{2,2}(a,b)$ denote the space of matrix functions $W(t)$ of size $2\,{\times}\,2$ with norm $\displaystyle\|W\|_{L_{2,2}^{2,2}(a,b)}=\biggl(\int_a^b\|W(t)\|\,dt\biggr)^{1/2}$. The operator $\mathbb{L}\mathbf{y}={B\mathbf{y}'+V\mathbf{y}}$ is regarded as a linear operator in the space $L_{2,2}(0,\pi)$ with domain $D(\mathbb{L})=\{\mathbf{y}\in W_1^1[0,\pi]\colon \mathbb{L}\mathbf{y}\in L_{2,2}(0,\pi),\, U_j(\mathbf{y})=0,\, j=1,2\}$. We let
$$
\begin{equation}
E(x,\lambda)=\begin{pmatrix} c_1(x,\lambda)&-s_2(x,\lambda) \\ s_1(x,\lambda)&c_2(x,\lambda) \end{pmatrix}
\end{equation}
\tag{1.3}
$$
denote the matrix of the fundamental system of solutions of equation (1.1) with boundary conditions $E(0,\lambda)=I$, where $I$ is the identity matrix, and $E_0(x,\lambda)$ denotes the matrix of the fundamental system of solutions of the nonperturbed equation $B\mathbf{y}'=\lambda\mathbf{y}$ with boundary conditions $E_0(0,\lambda)=I$. It is obvious that
$$
\begin{equation*}
E_0(x,\lambda)=\begin{pmatrix} \cos\lambda x&-\sin\lambda x \\ \sin\lambda x&\cos\lambda x \end{pmatrix}.
\end{equation*}
\notag
$$
It is well known that the entries of $E(x,\lambda)$ satisfy
$$
\begin{equation}
c_1(x,\lambda)c_2(x,\lambda)+s_1(x,\lambda)s_2(x,\lambda)=1
\end{equation}
\tag{1.4}
$$
for any $x$ and $\lambda$. Eigenvalues of the problem (1.1), (1.2) are roots of the characteristic equation where
$$
\begin{equation*}
\Delta(\lambda)= \biggl|\begin{matrix} U_1(E^{[1]}(\,\cdot\,,\lambda)) &U_1(E^{[2]}(\,\cdot\,,\lambda)) \\ U_2(E^{[1]}(\,\cdot\,,\lambda)) &U_2(E^{[2]}(\,\cdot\,,\lambda)) \end{matrix} \biggr|
\end{equation*}
\notag
$$
and $E^{[k]}(x,\lambda)$ is the $k$th column of the matrix (1.3). We let $J_{ij}$ denote the determinant formed of the $i$th and $j$th columns of the matrix $A$. We introduce the notation $J_0=J_{12}+J_{34}$, $J_1=J_{14}-J_{23}$ and $J_2=J_{13}+J_{24}$. It is well known that the characteristic function $\Delta(\lambda)$ of the problem (1.1), (1.2) is reducible to the form
$$
\begin{equation}
\begin{aligned} \, \notag \Delta(\lambda) &=J_{12}+J_{34}+J_{14}c_2(\pi,\lambda)-J_{23}c_1(\pi,\lambda)-J_{13}s_2(\pi,\lambda) -J_{24}s_1(\pi,\lambda) \\ &=\Delta_0(\lambda)+\int_0^\pi r_1(t)e^{-i\lambda t}\,dt+\int_0^\pi r_2(t)e^{i\lambda t}\,dt, \end{aligned}
\end{equation}
\tag{1.5}
$$
where
$$
\begin{equation}
\Delta_0(\lambda)=J_0+J_1\cos\pi\lambda-J_2\sin\pi\lambda=J_0+\frac{J_1+iJ_2}{2}\,e^{i\pi\lambda} +\frac{J_1-iJ_2}{2}\,e^{-i\pi\lambda}
\end{equation}
\tag{1.6}
$$
is the characteristic function of the nonperturbed problem
$$
\begin{equation}
B\mathbf{y}'=\lambda\mathbf{y}, \qquad U(\mathbf{y})=0,
\end{equation}
\tag{1.7}
$$
and the functions $r_j$ are in $L_2(0,\pi)$, $j=1,2$. The boundary conditions (1.2) can be divided into four basic types. Definition 1. The boundary conditions (1.2) are called regular if and strongly regular if, in addition, Definition 2. The boundary conditions (1.2) are called regular but not strongly regular if (1.8) holds but (1.9) does not, that is, Definition 3. The boundary conditions (1.2) are called irregular if or Definition 4. Boundary conditions (1.2) are called degenerate if We introduce the notation $c_j(\lambda)=c_j(\pi,\lambda)$ and $s_j(\lambda)=s_j(\pi,\lambda)$, $j=1,2$. We also let $\mathrm{PW}_\sigma$ denote the class of entire functions $f(z)$ of exponential type not exceeding $\sigma$ such that $\|f\|_{L_2(R)}<\infty$. It is known (see [8]) that the functions $c_j(\lambda)$ and $s_j(\lambda)$ admit the representations
$$
\begin{equation}
c_j(\lambda)= \cos\pi\lambda+g_j(\lambda)\quad\text{and} \quad s_j(\lambda)=\sin\pi\lambda+h_j(\lambda),
\end{equation}
\tag{1.10}
$$
where $g_j,h_j\in \mathrm{PW}_\pi$, $j=1,2$. Lemma (see [3]). Two entire functions $u(\lambda)$ and $v(\lambda)$ admit the representations The convergence of infinite products is understood in the sense of principal value. § 2. Characteristic functionIn this paper we consider the problem (1.1), (1.2) under the assumptions
$$
\begin{equation}
J_{13}\ne J_{24}, \quad J_{14}=0, \quad J_{13}\ne0\quad\text{and}\quad J_{24}\ne0.
\end{equation}
\tag{2.1}
$$
Relations (2.1) are satisfied by a wide class of boundary conditions, for example, conditions specified by a matrix
$$
\begin{equation*}
A=\begin{pmatrix} a_1&b_1&c_1 &d_1 \\ 0&b_2&c_2 &0 \end{pmatrix},
\end{equation*}
\notag
$$
such that $a_1d_1b_2c_2\ne0$ and $ b_2d_1\ne -a_1c_2$. These include: $\bullet$ strongly regular conditions for
$$
\begin{equation*}
A=\begin{pmatrix} 1&1&1 &1 \\ 0&1&2 &0 \end{pmatrix};
\end{equation*}
\notag
$$
$\bullet$ regular but not strongly regular conditions for
$$
\begin{equation*}
A=\begin{pmatrix} 1&1&1+\sqrt{2} &1 \\ 0&1&2 &0 \end{pmatrix};
\end{equation*}
\notag
$$
$\bullet$ irregular conditions for
$$
\begin{equation*}
A=\begin{pmatrix} 1&1&3-i &1 \\ 0&1&2 &0 \end{pmatrix};
\end{equation*}
\notag
$$
$\bullet$ degenerate conditions for
$$
\begin{equation*}
A=\begin{pmatrix} 1&1&\dfrac{1+3i}{2}&2 \\ 0&2&1 &0 \end{pmatrix}.
\end{equation*}
\notag
$$
We consider the Dirac system (1.1), (1.2), (2.1). It follows from (1.5), (1.6) that the characteristic function $\Delta(\lambda)$ of this problem reduces to
$$
\begin{equation}
\Delta(\lambda)=J_0-J_{23} c_1(\lambda)-J_{13}s_2(\lambda)-J_{24}s_1(\lambda)=\Delta_0(\lambda)+f(\lambda),
\end{equation}
\tag{2.2}
$$
where $\Delta_0(\lambda)=J_0-(J_{13}+J_{24})\sin\pi\lambda-J_{23} \cos\pi\lambda$ and $f\in \mathrm{PW}_\pi$. The converse assertion is also true. Theorem 1. For any function $f\in\mathrm{PW}_\pi$ there exists $V\in L_2(0,\pi)$ such that the characteristic function $\Delta(\lambda)$ of problem (1.1), (1.2), (2.1) with potential $V(x)$ satisfies (2.2). Proof. Let $f$ be an arbitrary function in the class $\mathrm{PW}_\pi$. The Paley-Wiener theorem and Lemma 1.3.1 in [9] imply that Let $\{\lambda_n\}$, $n\in\mathbb{Z}$, be a strictly monotonically increasing sequence of real numbers satisfying the conditions which we call conditions $(*)$.
We introduce the notation
$$
\begin{equation*}
c(\lambda)=\prod_{n=-\infty}^\infty \frac{\lambda_n-\lambda}{n-1/2}.
\end{equation*}
\notag
$$
The above lemma yields where $g\in \mathrm{PW}_\pi$. It follows from the Paley-Wiener theorem and Lemma 1.3.1 in [9] that
$$
\begin{equation*}
\lim_{|\lambda|\to\infty}e^{-\pi |{\operatorname{Im}\lambda}|}g(\lambda)=0;
\end{equation*}
\notag
$$
therefore,
$$
\begin{equation}
|c(\lambda)|\geqslant c_0e^{\pi |{\operatorname{Im}\lambda}|}
\end{equation}
\tag{2.5}
$$
for $|{\operatorname{Im}\lambda}|\geqslant M$, where $c_0>0$ and $M$ is some sufficiently large number.
Differentiating (2.4) we obtain
$$
\begin{equation}
\dot c(\lambda)=-\pi\sin\pi \lambda+\dot g(\lambda).
\end{equation}
\tag{2.6}
$$
Since the function $\dot g$ is in $\mathrm{PW}_\pi$, according to [3], we have where The last equality and the definition of $\lambda_n$ imply that where It follows that $\dot c(\lambda_n)>0$ for all even $n$ that are sufficiently large in absolute value. We can easily see that $\dot c(\lambda_n)\dot c(\lambda_{n+1})<0$ for all $n\in\mathbb{Z}$. This yields the inequality for all $n$. Note that (2.7) implies that
$$
\begin{equation}
\frac{1}{\dot c(\lambda_n)}=\frac{(-1)^n}{\pi}+\sigma_n,
\end{equation}
\tag{2.9}
$$
where
We set
$$
\begin{equation*}
\alpha=-J_{13}, \quad \beta=-J_{24}, \quad\gamma=-J_{23}\quad\text{and} \quad u_+(\lambda)=(\alpha +\beta)\sin\pi\lambda+\gamma \cos\pi\lambda+f(\lambda).
\end{equation*}
\notag
$$
Note that
$$
\begin{equation}
\alpha\ne0, \quad\beta\ne0 \quad\text{and} \quad\alpha\ne\beta.
\end{equation}
\tag{2.10}
$$
We consider the equation It has the roots
$$
\begin{equation*}
s_n^\pm=\frac{u_+(\lambda_n)\pm\sqrt{u_+^2(\lambda_n)-4\alpha\beta}}{2\alpha}.
\end{equation*}
\notag
$$
Substituting the expression for $u_+(\lambda_n)$ into the last equality we obtain
$$
\begin{equation*}
\begin{aligned} \, s_n^\pm &=\frac1{2\alpha}\bigr((\alpha +\beta)\sin\pi\lambda_n+\gamma \cos\pi\lambda_n+f(\lambda_n)\bigl) \\ &\qquad \pm\frac1{2\alpha}\bigl(((\alpha +\beta)\sin\pi\lambda_n+\gamma\cos\pi\lambda_n+f(\lambda_n))^2-4\alpha\beta\bigr)^{1/2} \\ &=\frac1{2\alpha}\bigl((\alpha +\beta)\sin\pi\lambda_n+\gamma \cos\pi\lambda_n+f(\lambda_n)\bigr) \\ &\qquad \pm\frac1{2\alpha}\bigl((\alpha-\beta)^2+2(\alpha +\beta)\sin\pi\lambda_n(\gamma \cos\pi\lambda_n+f(\lambda_n)) \\ &\qquad\qquad+(\gamma \cos\pi\lambda_n+f(\lambda_n))^2-(\alpha +\beta)^2\cos^2\pi\lambda_n\bigr)^{1/2}. \end{aligned}
\end{equation*}
\notag
$$
We introduce the following notation:
$$
\begin{equation*}
\begin{aligned} \, F^\pm(\lambda) &=\frac1{2\alpha}\bigl((\alpha +\beta)\sin\pi\lambda+\gamma \cos\pi\lambda+f(\lambda)\bigr) \\ &\qquad\pm\frac1{2\alpha}\bigl(((\alpha +\beta)\sin\pi\lambda+\gamma\cos\pi\lambda+f(\lambda))^2-4\alpha\beta\bigr)^{1/2} \\ &=\frac1{2\alpha}\bigl((\alpha +\beta)\sin\pi\lambda+\gamma \cos\pi\lambda+f(\lambda)\bigr) \\ &\qquad\pm\frac1{2\alpha}\bigl((\alpha-\beta)^2+2(\alpha +\beta)\sin\pi\lambda(\gamma \cos\pi\lambda+f(\lambda)) \\ &\qquad\qquad+(\gamma \cos\pi\lambda+f(\lambda))^2-(\alpha +\beta)^2\cos^2\pi\lambda\bigr)^{1/2} \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, F_0^\pm(\lambda) &=\frac1{2\alpha}\bigl((\alpha +\beta)\sin\pi\lambda+\gamma \cos\pi\lambda\bigr) \\ &\qquad\pm\frac1{2\alpha}\bigl(((\alpha +\beta)\sin\pi\lambda+\gamma\cos\pi\lambda)^2-4\alpha\beta\bigr)^{1/2} \\ &=\frac1{2\alpha}\bigl((\alpha +\beta)\sin\pi\lambda+\gamma \cos\pi\lambda\bigr) \\ &\qquad\pm\frac1{2\alpha}\bigl((\alpha-\beta)^2+(\alpha +\beta)\gamma \sin\pi\lambda\cos\pi\lambda+(\gamma^2-(\alpha +\beta)^2)\cos^2\pi\lambda\bigr)^{1/2}. \end{aligned}
\end{equation*}
\notag
$$
We also let $\Gamma(z,r)$ denote the circle with centre $z$ and radius $r$. Now we consider several cases. 1) $\operatorname{Im}({\beta}/{\alpha})\ne0$. Let $l_0$ be the straight line through the points $-1$ and ${\beta}/{\alpha}$ and $l$ be the line through the origin which is parallel to $l_0$. Clearly, there exists $\varepsilon_0$ such that the discs $\Gamma(-1,\varepsilon_0)$ and $\Gamma({\beta}/{\alpha},\varepsilon_0)$ lie strictly on the same side of $l$. It follows from (2.3) and (2.10) that there exists an odd positive integer $\widetilde N$ such that
$$
\begin{equation}
|F^\pm(\lambda)-F_0^\pm(\lambda)|<\frac{\varepsilon_0}2
\end{equation}
\tag{2.12}
$$
for all $\lambda=k-1/2$, $k\in \mathbb Z$, satisfying $|\lambda|\geqslant\widetilde N-1$. Inequalities (2.10) imply that there exists $\delta_0$, $0<\delta_0<1/4$, such that
$$
\begin{equation}
|F_0^\pm(\lambda)-F_0^\pm(\lambda+z)|<\frac{\varepsilon_0}2
\end{equation}
\tag{2.13}
$$
for $\lambda=\pm(\widetilde N-1/2)$ and $|z|<\delta_0$. We define a sequence $\{\lambda_n\}$, $n\in \mathbb Z$, as follows: $\lambda_n=n-1/2$ if $n\geqslant\widetilde N+1$, $\lambda_n=\widetilde N-1/2+\epsilon_n$, where $0<\epsilon_1<\dots<\epsilon_{\widetilde N}<\delta$, if $n=1,\dots, \widetilde N$, and $\lambda_n=-\lambda_{-n+1}$. It is obvious that $\{\lambda_n\}$ satisfies conditions $(*)$. It is easy to see that if $n\geqslant \widetilde N+1$ or $n\leqslant -\widetilde N$, then for odd $n$ the points $s_n^+$ are in the disc $\Gamma(1,\varepsilon_0)$, and for even $n$ the points $s_n^-$ are in the disc $\Gamma(-1,\varepsilon_0)$. Let $s_n=s_n^+$ for odd $n$ and $s_n=s_n^-$ for even $n$; thus, all points $(-1)^ns_n$ are in the disc $\Gamma(-1,\varepsilon_0)$.
Let $-\widetilde N+1\leqslant n\leqslant\widetilde N$. Then it follows from (2.12) and (2.13) that
$$
\begin{equation*}
\biggl|F^\pm(\lambda_n)-F_0^\pm\biggl(\widetilde N-\frac12\biggr)\biggr|<\varepsilon_0.
\end{equation*}
\notag
$$
We can easily see that, as $\widetilde N$ is odd, all points $s_n^+$ lie in the disc $\Gamma(1,\varepsilon_0)$, while all the $s_n^-$ lie in the disc $\Gamma({\beta}/{\alpha},\varepsilon_0)$. We set $s_n=s_n^+$ if $n$ is odd; then all points $(-1)^ns_n$ lie in $\Gamma(-1,\varepsilon_0)$. We set $s_n=s_n^-$ if $n$ is even; then the points $(-1)^ns_n$ lie in $\Gamma({\beta}/{\alpha},\varepsilon_0)$. Thus, all points $(-1)^ns_n$, $n\in \mathbb Z$, are strictly on one side of the line $l$.
2) $\operatorname{Im}(\beta/\alpha)=0$, $\operatorname{Re}(\beta/\alpha)<0$. There obviously exists a number $\varepsilon_0$ such that the discs $\Gamma(-1,\varepsilon_0)$ and $\Gamma(\beta/\alpha,\varepsilon_0)$ lie strictly to the left of the imaginary axis. Reasoning similarly to the previous case, we deduce that all points $(-1)^ns_n$ lie strictly to the left of the imaginary axis. 3) $\operatorname{Im}(\beta/\alpha)=0$ and $\operatorname{Re}(\beta/\alpha)>0$. We set $\widetilde \alpha=\alpha\overline\alpha$, $\widetilde \beta=\beta\overline\alpha$, $\widetilde \gamma=\gamma\overline\alpha$ and $\widetilde f(\lambda)=\overline\alpha f(\lambda)$; then $\widetilde \alpha>0$ and $\widetilde \beta>0$. It is easy to see that
$$
\begin{equation*}
\begin{aligned} \, s_n^\pm &=\frac1{2\widetilde\alpha}\bigl((\widetilde\alpha +\widetilde\beta)\sin\pi\lambda_n+\widetilde\gamma \cos\pi\lambda_n+\widetilde f(\lambda_n)\bigr) \\ &\qquad \pm\frac1{2\widetilde\alpha}\bigl(((\widetilde\alpha +\widetilde\beta)\sin\pi\lambda_n+\widetilde\gamma\cos\pi\lambda_n+\widetilde f(\lambda_n))^2-4\widetilde\alpha\widetilde\beta\bigr)^{1/2}. \end{aligned}
\end{equation*}
\notag
$$
We introduce the notation
$$
\begin{equation*}
R^\pm=\frac{\widetilde\gamma\pm\sqrt{\widetilde\gamma^2-4\widetilde\alpha\widetilde\beta}} {2\widetilde\alpha}.
\end{equation*}
\notag
$$
It is clear that $R^+R^->0$; consequently, $\operatorname{Im}R^+\operatorname{Im}R^-<0$ or $\operatorname{Im}R^+=0=\operatorname{Im}R^-=0$.
3.1) $\operatorname{Im}R^+\operatorname{Im}R^-<0$. In this case
$$
\begin{equation}
\widetilde\gamma^2-4\widetilde\alpha\widetilde\beta\ne0.
\end{equation}
\tag{2.14}
$$
For definiteness let $\operatorname{Im}R^+>0$; then $\operatorname{Im}R^-<0$. Inequality (2.14) shows that there exists $\varepsilon_1>0$ such that $\Gamma(R^+,\varepsilon_1)$ and $\Gamma(R^-,\varepsilon_1)$ lie strictly on opposite sides of some straight line $l$ through the origin which is distinct from the real axis. Clearly, there exists $\varepsilon_2$, where $0<\varepsilon_2<\varepsilon_1$, such that the disc $\Gamma(-1,\varepsilon_2)$ does not intersect $l$; therefore, $\Gamma(-1,\varepsilon_2)$ lies strictly on one side of $l$. It follows from (2.3) and (2.10) that there exists an odd positive integer $\widehat N$ such that
$$
\begin{equation}
|F^\pm(\lambda)-F_0^\pm(\lambda)|<\frac{\varepsilon_1}2
\end{equation}
\tag{2.15}
$$
for all $\lambda=k-1/2$, $k\in \mathbb Z$, satisfying $|\lambda|\geqslant\widehat N-1$. By virtue of (2.14) there exists $\delta_1$, $0<\delta_1<1/4$, such that
$$
\begin{equation}
|F_0^\pm(\lambda-F_0^\pm(\lambda+z)|<\frac{\varepsilon_2}2
\end{equation}
\tag{2.16}
$$
if $\lambda=\pm(\widehat N-1)$ and $|z|<\delta_1$.
We define a sequence $\{\lambda_n\}$, $n\in \mathbb Z$, as follows: $\lambda_n=n-1/2$ if $n\geqslant\widehat N+1$, $\lambda_n=\widehat N-1+\epsilon_n$, where $0<\epsilon_1<\dots<\epsilon_{\widehat N}<\delta_1$, if $n=1,\dots, \widehat N$; also let $\lambda_n=-\lambda_{-n+1}$. It is obvious that $\{\lambda_n\}$ satisfies conditions $(*)$. It is straightforward to see that if $n\geqslant \widehat N+1$ or $n\leqslant -\widehat N$, then $s_n^+$ lies in the disc $\Gamma(1,\varepsilon_1)$ if $n$ is odd, while $s_n^-$ lies in the disc $\Gamma(-1,\varepsilon_1)$ if $n$ is even. We set $s_n=s_n^+$ if $n$ is odd and $s_n=s_n^-$ if $n$ is even; thus, all points $(-1)^ns_n$ lie in $\Gamma(-1,\varepsilon_1)$. Let $-\widehat N+1\leqslant n\leqslant\widehat N$; then it follows from (2.15) and (2.16) that
$$
\begin{equation*}
|F^\pm(\lambda_n)-F_0^\pm(\widehat N-1)|<\varepsilon_1.
\end{equation*}
\notag
$$
Then the points $s_n^+$ belong to $\Gamma(R^+,\varepsilon_1)$, while the $s_n^-$ belong to $\Gamma(R^-,\varepsilon_1)$. Assume, for example, that the disc $\Gamma(R^-,\varepsilon_1)$ lies on the same side of $l$ as $\Gamma(-1,\varepsilon_0)$. We set $s_n=s_n^+$ if $n$ is odd and $s_n=s_n^-$ if $n$ is even; then all points $(-1)^ns_n$, $n\in \mathbb Z$, lie strictly on the same side of $l$.
3.2) $\operatorname{Im}R^+=\operatorname{Im}R^-=0$. Clearly, there exists $\varepsilon_3>0$ such that the disc $\Gamma(-1,\varepsilon_3)$ lies strictly below the line $l_1\colon y=-x$, whereas the disc $\Gamma\Bigl(i\sqrt{{\widetilde\beta}/{\widetilde\alpha}},\varepsilon_3\Bigr)$ lies strictly above and $\Gamma\Bigl(-i\sqrt{{\widetilde\beta}/{\widetilde\alpha}},\varepsilon_3\Bigr)$ strictly below $l_1$. It is obvious that ${\widetilde\gamma}/{\widetilde\alpha}$ is real; therefore, $\widetilde\gamma$ is real. We consider the equation
$$
\begin{equation*}
(\widetilde\alpha +\widetilde\beta)\sin t+\widetilde\gamma \cos t=0.
\end{equation*}
\notag
$$
Since $\widetilde\alpha +\widetilde\beta\ne0$, this equation has the roots $t_n=-\arctan({\widetilde\gamma}/(\widetilde\alpha +\widetilde\beta))+\pi n$, where $n\in \mathbb Z$. We introduce the notation
$$
\begin{equation*}
h_0=\frac{-\arctan(\widetilde\gamma/(\widetilde\alpha +\widetilde\beta))}{\pi}.
\end{equation*}
\notag
$$
Relations (2.3) and (2.10) imply that there exists a positive number $\widehat N$ such that
$$
\begin{equation}
|F^\pm(\lambda)-F_0^\pm(\lambda)|<\frac{\varepsilon_3}2
\end{equation}
\tag{2.17}
$$
for all $\lambda=k-1/2$, $k\in \mathbb Z$, satisfying $|\lambda|\geqslant\widetilde N-1$ and also for $\lambda =\widehat\lambda=t_{\widehat N-1}/\pi$ and that there also exists $\delta_3>0$ such that
$$
\begin{equation}
|F_0^\pm(\lambda)-F_0^\pm(\lambda+z)|<\frac{\varepsilon_3}2
\end{equation}
\tag{2.18}
$$
if $\lambda=\pm \widehat \lambda$ and $|z|<\delta_3$.
We define a sequence $\{\lambda_n\}$, $n\in \mathbb Z$, as follows: $\lambda_n=n - 1/2$ for $n \geqslant \widetilde N + 1$, $\lambda_n = \widehat\lambda + \epsilon_n$ for $1 \leqslant n \leqslant \widehat N$, where $0<\epsilon_1<\dots<\epsilon_{\widehat N}<\min(\delta,1/2-h_0)$, and $\lambda_n=-\lambda_{-n+1}$. It is clear that $\{\lambda_n\}$ satisfies conditions $(*)$. We can easily see that if $n\geqslant \widetilde N+1$ or $n\leqslant -\widetilde N$, then for odd $n$ all the $s_n^+$ lie in the disc $\Gamma(1,\varepsilon_0)$ while for even $n$ all the $s_n^-$ lie in $\Gamma(-1,\varepsilon_0)$. We set $s_n=s_n^+$ if $n$ is odd and $s_n=s_n^-$ if $n$ is even; thus, all points $(-1)^ns_n$ lie in $\Gamma(-1,\varepsilon_0)$. Let $-\widetilde N+1\leqslant n\leqslant\widetilde N$; then $s_n^+\in\Gamma(i\sqrt{\beta/\alpha}, \varepsilon_2)$ and $s_n^-\in\Gamma(-i\sqrt{\beta/\alpha}, \varepsilon_2)$. Let $s_n=s_n^+$ if $n$ is odd and $s_n=s_n^-$ if $n$ is even. Then all points $(-1)^ns_n$ lie strictly below the line $l$. Since $\{f(\lambda_n)\}\in l_2$ (see [10]), the definition of $s_n$ shows that where $\{\vartheta_n\}\in l_2$. It follows from the definition of $s_n$ and (2.8) that in all cases all points $z_n=s_n/{\dot c(\lambda_n)}$ lie strictly on the same side of some straight line passing through the origin, while it follows from (2.9) and (2.19) that where $\{\rho_n\}\in l_2$. Let $\beta_n=s_n-\sin\pi\lambda_n$; equality (2.19) implies that $\{\beta_n\}\in l_2$. We introduce the notation
$$
\begin{equation*}
h(\lambda)=c(\lambda)\sum_{n=-\infty}^\infty \frac{\beta_n}{\dot c(\lambda_n)(\lambda-\lambda_n)}.
\end{equation*}
\notag
$$
By Theorem 28 in [11] the function $h$ belongs to $\mathrm{PW}_\pi$ and $h(\lambda_n)=\beta_n$. We set $s(\lambda)=\sin\pi\lambda+h(\lambda)$. It is true that $s(\lambda_n)=s_n\ne0$; hence $s(\lambda)$ and $c(\lambda)$ have no common zeros.
Let
$$
\begin{equation*}
Y_0(x,\lambda)= \begin{pmatrix} \cos\lambda x \\ \sin\lambda x \end{pmatrix}.
\end{equation*}
\notag
$$
It was established in [7] that the system of vectors $Y_0(x,\lambda_n)$, $n\in\mathbb{Z}$, is complete in $L_{2,2}(0,\pi)$.
We set
$$
\begin{equation}
F(x,t)=-\sum_{n=-\infty}^\infty \biggl(\frac{s_n}{\dot c(\lambda_n)}\bigl(Y_0(x,\lambda_n) Y_0^T(t,\lambda_n)\bigr)+\frac{1}{\pi}Y_0\biggl(x,n-\frac12\biggr) Y_0^T\biggl(t,n-\frac12\biggr)\biggr).
\end{equation}
\tag{2.21}
$$
It follows from [12] and (2.20) that
$$
\begin{equation*}
\|F(\cdot,x)\|_{L_{2,2}^{2,2}(0,\pi)}+\|F(x,\cdot)\|_{L_{2,2}^{2,2}(0,\pi)}<C,
\end{equation*}
\notag
$$
where $C$ is independent of $x$.
We prove that for each $x\in[0,\pi]$ the homogeneous equation
$$
\begin{equation}
\mathbf{f}^T(t)+\int_0^x\mathbf{f}^T(s)F(s,t)\,ds=0,
\end{equation}
\tag{2.22}
$$
where $\mathbf{f}(t)=\operatorname{col}(f_1(t),f_2(t))$, $\mathbf{f}\in L_{2,2}(0,x)$ and $\mathbf{f}(t)=0$ for $x<t\leqslant\pi$, has only the trivial solution. Multiplying (2.22) by $\overline{\mathbf{f}^T(t)}$ and integrating the resulting equation over the interval $[0,x]$ we obtain
$$
\begin{equation}
\|\mathbf{f}\|^2_{L_{2,2}(0,x)}+\int_0^x\biggl\langle\int_0^x\mathbf{f}^T(s)F(s,t)\,ds, \overline{\mathbf{f}^T(t)}\biggr\rangle\, dt=0.
\end{equation}
\tag{2.23}
$$
Based on (2.21), we infer via simple calculations that
$$
\begin{equation}
\begin{aligned} \, \notag &\mathbf{f}^T(s)F(s,t) \\ \notag &=-\biggl\{\sum_{n=-\infty}^\infty \biggl\{z_n\bigl[f_1(s)\cos\lambda_ns\cos\lambda_nt+f_2(s)\sin\lambda_ns\cos\lambda_nt, \\ \notag &\qquad\qquad f_1(s)\cos\lambda_ns\sin\lambda_nt +f_2(s)\sin\lambda_ns\sin\lambda_nt\bigr] \\ \notag &\qquad +\frac{1}{\pi}\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s\cos \biggl(n-\frac12\biggr)t+f_2(s)\sin \biggl(n-\frac12\biggr)s\cos \biggl(n-\frac12\biggr)t, \\ \notag &\qquad\qquad f_1(s)\cos \biggl(n-\frac12\biggr)s\sin \biggl(n-\frac12\biggr)t +f_2(s)\sin \biggl(n-\frac12\biggr)s\sin \biggl(n-\frac12\biggr)t\biggr]\biggr\}\biggr\} \\ \notag &=-\biggl\{\sum_{n=-\infty}^\infty \biggl\{z_n\bigl[f_1(s)\cos\lambda_ns\cos\lambda_nt+f_2(s)\sin\lambda_ns\cos\lambda_nt\bigr] \\ \notag &\qquad+\frac{1}{\pi}\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s\cos \biggl(n-\frac12\biggr)t+f_2(s)\sin \biggl(n-\frac12\biggr)s\cos \biggl(n-\frac12\biggr)t\biggr], \\ \notag &\qquad\qquad z_n\bigl[f_1(s)\cos\lambda_ns\sin\lambda_nt+f_2(s)\sin\lambda_ns\sin\lambda_nt\bigr] \\ &\qquad+\frac{1}{\pi} \biggl[f_1(s)\cos \biggl(n-\frac12\biggr) s\sin\biggl(n-\frac12\biggr)t +f_2(s)\sin \biggl(n-\frac12\biggr)s\sin \biggl(n-\frac12\biggr)t\biggr]\biggr\}\biggr\}. \end{aligned}
\end{equation}
\tag{2.24}
$$
Substituting the right-hand side of (2.24) into the second term on the left-hand side of (2.23), transforming the iterated integrals into products of integrals, and using the fact that all numbers $\lambda_n$ are real we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_0^x\biggl\langle \int_0^x \mathbf{f}^T(s)F(s,t)\,ds,\overline{\mathbf{f}^T(t)}\bigg\rangle \,dt \\ &=-\biggl\{\sum_{n=-\infty}^\infty\int_0^x\biggr(\int_0^x\biggl\{z_n\bigl[f_1(s) \cos\lambda_ns\cos\lambda_nt+f_2(s)\sin\lambda_ns\cos\lambda_nt\bigr] \\ &\quad\qquad +\frac{1}{\pi}\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s\cos \biggl(n-\frac12\biggr)t \\ &\quad\qquad +f_2(s)\sin \biggl(n-\frac12\biggr)s\cos \biggl(n-\frac12\biggr)t\biggr]\biggr\}\,ds\biggr)\overline{f_1(t)}\,dt \\ &\quad +\sum_{n=-\infty}^\infty\int_0^x\biggr(\int_0^x \biggl\{z_n\bigl[f_1(s)\cos\lambda_ns\sin\lambda_nt+f_2(s)\sin\lambda_ns\sin\lambda_nt\bigr] \\ &\quad\qquad +\frac{1}{\pi} \biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s\sin \biggl(n-\frac12\biggr)t \\ &\quad\qquad+f_2(s)\sin \biggl(n-\frac12\biggr)s\sin \biggl(n-\frac12\biggr)t\biggr]\biggr\}ds\biggr)\overline{f_2(t)}\,dt\biggr\} \\ &=-\biggl\{\sum_{n=-\infty}^\infty\biggr(\int_0^x z_n\bigl[f_1(s)\cos\lambda_ns +f_2(s)\sin\lambda_ns\bigr]\,ds\int_0^x\cos\lambda_nt\overline{f_1(t)}\,dt \\ &\quad\qquad +\frac{1}{\pi}\int_0^x\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s +f_2(s)\sin \biggl(n-\frac12\biggr)s\biggr]\,ds\biggr) \\ &\quad\qquad\qquad\times\int_0^x\cos \biggl(n-\frac12\biggr)t\overline{f_1(t)}\,dt \\ &\quad\qquad +\sum_{n=-\infty}^\infty\biggr(\int_0^x z_n\bigl[f_1(s)\cos\lambda_ns+f_2(s)\sin\lambda_ns\bigr]\,ds \int_0^x\sin\lambda_nt\overline{f_2(t)}\,dt \\ &\quad\qquad\qquad +\frac{1}{\pi} \int_0^x\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s +f_2(s)\sin \biggl(n-\frac12\biggr)s\biggr]\,ds \\ &\quad\qquad\qquad\qquad\times\int_0^x\sin \biggl(n-\frac12\biggr)t\overline{f_2(t)}\,dt \biggr)\biggr\} \\ &=-\biggl\{\sum_{n=-\infty}^\infty\biggr(\int_0^x z_n[f_1(s)\cos\lambda_ns +f_2(s)\sin\lambda_ns]\,ds\int_0^x\cos\lambda_nt\overline{f_1(t)}\,dt \\ &\quad\qquad+\int_0^x z_n\bigl[f_1(s)\cos\lambda_ns+f_2(s)\sin\lambda_ns\bigr]\,ds\int_0^x \sin\lambda_nt\overline{f_2(t)}\,dt\biggr) \\ &\quad\qquad +\frac{1}{\pi}\sum_{n=-\infty}^\infty\biggr(\int_0^x\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s \\ &\quad\qquad +f_2(s)\sin \biggl(n-\frac12\biggr)s\biggr]\,ds\int_0^x \cos\biggl(n-\frac12\biggr)t\overline{f_1(t)}\,dt \\ &\quad\qquad+\int_0^x\biggl[f_1(s)\cos \biggl(n-\frac12\biggr)s +f_2(s)\sin \biggl(n-\frac12\biggr)s\biggr]\,ds \\ &\quad\qquad\qquad\times\int_0^x\sin \biggl(n-\frac12\biggr)t\overline{f_2(t)}\,dt \biggr)\biggr\} \\ &=-\biggl\{\sum_{n=-\infty}^\infty\biggr(\int_0^x z_n[f_1(t)\cos\lambda_nt +f_2(t)\sin\lambda_nt]\,dt\int_0^x\cos\lambda_nt\overline{f_1(t)}\,dt \\ &\quad\qquad+\int_0^x z_n\bigl[f_1(t)\cos\lambda_nt+f_2(t)\sin\lambda_nt\bigr]\,dt \int_0^x\sin\lambda_nt\overline{f_2(t)}\,dt\biggr) \\ &\quad\qquad+\frac{1}{\pi}\sum_{n=-\infty}^\infty\biggr(\int_0^x\biggl[f_1(t)\cos \biggl(n-\frac12\biggr)t \\ &\quad\qquad+f_2(t)\sin \biggl(n-\frac12\biggr)t\biggr]\,dt\int_0^x\cos \biggl(n-\frac12\biggr)t\overline{f_1(t)}\,dt \\ &\quad\qquad+\int_0^x\biggl[f_1(t)\cos nt+f_2(t)\sin \biggl(n-\frac12\biggr)t\biggr]\,dt\int_0^x\sin \biggl(n-\frac12\biggr)t\overline{f_2(t)}\,dt\biggr)\biggr\} \\ &=-\biggl\{\sum_{n=-\infty}^\infty z_n\int_0^x [f_1(t)\cos\lambda_nt+f_2(t)\sin\lambda_nt]\,dt \\ &\quad\qquad\qquad\times \int_0^x\bigl[\overline{f_1(t)}\cos\lambda_nt+\overline{f_2(t)}\sin\lambda_nt\bigr]\,dt \\ &\quad\qquad+\sum_{n=-\infty}^\infty \frac{1}{\pi}\int_0^x\biggl[f_1(t)\cos \biggl(n-\frac12\biggr)t+f_2(t)\sin \biggl(n-\frac12\biggr)t\biggr]\,dt \\ &\quad\qquad\qquad\times \int_0^x\biggl[\overline{f_1(t)}\cos \biggl(n-\frac12\biggr)t+\overline{f_2(t)}\sin \biggl(n-\frac12\biggr)t\biggr]\,dt\biggr\} \\ &=-\sum_{n=-\infty}^\infty z_n\biggl|\int_0^x\langle\mathbf{f}(t),Y_0(t,\lambda_n)\rangle \,dt\biggr|^2-\sum_{n=-\infty}^\infty \frac{1}{\pi}\biggl|\int_0^x\biggl\langle \mathbf{f}(t),Y_0\biggl(t,\biggl(n-\frac12\biggr)\biggr)\biggr\rangle \,dt\biggr|^2. \end{aligned}
\end{equation*}
\notag
$$
Parseval’s identity implies that
$$
\begin{equation*}
\|\mathbf{f}\|^2_{L_{2,2}(0,x)} =\sum_{n=-\infty}^\infty \frac{1}{\pi}\biggl|\int_0^x\biggl\langle \mathbf{f}(t),Y_0\biggl(t,\biggl(n-\frac12\biggr)\biggr)\biggr\rangle \,dt\biggr|^2;
\end{equation*}
\notag
$$
hence
$$
\begin{equation}
\sum_{n=-\infty}^\infty z_n\biggl|\int_0^x\langle \mathbf{f}(t),Y_0(t,\lambda_n)\rangle\, dt\biggr|^2=0.
\end{equation}
\tag{2.25}
$$
Since all numbers $z_n$ are strictly on one side of some straight line through the origin, it follows from (2.25) that $\displaystyle\int_0^x\langle \mathbf{f}(t),Y_0(t,\lambda_n)\rangle \,dt=0$. This fact and the completeness in $L_{2,2}(0,\pi)$ of the vector system $\{Y_0(t,\lambda_n)\}$ yield $\mathbf{f}(t)\equiv0$.
We infer from this (see [8]) that the functions $c(\lambda)$ and $-s(\lambda)$ are the entries in the first row of the monodromy matrix
$$
\begin{equation*}
\widetilde U(\pi,\lambda)= \begin{pmatrix} \widetilde c_1(\pi,\lambda) &-\widetilde s_2(\pi,\lambda) \\ \widetilde s_1(\pi,\lambda) &\widetilde c_2(\pi,\lambda) \end{pmatrix}
\end{equation*}
\notag
$$
of problem (1.1), (1.2), (2.1) with some potential $\widetilde V\in L_2(0,\pi)$, that is,
$$
\begin{equation}
c(\lambda)=\widetilde c_1(\pi,\lambda)\quad\text{and} \quad s(\lambda)=\widetilde s_2(\pi,\lambda).
\end{equation}
\tag{2.26}
$$
From (2.2) we derive that the corresponding characteristic determinant is given by
$$
\begin{equation*}
\widetilde\Delta(\lambda)=J_0+\gamma \widetilde c_1(\lambda)+\alpha\widetilde s_2(\lambda)+\beta \widetilde s_1(\lambda)=J_0+(\alpha +\beta)\sin\pi\lambda+\gamma \cos\pi\lambda+\widetilde f(\lambda),
\end{equation*}
\notag
$$
where $\widetilde f\in \mathrm{PW}_\pi$. Relations (1.4), (2.11) and (2.26) imply that
$$
\begin{equation*}
\begin{aligned} \, \widetilde\Delta(\lambda_n) &=J_0+\alpha\widetilde s_2(\pi,\lambda_n)+\beta \widetilde s_1(\pi,\lambda_n)=J_0+\frac{\beta}{\widetilde s_2(\pi,\lambda_n)}+\alpha\widetilde s_2(\pi,\lambda_n) \\ &=J_0+\frac{\beta}{s(\lambda_n)}+\alpha s(\lambda_n)=J_0+u_+(\lambda_n)=U(\lambda_n). \end{aligned}
\end{equation*}
\notag
$$
It follows that
$$
\begin{equation*}
\Phi(\lambda)=\frac{U(\lambda)-\widetilde\Delta(\lambda)}{c(\lambda)}=\frac{f(\lambda)-\widetilde f(\lambda)}{c(\lambda)}
\end{equation*}
\notag
$$
is an entire function on the whole complex plane. Since
$$
\begin{equation}
|f(\lambda)-\widetilde f(\lambda)|<c_1e^{\pi|{\operatorname{Im}\lambda}|},
\end{equation}
\tag{2.27}
$$
from (2.5) we obtain $|\Phi(\lambda)|\leqslant c_2$ for $|{\operatorname{Im}\lambda}|\geqslant M$. We let $H$ denote the union of the vertical line segments $\{z\colon |{\operatorname{Re} z}|=n,\ |{\operatorname{Im}\lambda}|\leqslant M\}$, where $|n|=N_0+1,N_0+2, \dots$ . Since $c(\lambda)$ is a sine-type function (see [13]), we have $|c(\lambda)|>\delta>0$ if $\lambda\in H$. We deduce from the last inequality, (2.27) and the maximum modulus principle for analytic functions that $|\Phi(\lambda)|<c_3$ in the strip $|{\operatorname{Im}\lambda}|\leqslant M$; therefore, the function $\Phi(\lambda)$ is bounded on the whole complex plane and, by Liouville’s theorem, is a constant. Let $|{\operatorname{Im}\lambda}|=M$. Then it follows from (2.3) that $\lim_{|\lambda|\to\infty}(f(\lambda)-\widetilde f(\lambda))=0$; hence $\Phi(\lambda)\equiv0$. Thus, $U(\lambda)\equiv\widetilde\Delta(\lambda)$.
Theorem 1 is proved. § 3. SpectrumIn addition to (2.1), we assume that the boundary conditions (1.2) are regular; thus, according to (1.8), we have Theorem 2. For a set $\Lambda$ to be the spectrum of a Dirac operator (1.1), (1.2), (2.1) with complex-valued potential $V\in L_2(0,\pi)$, it is necessary and sufficient that it consists of two sequences of eigenvalues $\lambda_{n,j}$ satisfying the condition where the $z_j$ are the roots of the equation the branch of the logarithm takes values in the strip $\operatorname{Im} \lambda\in(-\pi,\pi]$, and $\{\varepsilon_{n,j}\}\in l_2$, $j=1,2$, $n\in \mathbb{Z}$. Proof. Necessity was proved in [14].
Sufficiency. Assume that two sequences $\lambda_{n,j}$ satisfy condition (3.1). Clearly, there exists a constant $M$ such that
$$
\begin{equation}
\sup|\varepsilon_{n,j}|<M\quad\text{and} \quad \sum_{j=1,2,\, n\in \mathbb Z}|\varepsilon_{n,j}|^2<M.
\end{equation}
\tag{3.2}
$$
It follows from [14] that the eigenvalues of problem (1.1), (1.2), (2.1) with zero potential $V(x)$ are given by the formulae $j=1,2$, $n\in \mathbb Z$. Let $t_j={\ln z_j}/(i\pi)$. By Hadamard’s theorem the characteristic function $\Delta_0(\lambda)$ of the nonperturbed problem (1.7), (2.1) has the form
$$
\begin{equation*}
\Delta_0(\lambda)=J_0-(J_{13}+J_{24})\sin\pi\lambda-J_{23}\cos\pi\lambda=c\lambda^m\prod_{(n,j)\in T_1}\frac{\lambda_{n,j}^0-\lambda}{\lambda_{n,j}^0},
\end{equation*}
\notag
$$
where $c\ne0$ and $T_1$ is the set of pairs $(n,j)$ such that $\lambda_{n,j}^0\ne0$. We obviously have
$$
\begin{equation}
|\Delta_0(\lambda)|< c_1e^{\pi |{\operatorname{Im}\lambda}|}.
\end{equation}
\tag{3.3}
$$
Let $T_0$ be the set of pairs $(n,j)$ such that $\lambda_{n,j}^0=0$ and $m$ be the cardinality of $T_0$. It follows from [14] that $0\leqslant m\leqslant2$. We also set $T=T_0\cup T_1$ and
$$
\begin{equation*}
\Delta(\lambda)=c\prod_{(n,j)\in T_0}(\lambda-\lambda_{n,j})\prod_{(n,j)\in T_1}\frac{\lambda_{n,j}-\lambda}{\lambda_{n,j}^0}.
\end{equation*}
\notag
$$
Let $f(\lambda)=\Delta(\lambda)-\Delta_0(\lambda)$. We investigate the properties of the function $f(\lambda)$ basing on the following results.
Proposition 1. The function $f(\lambda)$ is an entire function of exponential type not exceeding $\pi$. Proof. We let $\Gamma$ denote the union of discs $\Gamma(2n+t_j,1/4)$, $n\in \mathbb Z$. If $\lambda\notin\Gamma$, then
$$
\begin{equation}
f(\lambda)=-\Delta_0(\lambda)\biggl(1-\frac{\Delta(\lambda)}{\Delta_0}\biggr) =-\Delta_0(\lambda)(1-\phi(\lambda)),
\end{equation}
\tag{3.4}
$$
where
$$
\begin{equation*}
\phi(\lambda)=\prod_{(n,j)\in T_0}\biggl(1-\frac{\lambda_{n,j}}{\lambda}\biggr) \prod_{(n,j)\in T_1}\biggl(1+\frac{\varepsilon_{n,j}}{\lambda_{n,j}^0-\lambda}\biggr) =\prod_{(n,j)\in T}\biggl(1+\frac{\varepsilon_{n,j}}{2n+t_j-\lambda}\biggr).
\end{equation*}
\notag
$$
We estimate the function $\phi(\lambda)$. Let $\alpha_{n,j}(\lambda)={\varepsilon_{n,j}}/(2n+t_j-\lambda)$. The inequalities (3.2) imply that
It is straightforward to see that for all $|n|>n_0$, where $n_0$ is sufficiently large, we have for any $\lambda\notin\Gamma$. If $|n|\leqslant n_0$, then (3.6) is valid for all sufficiently large $|\lambda|$; therefore, this inequality holds for all $|\lambda|\geqslant C_0$. We derive from (3.5), (3.6) and the elementary inequality which holds for $|z|\leqslant1/4$, that Below we choose the branch of $\ln(1+z)$ that vanishes at $z=0$. According to [15], Ch. V, § 1.72, we can rewrite the last relation as
$$
\begin{equation}
|\phi(\lambda)|\leqslant\prod_T|1+\alpha_{n,j}(\lambda)|\leqslant e^{c_4}.
\end{equation}
\tag{3.8}
$$
We infer from (3.3), (3.4) and (3.8) that
$$
\begin{equation}
|f(\lambda)|<c_5e^{\pi |{\operatorname{Im}\lambda}|}
\end{equation}
\tag{3.9}
$$
outside the domain $\Gamma'=\Gamma\cup\{|\lambda|<C_0\}$.
We set
$$
\begin{equation*}
D=\bigcup_{(n,j)\in T}\biggl[2n+\operatorname{Re}t_j-\frac 14,\, 2n+\operatorname{Re}t_j+\frac14\biggr]\quad\text{and} \quad D_0=(0,2)\setminus D.
\end{equation*}
\notag
$$
It is easy to see that $D_0$ is a union of finitely many line segments such that the sum of their lengths is at least $1$. Let $x_0$ be the midpoint of one of these segments. Then all points $x_0 + 2k$, $k\in \mathbb Z$, lie outside $D$. In particular, (3.9) is valid for $\lambda$ on the straight lines $\operatorname{Im}\lambda=\pm \widehat C_0$, where $\widehat C_0=C_0+|t_1|+|t_2|+1$, or vertical line segments with endpoints $(x_0+2k, -\widehat C_0)$ and $(x_0+2k, \widehat C_0)$, where $|2k-1|>C_0$, $k\in\mathbb{Z}$. By the maximum principle, (3.9) holds on the whole complex plane; hence $f(\lambda)$ is an entire function of exponential type not exceeding $\pi$.
Proposition 1 is proved. We set
$$
\begin{equation*}
W(\lambda)=\ln\phi(\lambda)=\sum_{(n,j)\in T}\ln(1+\alpha_{n,j}(\lambda));
\end{equation*}
\notag
$$
then
$$
\begin{equation}
f(\lambda)=-\Delta_0(\lambda)\bigl(1-e^{W(\lambda)}\bigr).
\end{equation}
\tag{3.10}
$$
We estimate the function $W(\lambda)$ if $\lambda\notin\Gamma'$. It follows from (3.2), (3.6) and (3.7) that
$$
\begin{equation*}
\begin{aligned} \, |W(\lambda)| &\leqslant\sum_{(n,j)\in T}|{\ln(1+\alpha_{n,j}(\lambda))}| \\ &\leqslant\frac{2M}{|\lambda|}+ \sum_{n=-\infty}^\infty \biggl(\frac{|\varepsilon{_n,1}|^2+|\varepsilon_{n,2}|^2}{10M}+ \frac{10M}{|2n-\lambda|^2}\biggr) \\ &\leqslant\frac{2M}{|\lambda|}+\frac{1}{10}+20M\sum_{n=0}^\infty\frac{1}{n^2+ |{\operatorname{Im}\lambda}|^2} \\ &\leqslant\frac{2M}{|\lambda|}+\frac{1}{10}+20M\biggl(\frac{2}{|{\operatorname{Im}\lambda}|^2} +\int_1^\infty\frac{dx}{x^2+|{\operatorname{Im}\lambda}|^2}\biggr) \\ &\leqslant\frac{2M}{|{\operatorname{Im}\lambda}|} +\frac{1}{10}+20M\biggl(\frac{2}{|{\operatorname{Im}\lambda}|^2} +\frac{\pi}{2|{\operatorname{Im}\lambda}|}\biggr). \end{aligned}
\end{equation*}
\notag
$$
The last inequality yields provided that $|{\operatorname{Im}\lambda}|\geqslant M_1=10(\pi+2+22M)+\widehat C_0$. From the elementary inequality which is valid for $|z|\leqslant1/4$, we derive that $|1-e^{W(\lambda)}|\leqslant2|W(\lambda)|$. The last inequality, (3.3) and (3.10) imply that for $\lambda\in l$, where $l$ is the line $\operatorname{Im}\lambda=M_1$. We prove that The elementary inequality $|{\ln(1+z)-z}|\leqslant|z|^2$, which is true for $|z|\leqslant1/2$, yields the relation where $|r(z)|\leqslant|z|^2$; hence where
$$
\begin{equation*}
S_1(\lambda)=\sum_{n=-\infty}^\infty (\alpha_{n,1}(\lambda)+\alpha_{n,2}(\lambda))\quad\text{and} \quad |S_2(\lambda)|\leqslant\sum_{n=-\infty}^\infty (|\alpha_{n,1}(\lambda)|^2+|\alpha_{n,2}(\lambda)|^2).
\end{equation*}
\notag
$$
It is obvious that
$$
\begin{equation}
|W(\lambda)|\leqslant|S_1(\lambda)|+|S_2(\lambda)|.
\end{equation}
\tag{3.14}
$$
We set $(m=1,2)$. First we consider the integral $I_1$. It follows from [16] that
$$
\begin{equation}
\begin{aligned} \, \notag I_1 &=\int_l\,\biggl|\sum_{n=-\infty}^\infty \biggl(\frac{\varepsilon_{n,1}}{2n+t_1-\lambda} +\frac{\varepsilon_{n,2}}{2n+t_2-\lambda}\biggr)\biggr|^2\,d\lambda \\ \notag &\leqslant 2\biggl(\int_l\,\biggl|\sum_{n=-\infty}^\infty \frac{\varepsilon_{n,1}}{2n+t_1-\lambda}\biggr|^2\,d\lambda+ \int_l\,\biggl|\sum_{n=-\infty}^\infty \frac{\varepsilon_{n,2}}{2n+t_2-\lambda}\biggr|^2\,d\lambda\biggr) \\ &=2\biggl(\int_{l_1}\biggl|\sum_{n=-\infty}^\infty \frac{\varepsilon_{n,1}}{2n-\lambda}\biggr|^2\,d\lambda+ \int_{l_2}\biggl|\sum_{n=-\infty}^\infty \frac{\varepsilon_{n,2}}{2n-\lambda}\biggr|^2\,d\lambda\biggr) <\infty, \end{aligned}
\end{equation}
\tag{3.15}
$$
where the $l_j$ are the lines $\operatorname{Im}\lambda=M_1-\operatorname{Im}t_j$, $j=1,2$.
We can clearly see that
$$
\begin{equation*}
|S_2(\lambda)|\leqslant\sum_{n=-\infty}^\infty \frac{|\varepsilon_{n,1}|^2}{|2n+t_1-\lambda|^2}+\sum_{n=-\infty}^\infty \frac{|\varepsilon_{n,2}|^2}{|2n+t_2-\lambda|^2}\leqslant c_7;
\end{equation*}
\notag
$$
therefore,
$$
\begin{equation}
\begin{aligned} \, \notag I_2 &\leqslant c_7 \int_l\biggl(\sum_{n=-\infty}^\infty \frac{|\varepsilon_{n,1}|^2}{|2n+t_1-\lambda|^2}+\sum_{n=-\infty}^\infty \frac{|\varepsilon_{n,2}|^2}{|2n+t_2-\lambda|^2}\biggr)\,d\lambda \\ & \leqslant c_8\sum_{n=-\infty}^\infty (|\varepsilon_{n,1}|^2+|\varepsilon_{n,2}|^2) \int_{\widetilde l}\frac{d\lambda}{|2n-\lambda|^2} \leqslant c_9\sum_{n=-\infty}^\infty (|\varepsilon_{n,1}|^2+|\varepsilon_{n,2}|^2)\leqslant c_{10}, \end{aligned}
\end{equation}
\tag{3.16}
$$
where $\widetilde l=l_1\cup l_2$. Relations (3.14)–(3.16) imply (3.13). It follows from (3.12), (3.13), and [12], Ch. 3, § 3.2.2, that
Proposition 2 is proved. Thus, the function $\Delta(\lambda)$ satisfies all assumptions of Theorem 1. Hence there exists a potential $V\in L_2(0,\pi)$ such that the spectrum of the corresponding problem (1.1), (1.2), (2.1) is as specified by (3.1). Theorem 2 is proved. AcknowledgementsThe author is grateful to the referee for useful remarks. 
Citation in format AMSBIB
\Bibitem{Mak23} |
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