On the Spectrum of Two-Point Boundary Value Problems for the Dirac Operator

Abstract

We consider the spectral problem for a Dirac operator with arbitrary two-point boundary conditions and an arbitrary complex-valued integrable potential. The existence of nontrivial boundary value problems of this type with an unbounded growth of the multiplicity of eigenvalues is established.

INTRODUCTION

In the present paper, we study the Dirac system

$$ B\mathbf {y}^{\prime }+V\mathbf {y}=\lambda \mathbf {y}, $$

(1)

where $\mathbf {y}=\mathrm {col}\thinspace (y_1(x),y_2(x))$, $\lambda \in \mathbb {C}$ is the spectral parameter,

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

and the functions $p, q\in L_1(0,\pi )$ are complex-valued, with the two-point boundary conditions

$$ U(\mathbf {y})\equiv C\mathbf {y}(0)+D\mathbf {y}(\pi )=0,$$

(2)

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 $a_{ij}$ can be any complex numbers, and the rows of the matrix

$$ A=(CD)=\begin {pmatrix} a_{{11}}&a_{12}&a_{13} &a_{14}\\ a_{21}&a_{{22}}&a_{23} &a_{24} \end {pmatrix}$$

are linearly independent.

We denote by $\|f\|=(|f_1|^2+|f_2|^2)^{1/2}$ the norm of an arbitrary vector $f=\mathrm {col}\thinspace (f_1,f_2)\in \mathbb {C}^2$ and set $\langle f,g\rangle =f_1g_1+f_2g_2$. We denote the norm of an arbitrary $2\times 2$ matrix $W$ by $\|W\|=\sup \limits _{\|f\|=1}\|Wf\|$. Let $L_{2,2}(a,b)$ be the space of two-dimensional vector functions $f(t)=\mathrm {col}\thinspace (f_1(t),f_2(t))$ with the norm $\|f\|_{L_{2,2}(a,b)}=(\int \nolimits _a^b\|f(t)\|\thinspace dt)^{1/2}$, and let $L_{2,2}^{2,2}(a,b)$ be the space of $2\times 2$ matrix functions $W(t)$ with the norm $\|W\|_{L_{2,2}^{2,2}(a,b)}=(\int \nolimits _a^b\|W(t)\|\thinspace dt)^{1/2}$. We treat the operator $\mathbb {L}\mathbf {y}=B\mathbf {y}^{\prime }+V\mathbf {y}$ as a linear operator in the space $L_{2,2}(0,\pi )$ with domain $D(\mathbb {L})=\{\mathbf {y}\in W_1^1[0,\pi ]:\mathbb {L}\mathbf {y}\in L_{2,2}(0,\pi )$, $U_j(\mathbf {y})=0$ $(j=1,2)\}$.

Let

$$ E(x,\lambda )=\begin {pmatrix} c_1(x,\lambda )&-s_2(x,\lambda )\\ s_1(x,\lambda )&c_2(x,\lambda ) \end {pmatrix}$$

be the fundamental matrix of Eq. (1) with the boundary condition $E(0,\lambda )=I$, where $I$ is the identity matrix, and let $E_0(x,\lambda )$ be the fundamental matrix of the unperturbed equation $B\mathbf {y}^{\prime }=\lambda \mathbf {y}$ with the boundary condition $E_0(0,\lambda )=I$. It is obvious that

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

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

$$ c_1(x,\lambda )c_2(x,\lambda )+s_1(x,\lambda )s_2(x,\lambda )=1$$

(3)

for any $x$ and $\lambda$. Let $J_{ij}$ be the determinant formed by the $i$th and $j$th columns of $A$. Set $J_0=J_{12}+J_{34}$, $J_1=J_{14}-J_{23}$, and $J_2=J_{13}+J_{24}$.

It was shown in [1] by the transformation operator method that the characteristic determinant $\Delta (\lambda )$ of problem (1), (2), which is equal to

$$ \Delta (\lambda ) =J_{12}+J_{34}+J_{14}c_2(\pi ,\lambda ) -J_{23}c_1(\pi ,\lambda )-J_{13}s_2(\pi ,\lambda ) -J_{24}s_1(\pi ,\lambda ), $$

(4)

can be reduced to the form

$$ \Delta (\lambda )=\Delta _0(\lambda )+\int _0^\pi r_1(t)e^{-i\lambda t}\thinspace dt+\int _0^\pi r_2(t)e^{i\lambda t}\thinspace dt=\Delta _0(\lambda )+R(\lambda ),$$

(5)

where the function

$$ \begin {aligned} \Delta _0(\lambda )&=J_0 +J_1\cos (\pi \lambda )-J_2\sin (\pi \lambda )\\ &=J_{12}+J_{34}+\frac {1}{2}\big (e^{i\pi \lambda } (J_1+iJ_2)+e^{-i\pi \lambda }(J_1-iJ_2)\big )=J_0+C_1e^{i\pi \lambda }+C_2e^{-i\pi \lambda }, \end {aligned}$$

(6)

$C_1=(J_1+iJ_2)/2$, $C_2=(J_1-iJ_2)/2$, is the characteristic determinant of the unperturbed problem

$$ B\mathbf {y}^{\prime }=\lambda \mathbf {y},\quad U(\mathbf {y})=0$$

(7)

and the functions $r_j$ belong to the space $L_1(0,\pi )$, $j=1,2$. If $p, q\in L_2(0,\pi )$ (for short, we write $V\in L_2(0,\pi )$), then $r_j\in L_2(0,\pi )$. It follows that the function $\Delta (\lambda )$ is an entire function of exponential type; therefore, we only have the following possibilities for the operator $\mathbb {L}$ of problem (1), (2):

• 1. The spectrum is empty.

• 2. The spectrum is a finite nonempty set.

• 3. The spectrum is a countable set without finite limit points.

• 4. The spectrum fills the entire complex plane.

Relations (5) and (6) imply that case 1 is realized for problem (7), for example, with the boundary conditions defined by the matrix

$$ A=\begin {pmatrix} 1&i&-1&i\\ 1&-i&1&i \end {pmatrix},$$

and case 4, with the boundary conditions defined by the matrix

$$ A=\begin {pmatrix} 1&-i&0&0\\ 0&0&i&1 \end {pmatrix}.$$

Let us prove that case 2 is impossible. Let the equation

$$ \Delta (\lambda )=0 $$

have finitely many roots $\lambda _k$, $k={1,\ldots ,n}$. If $C_1C_2\ne 0$, then conditions (2) are regular and problem (1), (2) has a countable set of eigenvalues; therefore, $C_1C_2=0$. Set $P(\lambda )=\prod _{k=1}^n(\lambda -\lambda _k)$. By [2],

$$ \Delta (\lambda )=P(\lambda )e^{a\lambda +b},$$

where $a$ and $b$ are some constants. Assume, for example, that $C_2=0$. Setting $\lambda =-iy$ in relation (5), where $y>0$, we obtain

$$ J_0+C_1e^{\pi y}+R(-iy)=P(-iy)e^{-iay+b},$$

which implies that

$$ J_0e^{-\pi y}+C_1+e^{-\pi y}R(-iy)=P(-iy)e^{b-i\thinspace \mathrm {Re}\thinspace a y}e^{(\mathrm {Im}\thinspace a-\pi )y}. $$

(8)

According to [3, p. 36], the expression on the left-hand side in relation (8) tends to $C_1$ as $y\to \infty$. If $\mathrm {Im}\thinspace a-\pi \ge 0$, then the expression on the right-hand side in relation (8) tends to infinity in absolute value, and if $\mathrm {Im}\thinspace a-\pi <0$, then it tends to zero. It follows that $C_1=0$. If $C_1=C_2=0$, then

$$ R(\lambda )=P(\lambda )e^{a\lambda +b}. $$

(9)

Obviously, the left-hand side of relation (9) is bounded on the real axis, while the right-hand side is not; that is, we arrive at a contradiction.

Definition.

We say that problem (1), (2) has the classical spectral asymptotics if its spectrum is a countable set and the multiplicities of the eigenvalues are uniformly bounded.

The present paper is aimed at constructing problems (1), (2) for which case 3 is realized and the multiplicities of the eigenvalues grow unboundedly, i.e., problems with nonclassical spectral asymptotics.

MAIN RESULTS

Set $c_j(\lambda )=c_j(\pi ,\lambda )$ and $s_j(\lambda )=s_j(\pi ,\lambda )$, $j=1,2$. In addition, let $PW_\sigma$ be the class of entire functions $f(z)$ of the exponential type $\le \sigma$ such that $\|f\|_{L_2(R)}<\infty$. It is well known [4] that the functions $c_j(\lambda )$ and $s_j(\lambda )$ admit the representation

$$ c_j(\lambda )=\cos (\pi \lambda )+g_j(\lambda ),\quad s_j(\lambda )=\sin (\pi \lambda )+h_j(\lambda ), $$

where $g_j,h_j\in PW_\pi$, $j=1,2$.

Lemma 1 [5].

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

$$ u(\lambda )=\sin (\pi \lambda )+h(\lambda ),\quad v(\lambda )=\cos (\pi \lambda )+g(\lambda ), $$

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

$$ u(\lambda )=-\pi (\lambda _0 -\lambda )\prod _{\substack {n=-\infty \\ n\ne 0}}^\infty \frac {\lambda _n-\lambda }{n}, $$

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

$$ v(\lambda )=\prod _{n=-\infty }^\infty \frac {\lambda _n-\lambda }{n-1/2}, $$

where $\lambda _n=n-1/2+\kappa _n$ and $\{\kappa _n\}\in l_2$ .

Consider the Dirac system with the boundary conditions defined by the matrix

$$ A=\begin {pmatrix} 1&0&0&1\\ 0&1&1&0 \end {pmatrix}. $$

(10)

We will assume that $V\in L_2(0,\pi )$. It follows from the representation (4) that the characteristic determinant $\Delta (\lambda )$ of problem (1), (2) with matrix $A$ defined in (10) can be reduced to the form

$$ \Delta (\lambda )=s_1(\lambda )-s_2(\lambda )=\int _{-\pi }^\pi r(t)e^{i\lambda t}\thinspace dt=f(\lambda ), $$

where $r\in L_2(0,\pi )$, and $f\in PW_\pi$. The converse statement holds true as well.

Theorem.

For each function $f\in PW_\pi$, there exists a potential $V\in L_2(0,\pi )$ such that the characteristic determinant $\Delta (\lambda )$ of problem (1), (2) with the matrix $A$ defined by relation (10) and the potential $V(x)$ is identically equal to $f(\lambda )$.

Proof. Let $f(\lambda )$ be an arbitrary function in the class $PW_\pi$. It follows from the Paley–Wiener theorem and [3, p. 36] that

$$ \lim \limits _{|\lambda |\to \infty }e^{-\pi |\mathrm {Im}\thinspace \lambda |}f(\lambda )=0;$$

(11)

consequently, there exists a positive integer $N_0$ so large that $|f(\lambda )|<1/100$ if $\mathrm {Im}\thinspace \lambda =0$ and $|\mathrm {Re}\thinspace \lambda |\ge N_0$.

Let $\{\lambda _n\}$ , $n\in \mathbb {Z}$ , be a strictly monotone increasing sequence of real numbers such that $N_0<\lambda _n<N_0+1/100$ if $1\le n\le N_0$ , $\lambda _n=n-1/2$ if $n>N_0$ , and $\lambda _n=-\lambda _{-n+1}$ for any $n$. Set

$$ c(\lambda )=\prod _{n=-\infty }^\infty \frac {\lambda _n-\lambda }{n-1/2}.$$

Lemma 1 implies the relation

$$ c(\lambda )=\cos (\pi \lambda )+g(\lambda ), $$

(12)

where $g\in PW_\pi$. It follows from the Paley–Wiener theorem and [3, p. 36] that

$$ \lim \limits _{|\lambda |\to \infty } e^{-\pi |\mathrm {Im}\thinspace \lambda |}g(\lambda )=0;$$

therefore,

$$ \big |c(\lambda )\big |\ge c_0e^{\pi |\mathrm {Im}\thinspace \lambda |}$$

(13)

($c_0=\mathrm {const}>0$) for $|\mathrm {Im}\thinspace \lambda |\ge M$, where $M$ is a sufficiently large number.

Differentiating relation (12), we obtain

$$ \dot c(\lambda )=-\pi \sin (\pi \lambda )+\dot g(\lambda ).$$

(14)

Since the function $\dot g$ belongs to the class $PW_\pi$, we have, according to [6],

$$ \dot c(\lambda _n)=-\pi \sin (\pi \lambda _n)+\tau _n, $$

where

$$ \sum _{n=-\infty }^\infty |\tau _n|^2<\infty . $$

Based on this, by the definition of the numbers $\lambda _n$, we obtain

$$ \dot c(\lambda _n)=\pi (-1)^n+\rho _n, $$

(15)

where

$$ \sum _{n=-\infty }^\infty |\rho _n|^2<\infty . $$

Consequently, for all even $n$ sufficiently large in modulus one has the inequality $\dot c(\lambda _n)>0$. One can readily see that the inequality $\dot c(\lambda _n)\dot c(\lambda _{n+1})<0$ holds for all $n\in \mathbb {Z}$. It follows that

$$ (-1)^n\dot c(\lambda _n)>0 $$

(16)

for all $n\in \mathbb {Z}$. Note that (15) implies the relation

$$ \frac {1}{\dot c(\lambda _n)}=\frac {(-1)^n}{\pi }+\sigma _n,$$

(17)

where

$$ \sum _{n=-\infty }^\infty |\sigma _n|^2<\infty . $$

Consider the quadratic equation

$$ w^2+f(\lambda _n)w-1=0.$$

(18)

It has the roots

$$ s_n^{\pm }=\frac {-f(\lambda _n)\pm \sqrt {f^2(\lambda _n)+4}}{2}. $$

By $\Gamma (z,r)$ we denote the disk of radius $r$ centered at point $z$. One can readily see that all numbers $s_n^{+}$ lie inside the disk $\Gamma (1,1/10)$ and all numbers $s_n^{-}$ lie inside the disk $\Gamma (-1,1/10)$. Let $s_n=s_n^{+}$ if $n$ is odd and $s_n=s_n^{-}$ if $n$ is even. Since [6] $\{f(\lambda _n)\}\in l_2$, it follows from the definition of the numbers $s_n$ that

$$ s_n=(-1)^{n+1}+\vartheta _n, $$

(19)

where $\{\vartheta _n\}\in l_2$. It also follows from the definition of the numbers $s_n$ and inequality (16) that all numbers $z_n={s_n}/{\dot c(\lambda _n)}$ lie strictly to the left of the imaginary axis, while (17) and (19) imply the relation

$$ z_n=-\frac {1}{\pi }+\rho _n,$$

where $\{\rho _n\}\in l_2$. Let $\beta _n=s_n-\sin (\pi \lambda _n)$; then $\{\beta _n\}\in l_2$ in view of (19). Set

$$ h(\lambda )=c(\lambda )\sum _{n=-\infty }^\infty \frac {\beta _n}{\dot c(\lambda _n)(\lambda -\lambda _n)}. $$

According to [7, p. 120], the function $h$ belongs to the class $PW_\pi$, and $h(\lambda _n)=\beta _n$. Set $s(\lambda )=\sin (\pi \lambda )+h(\lambda )$; then $s(\lambda _n)=s_n\ne 0$, and consequently, the functions $s(\lambda )$ and $c(\lambda )$ do not have common roots.

Set

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

In the subsequent exposition, we need the following elementary assertion.

Lemma 2.

If function systems $\{\varphi _n\}$ and $\{\psi _n\}$ are complete in $L_2(a,b)$ $(n\in \mathbb {N})$ , then the system of vectors

$$ \Psi _{n,n}=\begin {pmatrix} \{\varphi _n\}\\ \{\psi _n\} \end {pmatrix} \cup \begin {pmatrix} \{\varphi _n\}\\ \{-\psi _n\} \end {pmatrix} $$

is complete in $L_{2,2}(a,b)$ .

Proof. Assume that there exists a vector $f(x)=\mathrm {col}\thinspace (f_1(x),f_2(x))\ne 0$ such that

$$ \int _a^b\big (\varphi _n(x)\overline {f_1(x)}+\psi _n(x) \overline {f_2(x)}\thinspace \big )\thinspace dx=0,\quad \int _a^b\big (\varphi _n(x)\overline {f_1(x)}-\psi _n(x) \overline {f_2(x)}\thinspace \big )\thinspace dx=0 $$

for all $n\in \mathbb {N}$. Then

$$ \int _a^b\varphi _n(x)\overline {f_1(x)}\thinspace dx=0, \quad \int _a^b\psi _n(x)\overline {f_2(x)}\thinspace dx=0;$$

consequently, $f_1(x)\equiv f_2(x)\equiv 0$. The proof of the lemma is complete.

It follows from [8] that the function systems $\{\cos (\lambda _n x)\}$ and $\{\sin (\lambda _n x)\}$ $(n\in \mathbb {N})$ are complete in $L_2(0,\pi )$. Based on this, it follows from the definition of the numbers $\lambda _n$ and Lemma 2 that the system of vectors

$$ Y_0(x,\lambda _n)= \begin {pmatrix} \cos (\lambda _n x)\\ \sin (\lambda _n x) \end {pmatrix} $$

($n\in \mathbb {Z}$) is complete in $L_{2,2}(0,\pi )$. Set

$$ F(x,t)=-\sum _{n=-\infty }^\infty \biggl (\frac {s_n}{\dot c(\lambda _n)}(Y_0(x,\lambda _n) Y_0^{\mathrm {T}}(t,\lambda _n))+\frac {1}{\pi }Y_0(x,n-1/2) Y_0^{\mathrm {T}}(t,n-1/2)\biggr ). $$

(20)

It follows from [4] that

$$ \|F(\thinspace \cdot \thinspace ,x)\|_{L_{2,2}^{2,2}(0,\pi )}+\|F(x,\cdot \thinspace )\|_{L_{2,2}^{2,2}(0,\pi )}<C,$$

where $C$ is a constant independent of $x$. Let us prove that for each $x\in [0,\pi ]$ the homogeneous equation

$$ f^{\mathrm {T}}(t)+\int _0^xf^{\mathrm {T}}(s)F(s,t)\thinspace ds=0,$$

(21)

where $f(t)=\mathrm {col}\thinspace (f_1(t),f_2(t))$, $f\in L_{2,2}(0,x)$, $f(t)=0$ for $x<t\le \pi$, has only the trivial solution. Multiplying Eq. (21) by $\overline {f^{\mathrm {T}}(t)}$ and integrating the resulting equation over the segment $[0,x]$, we obtain

$$ \|f\|^2_{L_{2,2}(0,x)}+\int _0^x\left \langle \thinspace \int _0^xf^{\mathrm {T}}(s)F(s,t)\thinspace ds, \overline {f^{\mathrm {T}}(t)}\right \rangle \thinspace dt=0. $$

Taking into account definition (20), by simple calculations we find that

$$ \begin {aligned} &{}f^{\mathrm {T}}(s)F(s,t)\\ &\;{}=-\left \{\sum _{n=-\infty }^\infty \bigg \{z_n\big [f_1(s)\cos (\lambda _ns) \cos (\lambda _nt)+f_2(s)\sin (\lambda _ns)\cos (\lambda _nt),\right .\\ &\;\qquad \qquad \qquad {}f_1(s)\cos (\lambda _ns)\sin (\lambda _nt)+f_2(s)\sin (\lambda _ns)\sin (\lambda _nt)\big ]\\ &\;\qquad {}+\frac {1}{\pi }\Big [f_1(s)\cos \big ((n-1/2)s\big )\cos \big ((n-1/2)t\big )+f_2(s)\sin \big ((n-1/2)s\big )\cos \big ((n-1/2)t\big ),\\ &\;\qquad \qquad \qquad {}\left .f_1(s)\cos \big ((n-1/2)s\big )\sin \big ((n-1/2)t\big )+f_2(s)\sin \big ((n-1/2)s\big )\sin \big ((n-1/2)t\big )\Big ]\bigg \}\vphantom {\sum _{n=-\infty }^\infty }\right \}\\ &\;{}=-\left \{\sum _{n=-\infty }^\infty \bigg \{z_n\big [f_1(s)\cos (\lambda _ns)\cos (\lambda _nt)+f_2(s)\sin (\lambda _ns)\cos (\lambda _nt)\big ]\right .\\ &\;\qquad {}+\frac {1}{\pi }\Big [f_1(s)\cos \big ((n-1/2)s\big )\cos \big ((n-1/2)t\big )+f_2(s)\sin \big ((n-1/2)s\big )\cos \big ((n-1/2)t\big )\Big ],\\ &\;\qquad \qquad \qquad {}z_n\big [f_1(s)\cos (\lambda _ns)\sin (\lambda _nt)+f_2(s)\sin (\lambda _ns)\sin (\lambda _nt)\big ]\\ &\;\qquad {}+\left .\frac {1}{\pi } \Big [f_1(s)\cos \big (((n-1/2)s)\big )\sin \big ((n-1/2)t\big )f_2(s)\sin \big ((n-1/2)s\big )\sin \big ((n-1/2)t\big )\Big ]\bigg \}\vphantom {\sum _{n=-\infty }^\infty }\right \}, \end {aligned}$$

which implies that

$$ \begin {aligned} &\int _0^x\left \langle \int _0^x f^{\mathrm {T}}(s)F(s,t)\thinspace ds,\overline {f^{\mathrm {T}}(t)}\right \rangle \thinspace dt \\ &\quad {}=-\left \{\sum _{n=-\infty }^\infty \int _0^x\left (\int _0^x\bigg \{z_n\big [f_1(s)\cos (\lambda _ns)\cos (\lambda _nt)+f_2(s)\sin (\lambda _ns)\cos (\lambda _nt)\big ]\right .\right . \\[2ex] &\quad \quad \quad \quad \quad \quad {}+\frac {1}{\pi }\Big [f_1(s)\cos \big ((n-1/2)s\big )\cos \big ((n-1/2)t\big ) \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}+\left .f_2(s)\sin \big ((n-1/2)s\big )\cos \big ((n-1/2)t\big )\Big ]\bigg \}\thinspace ds\vphantom {\int _0^x}\right )\overline {f_1(t)}\thinspace dt \\ &\quad \quad \quad \quad {}+\sum _{n=-\infty }^\infty \int _0^x\left (\int _0^x \bigg \{z_n\big [f_1(s)\cos (\lambda _ns)\sin (\lambda _nt)+f_2(s)\sin (\lambda _ns)\sin (\lambda _nt)\big ]\right . \\[2ex] &\quad \quad \quad \quad \quad \quad {}+\frac {1}{\pi }\Big [f_1(s)\cos \big ((n-1/2)s\big )\sin \big ((n-1/2)t\big ) \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}+\left .\left .f_2(s)\sin \big ((n-1/2)s\big )\sin \big ((n-1/2)t\big )\Big ]\bigg \}\thinspace ds\vphantom {\int _0^x}\right )\overline {f_2(t)}\thinspace dt\right \} \\ &\quad {}=-\left \{\sum _{n=-\infty }^\infty \left (\int _0^xz_n\big [f_1(s)\cos (\lambda _ns)+f_2(s)\sin (\lambda _ns)\big ]\thinspace ds \int _0^x\cos (\lambda _nt)\overline {f_1(t)}\thinspace dt\right .\right . \\[1ex] &\quad \quad \quad \quad \quad \quad {}+\left .\frac {1}{\pi }\int _0^x\Big [f_1(s)\cos \big ((n\!-\!1/2)s\big )+f_2(s)\sin \big ((n\!-\!1/2)s\big )\Big ]\thinspace ds\int _0^x\cos \big ((n\!-\!1/2)t\big )\overline {f_1(t)}\thinspace dt\right ) \\ &\quad \quad \quad \quad {}+\sum _{n=-\infty }^\infty \left (\int _0^x z_n\big [f_1(s)\cos (\lambda _ns)+f_2(s)\sin (\lambda _ns)\big ]\thinspace ds\int _0^x\sin (\lambda _nt)\overline {f_2(t)}\thinspace dt\right . \\[1ex] &\quad \quad \quad \quad \quad \quad {}+\left .\left .\frac {1}{\pi }\int _0^x\Big [f_1(s)\cos \big ((n\!-\!1/2)s\big )\!+\!f_2(s)\sin \big ((n\!-\!1/2)s\big )\Big ]\thinspace ds\!\int _0^x\sin \big ((n\!-\!1/2)t\big )\overline {f_2(t)}\thinspace dt\right )\!\right \} \\ &\quad {}=-\left \{\sum _{n=-\infty }^\infty \left (\int _0^x z_n\big [f_1(s)\cos (\lambda _ns)+f_2(s)\sin (\lambda _ns)\big ]\thinspace ds \int _0^x\cos (\lambda _nt)\overline {f_1(t)}\thinspace dt\right .\right . \\ &\quad \quad \quad \quad \quad \quad \quad \quad {}+\left .\int _0^x\big [f_1(s)\cos (\lambda _ns)+f_2(s)\sin (\lambda _ns)\big ]\thinspace ds\int _0^x\sin (\lambda _nt)\overline {f_2(t)}\thinspace dt\right ) \\[1ex] &\quad \quad \quad \quad {}+\frac {1}{\pi }\!\sum _{n=-\infty }^\infty \!\left (\int _0^x\Big [f_1(s)\cos \big ((n\!-\!1/2)s\big )\!+\!f_2(s)\sin \big ((n\!-\!1/2)s\big )\Big ]\thinspace ds\!\right )\!\int _0^x\cos \big ((n\!-\!1/2)t\big )\overline {f_1(t)}\thinspace dt \\ &\quad \quad \quad \quad {}+\left .\left .\int _0^x\Big [f_1(s)\cos \big ((n-1/2)s\big )+f_2(s)\sin \big ((n-1/2)s\big )\Big ]\thinspace ds\int _0^x\sin \big ((n-1/2)t\big )\overline {f_2(t)}\thinspace dt \right )\right \} \\ &\quad {}=-\left \{\sum _{n=-\infty }^\infty \left (\int _0^x z_n\big [f_1(t)\cos (\lambda _nt)+f_2(t)\sin (\lambda _nt)\big ]\thinspace dt\int _0^x\cos (\lambda _nt)\overline {f_1(t)}\thinspace dt\right .\right . \\ &\quad \quad \quad \quad \quad \quad \quad \quad {}+\left .\int _0^x\big [f_1(t)\cos (\lambda _nt)+f_2(t)\sin (\lambda _nt)\big ]\thinspace dt\int _0^x\sin (\lambda _nt)\overline {f_2(t)}\thinspace dt\right ) \\ &\quad \quad \quad \quad {}+\frac {1}{\pi }\sum _{n=-\infty }^\infty \left (\int _0^x\Big [f_1(t)\cos \big ((n\!-\!1/2)t\big )+f_2(t)\sin \big ((n\!-\!1/2)t\big )\Big ]\thinspace dt\int _0^x\cos \big ((n\!-\!1/2)t\big )\overline {f_1(t)}\thinspace dt\right . \\ &\quad \quad \quad \quad \quad \quad \quad \quad {}+\left .\left .\int _0^x\Big [f_1(t)\cos (nt)+f_2(t)\sin \big ((n-1/2)t\big )\Big ]\thinspace dt\int _0^x\sin \big ((n-1/2)t\big )\overline {f_2(t)}\thinspace dt \right )\right \} \\ &\quad {}=-\left \{\sum _{n=-\infty }^\infty z_n\int _0^x \big [f_1(t)\cos (\lambda _nt)+f_2(t)\sin (\lambda _nt)\big ]\thinspace dt\int _0^x\big [\overline {f_1(t)}\cos (\lambda _nt)+\overline {f_2(t)}\sin (\lambda _nt)\big ]\thinspace dt\right . \\ &\quad \quad \quad \quad \quad \quad \quad \quad {}+\sum _{n=-\infty }^\infty \frac {1}{\pi }\int _0^x\Big [f_1(t)\cos \big ((n-1/2)t\big )+ f_2(t)\sin \big ((n-1/2)t\big )\Big ]\thinspace dt \\ &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {}\times \left .\int _0^x\Big [\overline {f_1(t)}\cos \big ((n-1/2)t\big )+\overline {f_2(t)}\sin \big ((n-1/2)t\big )\Big ]\thinspace dt\right \} \\ &\quad {}=-\sum _{n=-\infty }^\infty z_n\left |\thinspace \int _0^x\big \langle f(t),Y_0(t,\lambda _n)\big \rangle \thinspace dt\thinspace \right |^2-\sum _{n=-\infty }^\infty \frac {1}{\pi }\left |\thinspace \int _0^x\big \langle f(t),Y_0(t,(n-1/2))\big \rangle \thinspace dt\thinspace \right |^2. \end {aligned} $$

In view of Parseval’s identity, we obtain

$$ \|f\|^2_{L_{2,2}(0,x)}=\sum _{n=-\infty }^\infty \frac {1}{\pi }\left |\thinspace \int _0^x\big \langle f(t),Y_0(t,(n-1/2))\big \rangle \thinspace dt\thinspace \right |^2; $$

therefore,

$$ \sum _{n=-\infty }^\infty z_n\left |\thinspace \int _0^x\big \langle f(t),Y_0(t,\lambda _n)\big \rangle \thinspace dt\thinspace \right |^2=0.$$

(22)

Since $\mathrm {Re}\thinspace z_n<0$ for each $n$, Eq. (22) implies that $\int \nolimits _0^x\langle f(t),Y_0(t,\lambda _n)\rangle \thinspace dt=0$. The latter and the completeness of the system of vectors $\{Y_0(t,\lambda _n)\}$ in $L_{2,2}(0,\pi )$ imply the identity $f(t)\equiv 0$. The unique solvability of Eq. (21) implies [4] that the functions $c(\lambda )$ and $-s(\lambda )$ are the entries of the first row of the monodromy matrix

$$ \tilde U(\pi ,\lambda )= \begin {pmatrix} \tilde c_1(\pi ,\lambda )&-\tilde s_2(\pi ,\lambda )\\ \tilde s_1(\pi ,\lambda )&\tilde c_2(\pi ,\lambda ) \end {pmatrix} $$

of problem (1), (2) with the matrix $A$ defined in (10) and some potential $\tilde V\in L_2(0,\pi )$; i.e.,

$$ c(\lambda )=\tilde c_1(\pi ,\lambda ),\quad s(\lambda )=\tilde s_2(\pi ,\lambda ).$$

(23)

By virtue of (4), the characteristic determinant $\tilde \Delta (\lambda )$ of this problem has the form

$$ \tilde \Delta (\lambda )=\tilde s_1(\pi ,\lambda )-\tilde s_2(\pi ,\lambda )=\tilde f(\lambda ), $$

where $\tilde f\in PW_\pi$. Relations (3), (18), and (23) imply the equality

$$ \tilde \Delta (\lambda _n)=\tilde s_1(\pi ,\lambda _n)-\tilde s_2(\pi ,\lambda _n)= \frac {1}{\tilde s_2(\pi ,\lambda _n)}-\tilde s_2(\pi ,\lambda _n)=\frac {1}{s(\lambda _n)}-s(\lambda _n)=f(\lambda _n).$$

It follows from the last equality that the function

$$ \Phi (\lambda )=\frac {f(\lambda )-\tilde \Delta (\lambda )}{c(\lambda )}=\frac {f(\lambda )-\tilde f(\lambda )}{c(\lambda )} $$

is entire. Since

$$ \big |f(\lambda )-\tilde f(\lambda )\big |<c_1e^{\pi |\mathrm {Im}\thinspace \lambda |},\quad c_1=\mathrm {const}, $$

(24)

we conclude in view of inequality (13) that $|\Phi (\lambda )|\le c_2=\mathrm {const}$ if $|\mathrm {Im}\thinspace \lambda |\ge M$.

Let $H$ stand for the union of vertical segments $\{z:|\mathrm {Re}\thinspace z|=n,|\mathrm {Im}\thinspace z|\le M\}$, where $|n|=N_0+1, N_0+2,\ldots$ Since the function $c(\lambda )$ is a sine-type function [9], we have $|c(\lambda )|>\delta >0$ for $\lambda \in H$. The last inequality, the estimate (24), and the maximum principle imply the inequality $|\Phi (\lambda )|<c_3=\mathrm {const}$ in the strip $|\mathrm {Im}\thinspace \lambda |\le M$. Consequently, the function $\Phi (\lambda )$ is bounded in the entire complex plane and is constant by Liouville’s theorem. Let $|\mathrm {Im}\thinspace \lambda |=M$. Then, in view of relation (11), we have $\lim \limits _{|\lambda |\to \infty }(f(\lambda )-\tilde f(\lambda ))=0$; therefore, $\Phi (\lambda )\equiv 0$, and hence $f(\lambda )\equiv \tilde \Delta (\lambda )$. The proof of the theorem is complete.

Examples of functions in the class $PW_\pi$ with roots of arbitrarily high multiplicity are known in the literature (see, e.g., [10, 11]). Note that the existence of one-dimensional boundary value problems with an unboundedly increasing multiplicity of eigenvalues was previously established for the Sturm–Liouville operator and an ordinary differential operator of any even order [10,11,12].