On the spectrum of non-self-adjoint Dirac operators with quasi-periodic boundary conditions

Abstract

In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

1. Introduction

The spectral theory of boundary value problems for first-order systems of ordinary differential equations of the form

(1.1)\begin{equation} \frac{1}{i}B\frac{{\rm d}y}{{\rm d}x}+Q(x)y=\lambda y,\quad y={\rm col}(y_1,\ldots,y_n),\end{equation}

where $B$ is a nonsingular diagonal $n\times n$ matrix,

\[ B={\rm diag}(b_1^{{-}1}I_{n_1},\ldots,b_r^{{-}1}I_{n_r})\in \mathbb{C}^{n\times n},\quad n=n_1+\cdots n_r, \]

with complex entries $b_j\ne b_k$, and $Q(x)$ is a potential matrix takes its origin in the paper by Birkhoff and Langer [2]. Afterwards their investigations were developed in many directions. In particular, one of the important classes of inverse spectral problems is the problem of recovering a system of differential equations from spectral data. The solution of such problems is considered in many papers (see [14, 17, 21, 32–38] and the references therein).

The main aim of the present article is to find necessary and sufficient conditions for solvability of inverse spectral problems for one-dimensional Dirac operators on a finite interval under possibly least restrictive assumptions on their potentials. We will consider canonical Dirac system

(1.2)\begin{equation} B\mathbf{y}'+V\mathbf{y} =\lambda\mathbf{y},\end{equation}

where $\mathbf {y}={\rm col}(y_1(x),\,y_2(x))$,

\[ B=\begin{pmatrix} 0 & 1\\ - 1 & 0 \end{pmatrix},\quad V(x)=\begin{pmatrix} p(x) & q(x)\\ q(x) & -p(x) \end{pmatrix}, \]

the complex-valued functions $p,\, q\in L_2(0,\,\pi )$, with two-point boundary conditions

(1.3)\begin{equation} U(\mathbf{y})=C\mathbf{y}(0)+D\mathbf{y}(\pi)=0,\end{equation}

where

\[ C=\begin{pmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{pmatrix},\quad D=\begin{pmatrix} a_{13} & a_{14}\\ a_{23} & a_{24} \end{pmatrix}, \]

the coefficients $a_{ij}$ are arbitrary complex numbers, and rows of the matrix

\[ A=\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24} \end{pmatrix} \]

are linearly independent.

Inverse self-adjoint problems (1.2), (1.3) have been studied in detail. In the cases of the Dirichlet and the Neumann boundary conditions reconstruction of a continuous potential from two spectra was carried out in [7], from one spectrum and the norming constants in [6], and from the spectral function in [16]. The analogous results for Dirac operator with summable potentials were established in [1]. The case of more general separated boundary conditions was considered in [4]. In the case of unseparated boundary conditions (including periodic, antiperiodic and quasi-periodic conditions) the indicated problem was solved in [20, 22–25]. In non-self-adjoint case the problem of reconstructing the potential $V(x)$ from spectral data is much more complicated, since many methods successfully used to study self-adjoint operators are inapplicable. For example, the characterization of the spectra of the periodic (antiperiodic) problem for operator (1.2) with real coefficients is given in [18] in terms of special conformal mappings, which do not exist for complex-valued potentials. The property that the eigenvalues of corresponding Dirichlet problem and Neumann problem are interlaced, which is often used to prove the solvability of the basic equation, loses its meaning in the complex case. Non-self-adjoint inverse problems for system (1.2) with different types of boundary conditions with sufficiently smooth coefficients, which, however, could have singularities were investigated in [3, 9, 27, 31].

Questions of uniqueness in inverse problems for operators of type (1.1) on a finite interval were studied in several papers. In particular, the uniqueness of the inverse problem for general Dirac-type systems ($B=B^{*}$) of order $2n$ was established in [15, 16]. The most complete uniqueness result on general first-order systems (1.1) and (1.2) on a finite interval has been obtained recently in [17]. Also the solution to the inverse spectral problem (from the spectral matrix function) for Dirac-type operators on the axis and semiaxis was obtained in [11]. New inverse approach based on the A-function concept proposed by Gesztesy and Simon to Schrodinger operator has been recently extended in [8] to Dirac systems on the semiaxis.

In the present paper, we consider system (1.2), where complex-valued functions $p,\, q\in L_2(0,\,\pi )$ $(V\in L_2)$ with quasi-periodic boundary conditions

(1.4)\begin{equation} \mathbf{y}(0)={\rm e}^{it}\mathbf{y}(\pi),\end{equation}

where $t\in \mathbb {C},\, t\ne \pi k$, $k\in Z$. Section 2 contains some basic facts and definitions related to the studied problems. In § 3 by using a modified version of the Gelfand–Levitan–Marchenko method we prove solvability of the basic equation and establish necessary and sufficient conditions for an entire function to be the characteristic determinant of problems (1.2), (1.4). Furthermore, we obtain necessary and sufficient conditions for a set of complex numbers to be the spectrum of the mentioned problem.

2. Preliminaries

In what follows, we introduce the Euclidean norm $\|f\|=(|f_1|^{2}+|f_2|^{2})^{1/2}$ for vectors $f={\rm col}(f_1,\,f_2)\in \mathbb {C}^{2}$, and set $\langle f,\,g\rangle =f_1g_1+f_2g_2$. If $W$ is $2\times 2$ matrix, then we set $\|W\|= \sup _{\|f\|=1}\|Wf\|$ and denote by $L_{2,2}(a,\,b)$ and $L_{2,2}^{2,2}(a,\,b)$, respectively, the spaces of 2-coordinate vector functions $f(t)={\rm col}(f_1(t),\,f_2(t))$ and $2\times 2$ matrix functions $W(t)$ with finite norms

\[ \|f\|_{L_{2,2}(a,b)}=\left(\int_a^{b}\|f(t)\|^{2}{\rm d}t\right)^{1/2}, \quad \|W\|_{L_{2,2}^{2,2}(a,b)}=\left(\int_a^{b}\|W(t)\|^{2}{\rm d}t\right)^{1/2}. \]

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 the domain $D(\mathbb {L})=\{\mathbf {y}\in W_1^{1}[0,\,\pi ]\times W_1^{1}[0,\,\pi ]:\, \mathbb {L}\mathbf {y}\in L_{2,2}(0,\,\pi )$, $U_j(\mathbf {y})=0$ $(j=1,\,2)\}$.

Denote by

(2.1)\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}

the matrix of the fundamental solution system to equation (1.2) with boundary condition $E(0,\,\lambda )=I$, where $I$ is the unit matrix, and by $E_0(x,\,\lambda )$ the fundamental solution system to the equation $B\mathbf {y}'=\lambda \mathbf {y}$ with boundary condition $E_0(0,\,\lambda )=I$. Obviously,

\[ E_0(x,\lambda)=\begin{pmatrix} \cos\lambda x & -\sin\lambda x\\ \sin\lambda x & \cos\lambda x \end{pmatrix}. \]

Denote the second column of the matrix $E_0(x,\,\lambda )$ by

\[ Y_0(x,\lambda)= \begin{pmatrix} -\sin\lambda x\\ \cos\lambda x \end{pmatrix}. \]

It is well known that the entries of the matrix $E(x,\,\lambda )$ are related by the identity

(2.2)\begin{equation} c_1(x,\lambda)c_2(x,\lambda)+s_1(x,\lambda)s_2(x,\lambda)=1,\end{equation}

which is valid for any $x,\, \lambda$. The matrix $E(\pi,\,\lambda )$ is called the monodromy matrix of operator $\mathbb {L}\mathbf {y}$. For its entries we introduce the notation $c_j(\lambda )=c_j(\pi,\,\lambda )$, $s_j(\lambda )=s_j(\pi,\,\lambda )$, $j=1,\,2$. We denote also the class of entire functions $f(z)$ of exponential type $\le \sigma$ such that $\|f\|_{L_2(R)}<\infty$ by $PW_\sigma$. It is known [29] that the functions $c_j(\lambda ),\,s_j(\lambda )$ admit the representation

(2.3)\begin{equation} c_j(\lambda)= \cos\pi\lambda+g_j(\lambda), \quad s_j(\lambda)=\sin\pi\lambda+h_j(\lambda),\end{equation}

where $g_j,\,h_j\in PW_\pi$, $j=1,\,2$. For functions of type (2.3) the following statement is true:

Lemma 2.1 [20]

Entire functions $u(\lambda )$ and $v(\lambda )$ admit the representations

(2.4)\begin{equation} u(\lambda)=\sin\pi \lambda+h(\lambda),\quad v(\lambda)=\cos\pi\lambda+g(\lambda),\end{equation}

where $h,\,g\in PW_\pi,$ if and only if

\[ u(\lambda)={-}\pi(\lambda_0 -\lambda)\mathop{{\prod}'}_{n={-}\infty}^{\infty}\frac{\lambda_n-\lambda}{n}, \]

where $\lambda _n=n+\epsilon _n,\,\{\epsilon _n\}\in l_2$,

\[ v(\lambda)=\prod_{n={-}\infty}^{\infty}\frac{\lambda_n-\lambda}{n-1/2}, \]

where $\lambda _n=n-1/2+\kappa _n,\,\{\kappa _n\}\in l_2(\mathbb {Z})$. (The notation $\mathop {{\prod }'}$ means that $n=0$ is missing in the product.)

Notice, that lemma 2.1 is a generalization of lemma 3.4.2 from [19]. In what follows, we will repeatedly use the following statement.

Lemma 2.2 [30]

If $f\in PW_\pi,$ then for every sequence $\{\lambda _n\}$ $(n\in \mathbb {Z})$ with $\lambda _n-n=o(1)$ as $|n|\to \infty$ and every $R>0$ the condition

\[ \sum_{n={-}\infty}^{\infty} \max_{|t-\lambda_n|\le R}|f(t)|^{2}<\infty \]

if fulfilled. In particular,

\[ \sum_{n={-}\infty}^{\infty}|f(\lambda_n)|^{2}<\infty. \]

Denote by $J_{jk}$ the determinant composed of the $j$th and $k$th columns of the matrix $A$. Denote also $J_0=J_{12}+J_{34}$, $J_1=J_{14}-J_{23}$, $J_2=J_{13}+J_{24}$.

Definition 2.3 The boundary conditions (1.3) are called regular if

(2.5)\begin{equation} J_1^{2}+J_2^{2}=(J_{14}+J_{32})^{2}+(J_{13}+J_{24})^{2}\ne 0,\end{equation}

and strongly regular if additionally

(2.6)\begin{equation} J_0^{2}\ne J_1^{2}-J_2^{2}.\end{equation}

Definition 2.4 The boundary conditions (1.3) are called regular but not strongly regular if (2.5) holds but (2.6) fails, i.e.

(2.7)\begin{equation} J_0^{2}=J_1^{2}-J_2^{2}.\end{equation}

It is well known (see, for instance, [5]) that boundary conditions (1.4) are strongly regular, the characteristic determinant of problem (1.2), (1.4) can be reduced to the form

(2.8)\begin{equation} \Delta(\lambda)={-}\cos t+\frac{c_1(\lambda)+c_2(\lambda)}{2},\end{equation}

and the eigenvalues are specified by the asymptotic formulas

(2.9)\begin{equation} \lambda_n^{{\pm}}=2n\pm\frac{t}{\pi}+\varepsilon_n^{{\pm}},\end{equation}

where $\{\varepsilon _n^{\pm }\}\in l_2$, $n\in \mathbb {Z}$. Further $\Gamma (z,\,r)$ denotes a disc of radius $r$ centred at the point $z$.

Next, we establish the necessary and sufficient conditions that an entire function must satisfy in order to be the characteristic determinant of some problem (1.2), (1.4). Then, we give an intrinsic description of sequences which are spectrum of operator (1.2), (1.4).

3. Main results

3.1 Characteristic determinant

Theorem 3.1 For a function $U(\lambda )$ to be the characteristic determinant of problem (1.2), (1.4), it is necessary and sufficient that it can be represented in the form

\[ U(\lambda)={-}\cos t+\cos\pi\lambda+f(\lambda), \]

where $f\in PW_\pi,$ and

(3.1)\begin{equation} \sum_{n={-}\infty}^{\infty}|f(n)|<\infty.\end{equation}

Proof. Necessity. Assume that function $U$ is the characteristic determinant, i.e. $U(\lambda )=\Delta (\lambda )$. Evidently, relations (2.3), (2.8) imply that $f\in PW_\pi$. To check inequality (3.1) we consider the monodromy matrix of problem (1.2), (1.4). Let the corresponding function $s_2(\lambda )$ have the roots $\lambda _n$, hence by [30, lemma 2.2],

(3.2)\begin{equation} \lambda_n=n+\delta_n,\end{equation}

where $\{\delta _n\}\in l_2$, $n\in \mathbb {Z}$. Relation (2.3) implies

(3.3)\begin{equation} c_j(\lambda_n)=\cos\pi\lambda_n+g_j(\lambda_n),\end{equation}

it follows from (2.3) and lemma 2.2 that

(3.4)\begin{equation} \sum_{n={-}\infty}^{\infty}|g_j(\lambda_n)|^{2}<\infty.\end{equation}

Denote

(3.5)\begin{equation} \chi(\lambda)=U(\lambda)+\cos t=\cos\pi\lambda+f(\lambda).\end{equation}

By virtue of (2.8),

\[ c_1(\lambda_n)+c_2(\lambda_n)=2\chi(\lambda_n). \]

It follows from (2.2) that $c_1(\lambda _n)c_2(\lambda _n)=1,\,$ consequently the numbers $c_1(\lambda _n),\,c_2(\lambda _n)$ are the roots of the quadratic equation

(3.6)\begin{equation} w^{2}-2\chi(\lambda_n)w+1=0.\end{equation}

Therefore we have

(3.7)\begin{equation} \begin{aligned} c_1(\lambda_n),c_2(\lambda_n) & =\chi(\lambda_n)\pm\sqrt{\chi^{2}(\lambda_n)-1}\\ & =\cos\pi\lambda_n+f(\lambda_n)\pm\sqrt{(\cos\pi\lambda_n+f(\lambda_n))^{2}-1}\\ & = \cos\pi\lambda_n+f(\lambda_n)\pm\sqrt{\cos^{2}\pi\lambda_n+2\cos\pi\lambda_nf(\lambda_n)+f^{2}(\lambda_n)-1}\\ & =\cos\pi\lambda_n+f(\lambda_n)\pm\sqrt{2\cos\pi\lambda_nf(\lambda_n)+f^{2}(\lambda_n)-\sin^{2}\pi\lambda_n}. \end{aligned} \end{equation}

It follows from (3.3) and (3.7) that

\[ (g_1(\lambda_n)-f(\lambda_n))^{2}=2\cos\pi\lambda_nf(\lambda_n)+f^{2}(\lambda_n)-\sin^{2}\pi\lambda_n, \]

hence,

(3.8)\begin{equation} 2\cos\pi\lambda_nf(\lambda_n)=g_1^{2}(\lambda_n)-2g_1(\lambda_n)f(\lambda_n)+\sin^{2}\pi\delta_n. \end{equation}

It follows from (3.2) that for all sufficiently large $|n|$ the inequality $|\cos \pi \lambda _n|>1/2$ holds. This, together with (3.2), (3.4), and lemma 2.2 implies

(3.9)\begin{equation} \sum_{n={-}\infty}^{\infty}|f(\lambda_n)|<\infty.\end{equation}

Since $f'\in PW_\pi$, then

\[ |f(n)|\le|f(\lambda_n)|+|f(n)-f(\lambda_n)| \le|f(\lambda_n)|+|\delta_n||\tau_n| \le|f(\lambda_n)|+(|\delta_n|^{2}+|\tau_n|^{2})/2, \]

where

\[ \tau_n=\max_{\lambda\in\Gamma(n,|\delta_n|)}|f'(\lambda)|. \]

By lemma 2.2, $\{\tau _n\}\in l_2$. This and (3.9) imply (3.1).

Sufficiency. Let $f\in PW_\pi$ satisfy condition (3.1). It follows from the Paley–Wiener theorem and [19, lemma 1.3.1] that

(3.10)\begin{equation} \lim_{|\lambda|\to\infty}{\rm e}^{-\pi |{\rm Im}\,\lambda|}f(\lambda)=0, \end{equation}

hence there exists a positive integer $N_0$ large enough that $|f(\lambda )|<1/100$ if ${\rm Im}\,\lambda =0$, $|{\rm Re}\,\lambda |\ge N_0$. Let $\lambda _n$ $(n\in \mathbb {Z})$ be a strictly monotone increasing sequence of real numbers such that for any $n\ne 0$ $\lambda _n=\lambda _{-n}$, $|\lambda _n-(N_0+1/2)|<1/100$ if $0\le n\le N_0$, and $\lambda _n=n$ if $n>N_0$. Denote

(3.11)\begin{equation} s(\lambda)={-}\pi(\lambda_0-\lambda)\mathop{{\prod}'}_{n={-}\infty}^{\infty}\frac{\lambda_n-\lambda}{n}.\end{equation}

It follows from lemma 2.1 that

(3.12)\begin{equation} s(\lambda)=\sin\pi \lambda+h(\lambda),\end{equation}

where $h\in PW_\pi$, hence,

(3.13)\begin{equation} |s(\lambda)|\ge C_1{\rm e}^{\pi|{\rm Im}\,\lambda|}\end{equation}

if $|{\rm Im}\,\lambda |\ge M$, where $M$ is sufficiently large. It follows from (3.11) that

\[ \dot s(\lambda_0)=\pi\mathop{{\prod}'}_{n={-}\infty}^{\infty}\frac{\lambda_n-\lambda_0}{n}>0. \]

One can readily see that the inequality $\dot s(\lambda _n)\dot s(\lambda _{n+1})<0$ holds for all $n\in \mathbb {Z}$. It follows from two last inequalities that

(3.14)\begin{equation} ({-}1)^{n}\dot s(\lambda_n)>0.\end{equation}

Relation (3.12) and lemma 2.2 imply that

(3.15)\begin{equation} \dot s(\lambda_n)=\pi({-}1)^{n}+\tau_n,\end{equation}

where $\{\tau _n\}\in l_2$, hence,

(3.16)\begin{equation} \frac{1}{\dot s(\lambda_n)}=\frac{({-}1)^{n}}{\pi}+\sigma_n,\end{equation}

where $\{\sigma _n\}\in l_2$. Equation (3.6) has the roots

(3.17)\begin{equation} \begin{aligned} c_n^{{\pm}} & =\chi(\lambda_n)\pm\sqrt{\chi^{2}(\lambda_n)-1}=\cos\pi\lambda_n+f(\lambda_n)\pm\sqrt{(\cos\pi\lambda_n+f(\lambda_n))^{2}-1}\\ & = \cos\pi\lambda_n+f(\lambda_n)\pm\sqrt{\cos^{2}\pi\lambda_n+2\cos\pi\lambda_nf(\lambda_n)+f^{2}(\lambda_n)-1}\\ & =\cos\pi\lambda_n+f(\lambda_n)\pm\sqrt{2\cos\pi\lambda_nf(\lambda_n)+f^{2}(\lambda_n)-\sin^{2}\pi\lambda_n}. \end{aligned} \end{equation}

It follows from (3.17) that if $0<|n|\le N_0$ the numbers $c_n^{+}$ are contained within the disc $\Gamma (i,\,1/10)$, the numbers $c_n^{-}$ are contained within the disc $\Gamma (-i,\,1/10)$, and if $|n|>N_0$ the numbers $c_n^{\pm }$ are contained within the disc $\Gamma (1,\,1/10)$ for even $n$, the numbers $c_n^{\pm }$ are contained within the disc $\Gamma (-1,\,1/10)$ for odd $n$. Denote $c_n=c_n^{+}$ for even $n$ and $c_n=c_n^{-}$ for odd $n$. Denote also

\[ z_n=\frac{c_n}{\dot s(\lambda_n)}. \]

It follows from (3.14) that the numbers $z_n$ lie strictly above the line $l:{\rm Im}\,\lambda =-{\rm Re}\,\lambda$.

Evidently,

(3.18)\begin{equation} \lambda_n=n+\rho_n,\end{equation}

where $\{\rho _n\}\in l_2$. It follows from (3.17) and (3.18) that

(3.19)\begin{equation} c_n=({-}1)^{n}+\vartheta_n,\end{equation}

where $\{\vartheta _n\}\in l_2$. Let $\beta _n=c_n-\cos \pi \lambda _n$, then $\{\beta _n\}\in l_2$. Let us consider the function

\[ g(\lambda)=s(\lambda)\sum_{n={-}\infty}^{\infty}\frac{\beta_n}{\dot s(\lambda_n)(\lambda-\lambda_n)}. \]

By [12, p. 120] the function $g\in PW_\pi$ and $g(\lambda _n)=\beta _n$. Denote $c(\lambda )=\cos \pi \lambda +g(\lambda )$, then $c(\lambda _n)=c_n\ne 0$, hence the functions $s(\lambda )$ and $c(\lambda )$ have disjoint zero sets.

Denote

\[ F(x,t)=\sum_{n={-}\infty}^{\infty}\left(z_nY_0(x,\lambda_n) Y_0^{T}(t,\lambda_n)-\frac{1}{\pi}Y_0(x,n) Y_0^{T}(t,n)\right). \]

It follows from [29] that

\[ \|F({\cdot},x)\|_{L_{2,2}^{2,2}(0,\pi)}+\|F(x,\cdot)\|_{L_{2,2}^{2,2}(0,\pi)}< C_2, \]

where $C_2$ not depending on $x$.

Using the properties of the numbers $z_n$ established above, we prove that for every $x\in [0,\,\pi ]$ the homogeneous equation

(3.20)\begin{equation} \mathbf{f}^{T}(t)+\int_0^{x}\mathbf{f}^{T}(s)F(s,t){\rm d}s=0,\end{equation}

where $\mathbf {f}(t)={\rm col}(f_1(t),\,f_2(t))$, $\mathbf {f}\in L_{2,2}(0,\,x)$, $\mathbf {f}(t)=0$ if $x< t\le \pi$ has the trivial solution only.

Multiplying equation (3.20) by $\overline {\mathbf {f}^{T}(t)}$ and integrating the resulting equation over segment $[0,\,x]$, we obtain

(3.21)\begin{equation} \|\mathbf{f}\|_{L_{2,2}(0,x)}^{2}+\int_0^{x}\left\langle\int_0^{x}\mathbf{f}^{T}(s)F(s,t){\rm d}s,\mathbf{f}^{T}(t)\right\rangle {\rm d}t=0.\end{equation}

Simple computations show

(3.22)\begin{equation} \begin{aligned} & \mathbf{f}^{T}(s)F(s,t)\\ & \quad= (f_1(s),f_2(s))\sum_{n={-}\infty}^{\infty}\left(z_n \begin{pmatrix} \sin\lambda_n s\sin\lambda_n t & -\sin\lambda_n s\cos\lambda_n t\\ - \cos\lambda_n s\sin\lambda_n t & \cos\lambda_n s\cos\lambda_n t \end{pmatrix}\right.\\ & \left. \qquad -\,\frac{1}{\pi} \begin{pmatrix} \sin n s\sin n t & -\sin n s\cos n t\\ - \cos n s\sin n t & \cos n s\cos n t \end{pmatrix}\right)\\ & \quad=\sum_{n={-}\infty}^{\infty}\left\{\vphantom{\frac{1}{\pi}}z_n[f_1(s)\sin\lambda_ns\sin\lambda_nt-f_2(s)\cos\lambda_ns\sin\lambda_nt,\right.\\ & \qquad- f_1(s)\sin\lambda_ns\cos\lambda_nt +f_2(s)\cos\lambda_ns\cos\lambda_nt]\\ & \qquad- \frac{1}{\pi}[f_1(s)\sin ns\sin nt-f_2(s)\cos ns\sin nt,\\ & \qquad- \left.\vphantom{\frac{1}{\pi}}f_1(s)\sin ns\cos nt +f_2(s)\cos ns\cos nt]\right\}\\ & \quad=\sum_{n={-}\infty}^{\infty}\left\{\vphantom{\frac{1}{\pi}}z_n[f_1(s)\sin\lambda_ns\sin\lambda_nt-f_2(s)\cos\lambda_ns\sin\lambda_nt]\right.\\ & \qquad-\frac{1}{\pi}[f_1(s)\sin ns\sin nt-f_2(s)\cos ns\sin nt] ,\\ & \qquad\;\;\;\; z_n[{-}f_1(s)\sin\lambda_ns\cos\lambda_nt +f_2(s)\cos\lambda_ns\cos\lambda_nt]\\ & \qquad-\left.\frac{1}{\pi}[{-}f_1(s)\sin ns\cos nt +f_2(s)\cos ns\cos nt]\right\}, \end{aligned} \end{equation}

therefore, substituting the right-hand side of (3.22) into the second term in the left-hand side of (3.21), transforming the iterated integrals into products of integrals and using the reality of all numbers $\lambda _n$, we obtain

(3.23)\begin{align} & \int_0^{x}\left\langle\int_0^{x}\mathbf{f}^{T}(s)F(s,t){\rm d}s,\mathbf{f}^{T}(t)\right\rangle {\rm d}t\nonumber\\ & \quad=\sum_{n={-}\infty}^{\infty}\int_0^{x}\left(\int_0^{x}\{z_n[f_1(s)\sin\lambda_ns\sin\lambda_nt-f_2(s)\cos\lambda_ns\sin\lambda_nt]\right.\nonumber\\ & \qquad-\left.\frac{1}{\pi}[f_1(s)\sin ns\sin nt-f_2(s)\cos ns\sin nt]\}{\rm d}s\right)\overline{f_1(t)}{\rm d}t\nonumber\\ & \qquad+\sum_{n={-}\infty}^{\infty}\int_0^{x}\left(\int_0^{x}\{z_n[{-}f_1(s)\sin\lambda_ns\cos\lambda_nt +f_2(s)\cos\lambda_ns\cos\lambda_nt]\right.\nonumber\\ & \qquad- \left.\frac{1}{\pi}[{-}f_1(s)\sin ns\cos nt +f_2(s)\cos ns\cos nt]\}{\rm d}s\right)\overline{f_2(t)}{\rm d}t\nonumber\\ & \quad=\sum_{n={-}\infty}^{\infty}\left(z_n\int_0^{x}[f_1(s)\sin\lambda_ns-f_2(s)\cos\lambda_ns]{\rm d}s\int_0^{x}\sin\lambda_nt\overline{f_1(t)}{\rm d}t\right.\nonumber\\ & \qquad-\left.\frac{1}{\pi}\int_0^{x}[f_1(s)\sin ns-f_2(s)\cos ns]{\rm d}s\int_0^{x}\sin nt\overline{f_1(t)}{\rm d}t\right)\nonumber\\ & \qquad+ \sum_{n={-}\infty}^{\infty} \left(z_n \int_0^{x}[{-}f_1(s)\sin\lambda_ns +f_2(s)\cos\lambda_ns]{\rm d}s\int_0^{x}\cos\lambda_nt\overline{f_2(t)}{\rm d}t\right.\nonumber\\ & \qquad- \left.\frac{1}{\pi}\int_0^{x}[{-}f_1(s)\sin ns +f_2(s)\cos ns]{\rm d}s\int_0^{x}\cos nt\overline{f_2(t)}{\rm d}t\right)\nonumber\\ & \quad= \sum_{n={-}\infty}^{\infty} z_n\left(\int_0^{x}[f_1(s)\sin\lambda_ns-f_2(s)\cos\lambda_ns]{\rm d}s\int_0^{x}\sin\lambda_nt\overline{f_1(t)}{\rm d}t\right.\nonumber\\ & \qquad\left.+\int_0^{x}[{-}f_1(s)\sin\lambda_ns +f_2(s)\cos\lambda_ns]{\rm d}s\int_0^{x}\cos\lambda_nt\overline{f_2(t)}{\rm d}t\right)\nonumber\\ & \qquad-\sum_{n={-}\infty}^{\infty} \frac{1}{\pi}\left(\int_0^{x}[f_1(s)\sin ns-f_2(s)\cos ns]{\rm d}s\int_0^{x}\sin nt\overline{f_1(t)}{\rm d}t\right.\nonumber\\ & \qquad+\left.\int_0^{x}[{-}f_1(s)\sin ns +f_2(s)\cos ns]{\rm d}s\int_0^{x}\cos nt\overline{f_2(t)}{\rm d}t\right)\nonumber\\ & \quad=\sum_{n={-}\infty}^{\infty} z_n\left(\int_0^{x}[f_1(t)\sin\lambda_nt-f_2(t)\cos\lambda_nt]{\rm d}t\int_0^{x}\sin\lambda_nt\overline{f_1(t)}{\rm d}t\right.\nonumber\\ & \qquad+\left.\int_0^{x}[{-}f_1(t)\sin\lambda_nt +f_2(t)\cos\lambda_nt]{\rm d}t\int_0^{x}\cos\lambda_nt\overline{f_2(t)}{\rm d}t\right)\nonumber\\ & \qquad-\sum_{n={-}\infty}^{\infty} \frac{1}{\pi}\left(\int_0^{x}[f_1(t)\sin nt-f_2(t)\cos nt]{\rm d}t\int_0^{x}\sin nt\overline{f_1(t)}{\rm d}t\right.\nonumber\\ & \qquad\left.+\int_0^{x}[{-}f_1(t)\sin nt +f_2(t)\cos nt]{\rm d}t\int_0^{x}\cos nt\overline{f_2(t)}{\rm d}t\right)\nonumber\\ & \quad=\sum_{n={-}\infty}^{\infty} z_n\int_0^{x}[f_1(t)\sin\lambda_nt-f_2(t)\cos\lambda_nt]{\rm d}t\int_0^{x}[\overline{f_1(t)}\sin\lambda_nt-\overline{f_2(t)}\cos\lambda_nt]{\rm d}t\nonumber\\ & \qquad-\sum_{n={-}\infty}^{\infty}\frac{1}{\pi}\int_0^{x}[f_1(t)\sin nt-f_2(t)\cos nt]{\rm d}t\int_0^{x}[\overline{f_1(t)}\sin nt-\overline{f_2(t)}\cos nt]{\rm d}t\nonumber\\ & \quad=\sum_{n={-}\infty}^{\infty} z_n\left|\int_0^{x}\langle \mathbf{f}(t),Y_0(t,\lambda_n)\rangle {\rm d}t\right|^{2}-\sum_{n={-}\infty}^{\infty} \frac{1}{\pi}\left|\int_0^{x}\langle \mathbf{f}(t),Y_0(t,n)\rangle {\rm d}t\right|^{2}. \end{align}

It is well known that the function system $\{\frac {1}{\sqrt {\pi }}Y_0(t,\,n)\}$ $(n\in \mathbb {Z})$ is an orthonormal basis in $L_{2,2}(0,\,\pi )$, hence it follows from the Parseval equality that

(3.24)\begin{equation} \|\mathbf{f}\|_{L_{2,2}(0,x)}^{2}=\sum_{n={-}\infty}^{\infty} \frac{1}{\pi}\left|\int_0^{x}\langle \mathbf{f}(t),Y_0(t,n)\rangle {\rm d}t\right|^{2}.\end{equation}

It follows from (3.21),(3.23) and (3.24) that

\[ \sum_{n={-}\infty}^{\infty} z_n\left|\int_0^{x}\langle \mathbf{f}(t),Y_0(t,\lambda_n)\rangle {\rm d}t\right|^{2}=0. \]

Since all the numbers $z_n$ are located strictly in the same half-plane relative to a line which passes through the origin, we see that

\[ \int_0^{x}\langle \mathbf{f}(t),Y_0(t,\lambda_n)\rangle {\rm d}t=0 \]

for all $n\in \mathbb {Z}$. It follows from (3.12) that the function $s(\lambda )$ is a sin-type function [13], therefore [1, lemma 5.3], the system $Y_0(t,\,\lambda _n)$ is a Riesz basis of $L_{2,2}(0,\,\pi )$, hence the system $Y_0(t,\,\lambda _n)$ is complete in $L_{2,2}(0,\,\pi )$, it follows now that $\mathbf {f}(t)\equiv 0$.

By [29, theorem 5.1], the functions $c(\lambda )$ and $-s(\lambda )$ are the entries of the first line of the monodromy matrix

\[ \tilde E(\pi,\lambda)= \begin{pmatrix} \tilde c_1(\pi,\lambda) & -\tilde s_2(\pi,\lambda)\\ \tilde s_1(\pi,\lambda) & \tilde c_2(\pi,\lambda) \end{pmatrix} \]

for problem (1.2), (1.4) with a potential $\tilde V\in L_2$, i.e.

(3.25)\begin{equation} c(\lambda)=\tilde c_1(\pi,\lambda), s(\lambda)=\tilde s_2(\pi,\lambda).\end{equation}

The corresponding characteristic determinant

\[ \tilde\Delta(\lambda)={-}\cos t+(\tilde c_1(\pi,\lambda)+\tilde c_2(\pi,\lambda))/2={-}\cos t+\cos\pi\lambda+\tilde f(\lambda), \]

where $\tilde f\in PW_\pi$. It follows from (2.2), (3.5), (3.6), (3.25) that

\begin{align*} \tilde\Delta(\lambda_n)& ={-}\cos t+(\tilde c_1(\pi,\lambda_n)+\tilde c_2(\pi,\lambda_n))/2\\ & ={-}\cos t+\left(\tilde c_1(\pi,\lambda_n)+\frac{1}{\tilde c_1(\pi,\lambda_n)}\right)/2={-}\cos t+\left(c(\lambda_n)+\frac{1}{c(\lambda_n)}\right)/2\\ & ={-}\cos t+\chi(\lambda_n)=U(\lambda_n). \end{align*}

This implies that the function

\[ \Phi(\lambda)=\frac{U(\lambda)-\tilde\Delta(\lambda)}{s(\lambda)}=\frac{f(\lambda)-\tilde f(\lambda)}{s(\lambda)} \]

is an entire function in the whole complex plane. Since by the Paley–Wiener theorem

(3.26)\begin{equation} |f(\lambda)-\tilde f(\lambda)|< C_3{\rm e}^{\pi|{\rm Im}\,\lambda|},\end{equation}

then by (3.13) $|\Phi (\lambda )|\le C_4$ if $|{\rm Im}\,\lambda |\ge M$. We denote by $\Omega$ the set

\[ \Gamma(N_0+1/2,1/10)\bigcup\Gamma({-}N_0-1/2,1/10))\bigcup\Gamma_{|n|>N_0}(n,1/10). \]

Since the function $s(\lambda )$ is a sin-type function [13], then $|s(\lambda )|>C_5>0$ if $\lambda \notin \Omega$. From this inequality, (3.26) and the maximum principle we obtain that $|\Phi (\lambda )|< C_6$ in the strip $|{\rm Im}\,\lambda |\le M$, hence the function $\Phi (\lambda )$ is bounded in the whole complex plane and, by virtue of Liouville theorem, it is a constant. Let $|{\rm Im}\,\lambda |=M$, then it follows from (3.10) that $\lim _{|\lambda |\to \infty }(f(\lambda )-\tilde f(\lambda ))=0$, consequently $\Phi (\lambda )\equiv 0$, therefore $U(\lambda )\equiv \tilde \Delta (\lambda )$.

3.2 Spectrum

Theorem 3.2 For a set $\Lambda$ to be the spectrum of some Dirac operator (1.2), (1.4) with a complex-valued potential $V\in L_2(0,\,\pi )$ it is necessary and sufficient that it consists of two sequences of eigenvalues $\lambda _n^{\pm }$ satisfying condition (2.9) and the inequality

(3.27)\begin{equation} \sum_{k={-}\infty}^{\infty}\left|\sum_{n={-}\infty}^{\infty}\left(\frac{\varepsilon_n^{+}}{2n+t/\pi-k}+\frac{\varepsilon_n^{-}}{2n-t/\pi-k}\right)\right|<\infty.\end{equation}

Proof. Sufficiency. Let two sequences $\lambda _n^{\pm }$ satisfy conditions (2.9) and (3.27). Evidently, there exists a constant $M$ such that

(3.28)\begin{equation} \sup|\varepsilon_n^{{\pm}}|< M,\quad \sum_{n={-}\infty}^{\infty}|\varepsilon_n^{{\pm}}|^{2}< M.\end{equation}

It is well known that

\[ \sin\pi \lambda=\pi \lambda\mathop{{\prod}'}_{n={-}\infty}^{\infty}\frac{n-\lambda}{n}=\pi \lambda\mathop{{\prod}'}_{n={-}\infty}^{\infty}\left(1-\frac{\lambda}{n}\right), \]

therefore the function $\Delta _0(\lambda )=\cos \pi \lambda -\cos t$ has the representation

\begin{align*} & \displaystyle\Delta_0(\lambda)={-}2\sin\dfrac{\pi\lambda+t}{2}\sin\dfrac{\pi\lambda-t}{2}={-}\dfrac{\pi^{2}(\lambda^{2}-(t/\pi)^{2})}{2}\\ & \qquad\qquad \mathop{{\prod}'}_{n={-}\infty}^{\infty}\dfrac{(2n+t/\pi-\lambda)(2n-t/\pi-\lambda)}{4n^{2}}. \end{align*}

Denote

\[ \Delta(\lambda)={-}\frac{\pi^{2}}{2}(\lambda_0^{+}-\lambda)(\lambda_0^{-}-\lambda)\mathop{{\prod}'}_{n={-}\infty}^{\infty}\frac{(\lambda_n^{+}-\lambda)(\lambda_n^{-}-\lambda)}{4n^{2}}. \]

Evidently,

(3.29)\begin{equation} |\Delta_0(\lambda)|< c_1{\rm e}^{\pi |{\rm Im}\,\lambda|}.\end{equation}

Let $f(\lambda )=\Delta (\lambda )-\Delta _0(\lambda )$. Investigation of the properties of the function $f(\lambda )$ is based on the following propositions.

Proposition 3.3 The function $f(\lambda )$ is an entire function of exponential type not exceeding $\pi$.

Denote $\Gamma$ the union of the discs $\Gamma (2n\pm t/\pi,\,1/4)$ $(n\in Z)$. If $\lambda \notin \Gamma,$ then

(3.30)\begin{equation} f(\lambda)={-}\Delta_0(\lambda)\left(1-\frac{\Delta(\lambda)}{\Delta_0}\right)={-}\Delta_0(\lambda)(1-\phi(\lambda)),\end{equation}

where

\begin{align*} \phi(\lambda)& = \frac{(\lambda_0^{+}-\lambda)(\lambda_0^{-}-\lambda)} {(\lambda^{2}-(t/\pi)^{2})}\mathop{{\prod}'}_{n={-}\infty}^{\infty}\frac{(\lambda_n^{+}-\lambda)(\lambda_n^{-}-\lambda)}{(2n+t/\pi-\lambda)(2n-t/\pi-\lambda)}\\ & =\prod_{n={-}\infty}^{\infty}\left(1+\frac{\varepsilon_n^{+}}{2n+t/\pi-\lambda}\right)\left(1+\frac{\varepsilon_n^{-}}{2n-t/\pi-\lambda}\right). \end{align*}

Let us estimate the function $\phi (\lambda )$. Denote $\alpha _n^{\pm }(\lambda )=\frac {\varepsilon _n^{\pm }}{2n\pm t/\pi -\lambda }$. It follows from (3.28) that

(3.31)\begin{align} & \sum_{n={-}\infty}^{\infty}(|\alpha_n^{+}(\lambda)|+|\alpha_n^{-}(\lambda)|)\le \sum_{n={-}\infty}^{\infty}(|\varepsilon_n^{+}|^{2}+|\varepsilon_n^{-}|^{2}+|2n+t/\pi-\lambda|^{{-}2}\nonumber\\ & \quad +|2n-t/\pi-\lambda|^{{-}2})/2< c_3.\end{align}

It is easy to see that for all $|n|>n_0$, where $n_0$ is a sufficiently large number, we have

(3.32)\begin{equation} |\alpha_n^{{\pm}}(\lambda)|<1/4\end{equation}

for all $\lambda \notin \Gamma$. If $|n|\le n_0$, then inequality (3.32) holds for all sufficiently large $|\lambda |$, hence inequality (3.32) is valid for all $|\lambda |\ge C_0$. It follows from (3.31), (3.32) and elementary inequality

(3.33)\begin{equation} |\ln(1+z)|\le 2|z|,\end{equation}

which is valid if $|z|\le 1/4$ that

\[ \sum_{n={-}\infty}^{\infty}(|\ln(1+\alpha_n^{+}(\lambda)|+|\ln(1+\alpha_n^{-}(\lambda))|\le c_4. \]

Here and throughout the following, we choose the branch of $\ln (1+z)$ that is zero for $z=0$. In view of [10, p. 433], we rewrite the last relation in the form

(3.34)\begin{equation} |\phi(\lambda)|\le\prod_{n={-}\infty}^{\infty}|1+\alpha_n^{+}(\lambda)|1+\alpha_n^{-}(\lambda)|\le {\rm e}^{c_4}.\end{equation}

It follows from (3.29), (3.30), (3.34) that

(3.35)\begin{equation} |f(\lambda)|< c_5{\rm e}^{\pi |{\rm Im}\,\lambda|}\end{equation}

outside the domain $\Gamma '=\Gamma \cup \{|\lambda |< C_0\}$.

Denote $x_0^{\pm }=|{\rm Re}\, t/\pi |\pm 1/3$, $T^{+}=\cup _n[2n+|{\rm Re}\, t/\pi |-1/4,\,2n+|{\rm Re}\, t/\pi |+1/4]$, $T^{-}=\cup _n[2n-|{\rm Re}\, t/\pi |-1/4,\,2n-|{\rm Re}\, t/\pi |+1/4]$. It easy to see that the points $x_0^{\pm }\notin T^{+}$ and at least one of these point does not belong $T^{-}$ since $x_0^{+}-x_0^{-}=2/3>1/2$. Denote this point by $x_0$ then all points $x_0+2k$, $k\in Z$ lie outside the set $T^{+}\cup T^{-}$.

In particular, inequality (3.35) is valid if $\lambda$ belongs lines ${\rm Im}\,\lambda =\pm \hat C_0$, where $\hat C_0=C_0+|t|$, and vertical segments with vertexes $(x_0+2k,\, -\hat C_0),\,(x_0+2k,\, \hat C_0)$, $|2k-1|>C_0$, $k\in \mathbb {Z}$. By the maximum principle, inequality (3.35) holds in whole complex plane, hence the function $f(\lambda )$ is an entire function of exponential type not exceeding $\pi$.

Proposition 3.4 The function $f$ belongs to $PW_\pi$.

Denote

\[ W(\lambda)=\ln\phi(\lambda)=\sum_{n={-}\infty}^{\infty}(\ln(1+\alpha_n^{+}(\lambda)+\ln(1+\alpha_n^{-}(\lambda)), \]

then

(3.36)\begin{equation} f(\lambda)={-}\Delta_0(\lambda)\left(1-{\rm e}^{W(\lambda)}\right).\end{equation}

Let us estimate the function $W(\lambda )$ if $\lambda \notin \Gamma '$. It follows from (3.28), (3.32), (3.33) that

\begin{align*} |W(\lambda)|& \le\sum_{n={-}\infty}^{\infty}(|\ln(1+\alpha_n^{+}(\lambda)|+|\ln(1+\alpha_n^{-}(\lambda))| \\ & \le\frac{2M}{|\lambda|}+ \sum_{n={-}\infty}^{\infty}\left(\frac{|\varepsilon_n^{+}|^{2}+|\varepsilon_n^{-}|^{2}}{10M}+\frac{10M}{|2n-\lambda|^{2}}\right)\\ & \le\frac{2M}{|\lambda|}+\frac{1}{10}+20M\sum_{n=0}^{\infty}\frac{1}{n^{2}+ |{\rm Im}\,\lambda|^{2}}\\ & \le\frac{2M}{|\lambda|}+\frac{1}{10}+20M\left(\frac{2}{|{\rm Im}\,\lambda|^{2}}+\int_1^{\infty}\frac{{\rm d}x}{x^{2}+|{\rm Im}\,\lambda|^{2}}\right)\\ & \le\frac{2M}{|{\rm Im}\,\lambda|} +\frac{1}{10}+20M\left(\frac{2}{|{\rm Im}\,\lambda|^{2}}+\frac{\pi}{2|{\rm Im}\,\lambda|}\right). \end{align*}

The last inequality implies that

(3.37)\begin{equation} |W(\lambda)|<1/4\end{equation}

if $|{\rm Im}\,\lambda |\ge M_1=10(\pi +2+22M)+\hat C_0$. Then from the trivial inequality

(3.38)\begin{equation} \frac{|z|}{2}\le |1-{\rm e}^{z}|\le 2|z|,\end{equation}

which holds for $|z|\le 1/4$, we obtain the inequality $|1-{\rm e}^{W(\lambda )}|\le 2|W(\lambda )|$, which, together with (3.29) and (3.36) implies that

(3.39)\begin{equation} |f(\lambda)|\le c_6|W(\lambda)|\end{equation}

for $\lambda \in l$, where $l$ is the line ${\rm Im}\,\lambda =M_1$. Let us prove that

(3.40)\begin{equation} \int_l|W(\lambda)|^{2}{\rm d}\lambda<\infty.\end{equation}

From the elementary inequality $|\ln (1+z)-z|\le |z|^{2}$ true for $|z|\le 1/2$, we obtain

\[ \ln(1+z)-z=r(z), \]

where $|r(z)|\le |z|^{2}$, hence,

(3.41)\begin{equation} W(\lambda)=S_1(\lambda)+S_2(\lambda),\end{equation}

where

\begin{align*} & S_1(\lambda)=\sum_{n={-}\infty}^{\infty} (\alpha_n^{+}(\lambda)+\alpha_n^{-}(\lambda)),\\ & |S_2(\lambda)|\le\sum_{n={-}\infty}^{\infty} (|\alpha_n^{+}(\lambda)|^{2}+|\alpha_n^{-}(\lambda)|^{2}). \end{align*}

Evidently,

(3.42)\begin{equation} |W(\lambda)|\le|S_1(\lambda)|+|S_2(\lambda)|.\end{equation}

Set

\[ I_m=\int_l|S_m(\lambda)|^{2}{\rm d}\lambda \]

$(m=1,\,2)$. First consider the integral $I_1$. It follows from [28, p. 221] that

(3.43)\begin{equation} \begin{aligned} I_1 & =\int_l\left|\sum_{n={-}\infty}^{\infty}\left(\frac{\varepsilon_n^{+}}{2n+t/\pi-\lambda}+\frac{\varepsilon_n^{-}}{2n-t/\pi-\lambda}\right)\right|^{2}{\rm d}\lambda\\ & \le 2\left(\int_l\left|\sum_{n={-}\infty}^{\infty}\frac{\varepsilon_n^{+}}{2n+t/\pi-\lambda}\right|^{2}{\rm d}\lambda+ \int_l\left|\sum_{n={-}\infty}^{\infty}\frac{\varepsilon_n^{+}}{2n-t/\pi-\lambda}\right|^{2}{\rm d}\lambda\right)\\ & = 2\left(\int_{l^{+}}\left|\sum_{n={-}\infty}^{\infty}\frac{\varepsilon_n^{+}}{2n-\lambda}\right|^{2}{\rm d}\lambda+ \int_{l^{-}}\left|\sum_{n={-}\infty}^{\infty}\frac{\varepsilon_n^{-}}{2n-\lambda}\right|^{2}{\rm d}\lambda\right) <\infty, \end{aligned} \end{equation}

where $l^{\pm }$ are the lines ${\rm Im}\,\lambda =M_1\mp t/\pi$ correspondingly.

It is readily seen that

\[ |S_2(\lambda)|\le \sum_{n={-}\infty}^{\infty}\frac{|\varepsilon_n^{+}|^{2}}{|2n+t/\pi-\lambda|^{2}}+\sum_{n={-}\infty}^{\infty}\frac{|\varepsilon_n^{-}|^{2}}{|2n-t/\pi-\lambda|^{2}}\le c_7, \]

hence,

(3.44)\begin{equation} \begin{aligned} I_2 & \le c_7\int_l\left(\sum_{n={-}\infty}^{\infty}\frac{|\varepsilon_n^{+}|^{2}}{|2n+t/\pi-\lambda|^{2}}+\sum_{n={-}\infty}^{\infty}\frac{|\varepsilon_n^{-}|^{2}}{|2n-t/\pi-\lambda|^{2}}\right){\rm d}\lambda\\ & \le c_8\sum_{n={-}\infty}^{\infty}(|\varepsilon_n^{+}|^{2}+|\varepsilon_n^{-}|^{2})\int_{\tilde l}\frac{{\rm d}\lambda}{|2n-\lambda|^{2}}< c_9\sum_{n={-}\infty}^{\infty}(|\varepsilon_n^{+}|^{2}+|\varepsilon_n^{-}|^{2})< c_{10}, \end{aligned} \end{equation}

where $\tilde l=l^{+}\cup l^{-}$. Relations (3.42)–(3.44) imply (3.40). It follows from (3.39), (3.40) and [26, p. 115] that

(3.45)\begin{equation} \int_R|f(\lambda)|^{2}{\rm d}\lambda<\infty.\end{equation}

Proposition 3.5 The function $f(\lambda )$ satisfies condition (3.1).

Let $k\in \mathbb {Z}$. Obviously,

(3.46)\begin{equation} 0< c_{11}<|\Delta_0(k)|< c_{12}.\end{equation}

Denote

\[ \epsilon_n=\max(|\varepsilon_n^{+}|,|\varepsilon_n^{-}|). \]

There exists a number $n_0>0$ such that

\[ \sum_{|n|>n_0}\epsilon_n^{2}<1/1000, \]

and for any $|n|>n_0$ the inequality $\epsilon _n^{2/3}<1/1000$ holds. Let $\lambda \notin \Gamma '$. Supplementary suppose that

\[ |\lambda|>M_2=1000(2n_0+1)n_0M. \]

Then, using the well-known inequality $ab\le \frac {a^{p}}{p}+\frac {b^{q}}{q}$ $(a,\,b>0,\, p,\,q>1,\, 1/p+1/q=1)$, we obtain

(3.47)\begin{align} \sum_{n={-}\infty}^{\infty}(|\alpha_n^{+}(\lambda)|+|\alpha_n^{-}(\lambda)|) & \le\sum_{|n|\le n_0}\left(\frac{\epsilon_n}{|2n+t/\pi-\lambda|}+\frac{\epsilon_n}{|2n-t/\pi-\lambda|}\right)\nonumber\\ & \quad +\sum_{|n|> n_0}\left(\frac{\epsilon_n}{|2n+t/\pi-\lambda|}+\frac{\epsilon_n}{|2n-t/\pi-\lambda|}\right) \nonumber\\ & \le 2M\sum_{|n|\le n_0}\frac{1}{|2n-\lambda|}+2\sum_{|n|> n_0}\left(\epsilon_n^{2}+\frac{\epsilon_n^{2/3}}{|2n-\lambda|^{4/3}}\right)\nonumber\\ & \le \frac{1}{50}+\frac{1}{500}\sum_{n=1}^{\infty}\frac{1}{n^{4/3}}<\frac{1}{10}, \end{align}

hence inequality (3.37) is valid for all $\lambda$ belonging to the considered domain. Arguing as above, we see that

\[ |f(\lambda)|\le c_{13}\left(\left| \sum_{n={-}\infty}^{\infty} (\alpha_n^{+}(\lambda)+\alpha_n^{-}(\lambda))\right|+\sum_{n={-}\infty}^{\infty} (|\alpha_n^{+}(\lambda)|^{2}+|\alpha_n^{-}(\lambda)|^{2})\right). \]

The last inequality implies that for all $|k|>k_0$, where $k_0=\max (C_0,\, M_2)$,

(3.48)\begin{align} |f(k)|& \le c_{14}\left(\left|\sum_{n={-}\infty}^{\infty}\left(\frac{\varepsilon_n^{+}}{2n+t/\pi-k}+\frac{\varepsilon_n^{-}}{2n-t/\pi-k}\right)\right|\right.\nonumber\\ & \quad \left. +\, \sum_{n={-}\infty}^{\infty}\left(\frac{|\varepsilon_n^{+}|^{2}}{|2n+t/\pi-k|^{2}}+\frac{|\varepsilon_n^{-}|^{2}}{|2n-t/\pi-k|^{2}}\right)\right). \end{align}

Clearly,

(3.49)\begin{equation} \begin{aligned} & \sum_{k={-}\infty}^{\infty}\sum_{n={-}\infty}^{\infty}\left(\frac{|\varepsilon_n^{+}|^{2}}{|2n+t/\pi-k|^{2}}+\frac{|\varepsilon_n^{-}|^{2}}{|2n-t/\pi-k|^{2}}\right)\\ & \quad=\sum_{n={-}\infty}^{\infty}|\varepsilon_n^{+}|^{2}\sum_{k={-}\infty}^{\infty}\frac{1}{|2n+t/\pi-k|^{2}}+ \sum_{n={-}\infty}^{\infty}|\varepsilon_n^{-}|^{2}\sum_{k={-}\infty}^{\infty}\frac{1}{|2n-t/\pi-k|^{2}}\\ & \quad< c_{15}\sum_{n={-}\infty}^{\infty}(|\varepsilon_n^{+}|^{2}+|\varepsilon_n^{-}|^{2})< c_{16}. \end{aligned} \end{equation}

It follows from (3.27), (3.46), (3.48), (3.49) that (3.1) holds.

Necessity. If a set $\{\Lambda \}$ is the spectrum of a Dirac operator (1.2), (1.4), then relation (2.9) takes place [5]. Let us prove that condition (3.27) holds. Since $f(\lambda )=\Delta (\lambda )-\Delta _0(\lambda )$, then by theorem 3.1 relation (3.1) is valid.

Let $\lambda =k$, $k\in \mathbb {Z}$, $|k|>k_0$, hence inequality (3.47) holds. It follows from (3.36), (3.38) and (3.46) that

\[ |W(k)|\le|f(k)|. \]

This, together with (3.41) implies

(3.50)\begin{equation} |S_1(k)|\le|f(k)|+\sum_{n={-}\infty}^{\infty}(|\alpha_n^{+}(k)|^{2}+|\alpha_n^{-}(k)|^{2}).\end{equation}

Using (3.49), we find that

(3.51)\begin{equation} \sum_{n={-}\infty}^{\infty}(|\alpha_n^{+}(k)|^{2}+|\alpha_n^{-}(k)|^{2}) < c_{17}.\end{equation}

It follows from (3.50), (3.51) and (3.1) that

\[ \sum_{|k|>k_0}|S_1(k)|< c_{18}. \]

It is easy to see that

\[ \sum_{|k|\le k_0}|S_1(k)|< k_0c_{19}. \]

The last two inequalities imply (3.27).