1. Introduction
The spectral theory of boundary value problems for first-order systems of ordinary differential equations of the form
where $B$ is a nonsingular diagonal $n\times n$ matrix,
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 [Reference Birkhoff and Langer2]. 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 [Reference Levitan and Sargsyan14, Reference Malamud17, Reference Mykytyuk and Puyda21, Reference Yurko32–Reference Yurko38] 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
where $\mathbf {y}={\rm col}(y_1(x),\,y_2(x))$,
the complex-valued functions $p,\, q\in L_2(0,\,\pi )$, with two-point boundary conditions
where
the coefficients $a_{ij}$ are arbitrary complex numbers, and rows of the matrix
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 [Reference Gasymov and Dzabiev7], from one spectrum and the norming constants in [Reference Dzabiev6], and from the spectral function in [Reference Malamud16]. The analogous results for Dirac operator with summable potentials were established in [Reference Albeverio, Hryniv and Mykytyuk1]. The case of more general separated boundary conditions was considered in [Reference Daskalov and Khristov4]. In the case of unseparated boundary conditions (including periodic, antiperiodic and quasi-periodic conditions) the indicated problem was solved in [Reference Misyura20, Reference Nabiev22–Reference Nabiev25]. 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 [Reference Mamedov18] 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 [Reference Bondarenko and Buterin3, Reference Gorbunov and Yurko9, Reference Ning27, Reference Yang and Yurko31].
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 [Reference Malamud15, Reference Malamud16]. The most complete uniqueness result on general first-order systems (1.1) and (1.2) on a finite interval has been obtained recently in [Reference Malamud17]. 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 [Reference Lesch and Malamud11]. New inverse approach based on the A-function concept proposed by Gesztesy and Simon to Schrodinger operator has been recently extended in [Reference Gesztesy and Sakhnovich8] 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
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
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
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,
Denote the second column of the matrix $E_0(x,\,\lambda )$ by
It is well known that the entries of the matrix $E(x,\,\lambda )$ are related by the identity
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 [Reference Tkachenko29] that the functions $c_j(\lambda ),\,s_j(\lambda )$ admit the representation
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 [Reference Misyura20]
Entire functions $u(\lambda )$ and $v(\lambda )$ admit the representations
where $h,\,g\in PW_\pi,$ if and only if
where $\lambda _n=n+\epsilon _n,\,\{\epsilon _n\}\in l_2$,
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 [Reference Marchenko19]. In what follows, we will repeatedly use the following statement.
Lemma 2.2 [Reference Tkachenko30]
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
if fulfilled. In particular,
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
and strongly regular if additionally
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.
It is well known (see, for instance, [Reference Djakov and Mityagin5]) that boundary conditions (1.4) are strongly regular, the characteristic determinant of problem (1.2), (1.4) can be reduced to the form
and the eigenvalues are specified by the asymptotic formulas
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
where $f\in PW_\pi,$ and
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 [Reference Tkachenko30, lemma 2.2],
where $\{\delta _n\}\in l_2$, $n\in \mathbb {Z}$. Relation (2.3) implies
it follows from (2.3) and lemma 2.2 that
Denote
By virtue of (2.8),
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
Therefore we have
It follows from (3.3) and (3.7) that
hence,
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
Since $f'\in PW_\pi$, then
where
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 [Reference Marchenko19, lemma 1.3.1] that
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
It follows from lemma 2.1 that
where $h\in PW_\pi$, hence,
if $|{\rm Im}\,\lambda |\ge M$, where $M$ is sufficiently large. It follows from (3.11) that
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
Relation (3.12) and lemma 2.2 imply that
where $\{\tau _n\}\in l_2$, hence,
where $\{\sigma _n\}\in l_2$. Equation (3.6) has the roots
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
It follows from (3.14) that the numbers $z_n$ lie strictly above the line $l:{\rm Im}\,\lambda =-{\rm Re}\,\lambda$.
Evidently,
where $\{\rho _n\}\in l_2$. It follows from (3.17) and (3.18) that
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
By [Reference Levin12, 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
It follows from [Reference Tkachenko29] that
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
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
Simple computations show
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
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
It follows from (3.21),(3.23) and (3.24) that
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
for all $n\in \mathbb {Z}$. It follows from (3.12) that the function $s(\lambda )$ is a sin-type function [Reference Levin and Ostrovskii13], therefore [Reference Albeverio, Hryniv and Mykytyuk1, 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 [Reference Tkachenko29, theorem 5.1], the functions $c(\lambda )$ and $-s(\lambda )$ are the entries of the first line of the monodromy matrix
for problem (1.2), (1.4) with a potential $\tilde V\in L_2$, i.e.
The corresponding characteristic determinant
where $\tilde f\in PW_\pi$. It follows from (2.2), (3.5), (3.6), (3.25) that
This implies that the function
is an entire function in the whole complex plane. Since by the Paley–Wiener theorem
then by (3.13) $|\Phi (\lambda )|\le C_4$ if $|{\rm Im}\,\lambda |\ge M$. We denote by $\Omega$ the set
Since the function $s(\lambda )$ is a sin-type function [Reference Levin and Ostrovskii13], 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
Proof. Sufficiency. Let two sequences $\lambda _n^{\pm }$ satisfy conditions (2.9) and (3.27). Evidently, there exists a constant $M$ such that
It is well known that
therefore the function $\Delta _0(\lambda )=\cos \pi \lambda -\cos t$ has the representation
Denote
Evidently,
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
where
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
It is easy to see that for all $|n|>n_0$, where $n_0$ is a sufficiently large number, we have
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
which is valid if $|z|\le 1/4$ that
Here and throughout the following, we choose the branch of $\ln (1+z)$ that is zero for $z=0$. In view of [Reference Lavrentiev and Shabat10, p. 433], we rewrite the last relation in the form
It follows from (3.29), (3.30), (3.34) that
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
then
Let us estimate the function $W(\lambda )$ if $\lambda \notin \Gamma '$. It follows from (3.28), (3.32), (3.33) that
The last inequality implies that
if $|{\rm Im}\,\lambda |\ge M_1=10(\pi +2+22M)+\hat C_0$. Then from the trivial inequality
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
for $\lambda \in l$, where $l$ is the line ${\rm Im}\,\lambda =M_1$. Let us prove that
From the elementary inequality $|\ln (1+z)-z|\le |z|^{2}$ true for $|z|\le 1/2$, we obtain
where $|r(z)|\le |z|^{2}$, hence,
where
Evidently,
Set
$(m=1,\,2)$. First consider the integral $I_1$. It follows from [Reference Sansug and Tkachenko28, p. 221] that
where $l^{\pm }$ are the lines ${\rm Im}\,\lambda =M_1\mp t/\pi$ correspondingly.
It is readily seen that
hence,
where $\tilde l=l^{+}\cup l^{-}$. Relations (3.42)–(3.44) imply (3.40). It follows from (3.39), (3.40) and [Reference Nikolskii26, p. 115] that
Proposition 3.5 The function $f(\lambda )$ satisfies condition (3.1).
Let $k\in \mathbb {Z}$. Obviously,
Denote
There exists a number $n_0>0$ such that
and for any $|n|>n_0$ the inequality $\epsilon _n^{2/3}<1/1000$ holds. Let $\lambda \notin \Gamma '$. Supplementary suppose that
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
hence inequality (3.37) is valid for all $\lambda$ belonging to the considered domain. Arguing as above, we see that
The last inequality implies that for all $|k|>k_0$, where $k_0=\max (C_0,\, M_2)$,
Clearly,
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 [Reference Djakov and Mityagin5]. 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
This, together with (3.41) implies
Using (3.49), we find that
It follows from (3.50), (3.51) and (3.1) that
It is easy to see that
The last two inequalities imply (3.27).
Acknowledgement
The author expresses his deep gratitude to the referee.