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This article is cited in 1 scientific paper (total in 1 paper) On completeness of the root function system of the A. S. Makin Institute of Applied Mathematics and Mechanics, Donetsk
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    § 1. IntroductionIn the present paper, we study the Dirac system with two-point boundary conditions
$$
\begin{equation}
\begin{aligned} \, U_1(\mathbf{y}) &= a_{11}y_1(0)+a_{12}y_2(0)+a_{13}y_1(\pi)+ a_{14}y_2(\pi)=0, \\ U_2(\mathbf{y}) &= a_{21}y_1(0)+a_{22}y_2(0)+a_{23}y_1(\pi)+ a_{24}y_2(\pi)=0, \end{aligned}
\end{equation}
\tag{2}
$$
where $\mathbf{y}=\operatorname{col}(y_1(x),y_2(x))$,
$$
\begin{equation*}
B=\begin{pmatrix} -i&0\\ 0&i \end{pmatrix},\qquad V=\begin{pmatrix} 0&P(x)\\ Q(x)&0 \end{pmatrix},
\end{equation*}
\notag
$$
the functions $P, Q$ lie in $L_1(0,\pi)$, the coefficients $a_{jk}$ are arbitrary complex numbers, and the rows of the matrix
$$
\begin{equation*}
A=\begin{pmatrix} a_{11}&a_{12}&a_{13} &a_{14} \\ a_{21}&a_{22}&a_{23} &a_{24} \end{pmatrix}
\end{equation*}
\notag
$$
are linearly independent. The operator $\mathbb{L}\mathbf{y}=B\mathbf{y}'+V\mathbf{y}$ is as a linear operator in $\mathbb{H}=L_2(0,\pi)\oplus L_2(0,\pi)$, with domain $D(\mathbb{L})=\{\mathbf{y}\in W_1^1[0,\pi]\oplus W_1^1[0,\pi]\colon \mathbb{L}\mathbf{y}\in \mathbb{H},\, U_j(\mathbf{y})=0\, (j=1,2)\}$. Let $A_{jk}$ $(1\leqslant j<k\leqslant4)$ be the determinant composed of the $j$th and $k$th columns of the matrix $A$. Boundary conditions (2) are called regular if otherwise, they are non-regular.Birkhoff and Langer [1] were the first to study the general spectral problem for $(n\times n)$ first-order systems of ordinary differential equations (ODEs) on a finite interval. More precisely, they introduced the concepts of regular and strictly regular boundary conditions, investigated the asymptotic behaviour of the eigenvalues and eigenfunctions, and proved that the spectral decompositions for the corresponding differential operator converge pointwise. The first result on completeness of the root functions is due to Ginzburg [2], who studied the case $B = I_n$, $V(\,{\cdot}\,) = 0$. Marchenko [3] showed that the system of root functions of the operator $\mathbb{L}$ with regular boundary conditions and continuous matrix potential $V$ is complete. This restriction occurs because the transformation operators used for the proof were constructed in [3] only for continuous potentials. Later, Malamud and Oridoroga [4] verified completeness for $B$-weakly regular boundary value problems for arbitrary $(n\times n)$ first-order systems of ODEs with integrable matrix potential $V\in L^1([0,\pi];\mathbb{C}^{n\times n})$ (this result was announced in [5] in 2000). Note that if conditions (2) are not regular, then the completeness property depends essentially on the potential $V$; in particular, in this case, the root function system of the unperturbed operator is not complete in $\mathbb{H}$ (see [3]).The first result on completeness for the $(2\times 2)$-Dirac-type operator $\mathbb{L}$ with matrix $B = \operatorname{diag}(b_1, b_2)$ and non-regular boundary conditions was obtained in [4]. Namely, it was shown that, with smooth $P,Q\in C^1[0,\pi]$, the system of root functions of the operator $\mathbb{L}$ is complete under the conditions
$$
\begin{equation}
\begin{aligned} \, |A_{32}| + |b_1A_{13}P(0) + b_2A_{42}Q(\pi)| &\ne0, \\ |A_{14}| + |b_1A_{13}P(\pi) + b_2A_{42}Q(0)| &\ne0. \end{aligned}
\end{equation}
\tag{4}
$$
In [6], similar results were obtained in the case of analytic $B\ne B^*$, $P$ and $Q$. The results in [6] and 4 depend on the method of transformation operators. Lunyov and Malamud [7], [8] generalized the results of [4] to establish potential-specific results on completeness and spectral synthesis results for the system of root functions of the $(n\times n$) system with non-weakly-regular boundary conditions assuming that the $n\times n$ potential matrix $V(\,{\cdot}\,)$ is continuous only at the end-points $0$ and $\pi$. In [9], Lunyov and Malamud extended completeness results from [4] for the $(2\times 2)$-Dirac-type operators $\mathbb{L}$ that involve boundary values $V^{(k)}(0)$ and $V^{(k)}(\pi)$, $k\in{0, 1,\ldots, m-1}$, of the derivatives of the potential $V \in W^m_2([0, \pi];C^{2\times 2})$. Kosarev and Shkalikov [10] established completeness results in the case of $(2\times 2)$-Dirac-type operators with non-constant matrix $B = \operatorname{diag}(b_1(x), b_2(x))$ and degenerate boundary conditions of special form $(y_1(0) = y_2(\pi) = 0)$; these results are natural extensions of the corresponding results from [8] and [4]. So, the system of root functions of the operator $\mathbb{L}$ is complete whenever the functions $b_1$, $b_2$, $P$, and $Q$ are absolutely continuous and satisfy $P(\pi)Q(0)\ne0$. Recently, Makin [11] obtained sufficient conditions for completeness for the root function system of problem (1), (2) with $A_{14}A_{23}=0$, $|A_{13}|+|A_{42}|>0$ and with potential $V$ from $L_1(0,\pi)$. In [9], Lunyov and Malamud strengthened their results announced in [9]. For a complete system of root functions, the question is whether this system forms a basis. Most complete result on the Riesz basis property of boundary value problems for $(2\times2$)-Dirac systems with $V(\,{\cdot}\,)\in L^1([0,1]; \mathbb{C}^{2\times2})$ and strongly regular boundary conditions were obtained independently and at the same time, but with different methods, by Savchuk and Shkalikov [13], and by Lunyov and Malamud [13] and [14]. The block Riesz basis property in the case of an $L_1$-potential matrix and regular boundary conditions was first proved in [13]. In [16], the author of the present paper, for spectral problems for the Dirac operator with regular but not strongly regular boundary conditions and complex-valued summable potential, obtained conditions under which the root function system forms a usual Riesz basis rather than a block Riesz basis. The present paper is concerned with the completeness problem in the case $A_{13}=A_{42}=0$. § 2. PreliminariesLet
$$
\begin{equation}
E(x,\lambda)=\begin{pmatrix} e_{11}(x,\lambda)&e_{12}(x,\lambda) \\ e_{21}(x,\lambda)&e_{22}(x,\lambda) \end{pmatrix}
\end{equation}
\tag{5}
$$
be the matrix of the fundamental solution system to equation (1) with boundary condition $E(0,\lambda)=I$, where $I$ is the identity matrix. It is well known (see [13], [14]) that
$$
\begin{equation}
\begin{alignedat}{2} e_{11}(x,\lambda) &=e^{ix\lambda}(1+o(1))+e^{-ix\lambda}o(1), &\qquad e_{12}(x,\lambda) &=e^{ix\lambda}o(1)+e^{-ix\lambda}o(1), \\ e_{21}(x,\lambda) &=e^{ix\lambda}o(1)+e^{-ix\lambda}o(1), &\qquad e_{22}(x,\lambda) &=e^{ix\lambda}o(1)+e^{-ix\lambda}(1+o(1)) \end{alignedat}
\end{equation}
\tag{6}
$$
as $\lambda\to\infty$ uniformly with respect to $x\in[0,\pi]$. The eigenvalues of problem (1), (2) are the roots of the characteristic equation where
$$
\begin{equation*}
\Delta(\lambda)= \begin{vmatrix} U_1(E^{[1]}(\,{\cdot}\,,\lambda))&U_1(E^{[2]}(\,{\cdot}\,,\lambda)) \\ U_2(E^{[1]}(\,{\cdot}\,,\lambda))&U_2(E^{[2]}(\,{\cdot}\,,\lambda)) \end{vmatrix},
\end{equation*}
\notag
$$
and $E^{[k]}(x,\lambda)$ is the $k$th column of matrix (5). The machinery of triangular transformation operators was used in [14] to show that the characteristic determinant $\Delta(\lambda)$ of problem (1), (2) satisfies
$$
\begin{equation}
\begin{aligned} \, \Delta(\lambda) &=A_{12}+A_{34}+A_{32}e_{11}(\pi,\lambda)+A_{14}e_{22}(\pi,\lambda)+A_{13}e_{12}(\pi,\lambda) +A_{42}e_{21}(\pi,\lambda) \nonumber \\ &=\Delta_0(\lambda)+\int_0^\pi r_1(t)e^{-i\lambda t}\, dt+\int_0^\pi r_2(t)e^{i\lambda t}\, dt, \end{aligned}
\end{equation}
\tag{7}
$$
where the function
$$
\begin{equation*}
\Delta_0(\lambda)=A_{12}+A_{34}-A_{23}e^{i\pi\lambda}+A_{14}e^{-i\pi\lambda}
\end{equation*}
\notag
$$
is the characteristic determinant of problem (3), and $r_j\in L_1(0,\pi)$, $j=1,2$. A representation similar to (7) was obtained in the recent paper [17] for the characteristic determinant for general first-order $(n\times n)$-systems of ODEs. For convenience, we recall some commonly used relations from [11]. Let $\lambda$ be a complex number, $\operatorname{Im}\lambda\ne0$, $\rho>0$. Suppose that $\tau$ is continuous on $[0,\pi]$. Then, for any $b\in[0,\pi]$,
$$
\begin{equation}
\biggl|\int_0^b x^\rho e^{-2|{\operatorname{Im}\lambda}| x}\tau(x)\, dx\biggr| \leqslant\frac{c}{|{\operatorname{Im}\lambda}|^{\rho+1}},
\end{equation}
\tag{8}
$$
where $c$ is independent of $b$. If $\tau\in L_1(0,\pi)$, then
$$
\begin{equation}
\biggl|\int_0^\pi x^\rho e^{-2|{\operatorname{Im}\lambda}| x}\tau(x)\, dx\biggr| =\frac{o(1)}{|{\operatorname{Im}\lambda}|^{\rho}}
\end{equation}
\tag{9}
$$
as $|\operatorname{Im}\mu|\to\infty$. In addition, simple computations show that if $\rho>0$, $\lambda>0$, $\rho\leqslant \pi\lambda$, then
$$
\begin{equation}
\max_{0\leqslant x\leqslant \pi}x^\rho e^{-\lambda x}=\frac{\rho^\rho}{\lambda^\rho}e^{-\rho}.
\end{equation}
\tag{10}
$$
We set
$$
\begin{equation}
g_1(t,\lambda) =\int_0^t e^{-2i\lambda t_1}P(t_1)\, dt_1\int_0^{t_1}e^{2i\lambda t_2}Q(t_2) \, dt_2,
\end{equation}
\tag{12}
$$
$$
\begin{equation}
\begin{aligned} \, g_n(t,\lambda) &=\int_0^t e^{-2i\lambda t_1}P(t_1) \, dt_1\int_0^{t_1}e^{2i\lambda t_2}Q(t_2) \,dt_2 \nonumber \\ &\qquad\times\dots\times \int_0^{t_{2n-2}}e^{-2i\lambda t_{2n-1}}P(t_{2n-1})\, dt_{2n-1} \int_0^{t_{2n-1}}e^{2i\lambda t_{2n}}Q(t_{2n})\, dt_{2n} \end{aligned}
\end{equation}
\tag{13}
$$
and, similarly,
$$
\begin{equation}
h_1(t,\lambda) =\int_0^t e^{2i\lambda t_1}Q(t_1)\, dt_1\int_0^{t_1}e^{-2i\lambda t_2}P(t_2)\, dt_2,
\end{equation}
\tag{15}
$$
$$
\begin{equation}
\begin{aligned} \, h_n(t,\lambda) &=\int_0^t e^{2i\lambda t_1}Q(t_1) \, dt_1\int_0^{t_1}e^{-2i\lambda t_2}P(t_2) \, dt_2 \nonumber \\ &\qquad\times\dots\times \int_0^{t_{2n-2}} e^{2i\lambda t_{2n-1}}Q(t_{2n-1})\, dt_{2n-1} \int_0^{t_{2n-1}}e^{-2i\lambda t_{2n}}P(t_2) \, dt_{2n}. \end{aligned}
\end{equation}
\tag{16}
$$
Lemma 1 (see [11]). The following representations hold: § 3. The main resultsLet $0<\varepsilon<\pi/10$. Let $\Omega^+_\varepsilon$ and $\Omega^-_\varepsilon$ denote, respectively, the domains $\varepsilon\leqslant \arg\lambda\leqslant\pi-\varepsilon$ and $-\pi+\varepsilon\leqslant \arg\lambda\leqslant-\varepsilon$. For brevity, we write $\|f\|:=\|f\|_{L_1(0,\pi)}$. To prove that the system of root functions is complete, we first estimate the functions $e_{jj}(\pi,\lambda)$ $(j=1,2)$ in the domains $\Omega^+_\varepsilon$ and $\Omega^-_\varepsilon$, and then find a lower estimate for the characteristic determinant $\Delta(\lambda)$ in the domain $\Omega_\varepsilon=\Omega_\varepsilon^-\cup\Omega_\varepsilon^+$. Lemma 2. Suppose that where $\rho_1>0$, $\rho_2>0$. Then, in the domain $\Omega_\varepsilon^+$, where $c_1>0$. Proof. For later purposes, we need some inequalities. From (19) we have where $\tau_2(h)\to0$ as $h\to0$, and the function $\tau_2(\,{\cdot}\,)$ is continuous. We set $\widehat P(x)= P(\pi-x)$. A similar analysis shows that where $\tau_1(h)\to0$ as $h\to0$, and the function $\tau_1(\,{\cdot}\,)$ continuous.
Let $0\leqslant t\leqslant\pi$. Integrating by parts, we obtain
$$
\begin{equation*}
\begin{aligned} \, \int_0^t e^{2i\lambda y}Q(y) \, dy &= e^{2it\lambda}\int_0^t Q(y) \, dy - 2i\lambda \int_0^t e^{2i\lambda y}\, dy\, \biggl(\int_0^y Q(t_1) \, dt_1\biggr) \\ &=t^{\rho_2} e^{2it\lambda}(\nu_2 + \tau_2(t))-2i\lambda \int_0^t y^{\rho_2} e^{2i\lambda y}(\nu_2 + \tau_2(y)) \, dy. \end{aligned}
\end{equation*}
\notag
$$
This together with (8) and (10) implies that
$$
\begin{equation}
\biggl|\int_0^{t} e^{2i\lambda x}Q(x)\, dx\biggr|\leqslant\frac{c_2}{|{\operatorname{Im}\lambda}|^{\rho_2}},
\end{equation}
\tag{21}
$$
where $c_2$ is independent of $t$.
Setting $y=\pi-s$, integrating by parts, and substituting $\pi-t=v$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_t^\pi e^{-2i\lambda y}P(y) \, dy = e^{-2i\pi\lambda}\int_0^{\pi-t}e^{2is\lambda}\widehat P(s) \, ds \\ &\qquad=e^{-2it\lambda}\int_0^{\pi-t}\widehat P(x) \, dx -2i\lambda e^{-2i\pi\lambda} \int_0^{\pi-t} e^{2i\lambda s} \, ds\, \biggl(\int_0^s\widehat P(x)\, ds\biggr) \\ &\qquad=(\pi-t)^{\rho_1} e^{-2it\lambda}(\nu_1 + \tau_1(\pi-t))-2i\lambda e^{-2i\pi\lambda}\int_0^{\pi-t} s^{\rho_1} e^{2i\lambda s}(\nu_1 + \tau_1(s)) \, ds \\ &\qquad= e^{-2i\pi\lambda}(v^{\rho_1} e^{2iv\lambda}(\nu_1 + \tau_1(v)))-2i\lambda\int_0^{\pi-t} s^{\rho_1} e^{2i\lambda s}(\nu_1 + \tau_1(s)) \, ds. \end{aligned}
\end{equation*}
\notag
$$
The last equality together with (8) and (10) implies that
$$
\begin{equation}
\biggl|\int_{t}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr| \leqslant \frac{c_3e^{2\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_1}}
\end{equation}
\tag{22}
$$
and
$$
\begin{equation}
\biggl|\int_0^{\pi-t}e^{2i\lambda s}\widehat P(s)\, ds\biggr| \leqslant \frac{c_3}{|{\operatorname{Im}\lambda}|^{\rho_1}},
\end{equation}
\tag{23}
$$
where in both cases $c_3$ is independent of $t$.
Let us estimate the function
$$
\begin{equation*}
g_1(\pi,\lambda)=\int_0^\pi e^{-2i\lambda t}P(t)\, dt\int_0^{t}e^{2i\lambda x}Q(x)\, dx.
\end{equation*}
\notag
$$
Changing the order of integration, we obtain
$$
\begin{equation*}
\begin{aligned} \, g_1(\pi,\lambda) &=\int_0^\pi e^{2i\lambda x}Q(x) \, dx\int_x^\pi e^{-2i\lambda t}P(t)\, dt \\ &=\int_0^\pi e^{2i\lambda x}Q(x)\, dx\, \biggl(\int_0^\pi e^{-2i\lambda t}P(t)\, dt-\int_0^x e^{-2i\lambda t} P(t)\, dt\biggr) \\ &=\int_0^\pi e^{2i\lambda x}Q(x)\, dx\int_0^\pi e^{-2i\lambda t}P(t)\, dt -\int_0^\pi e^{2i\lambda x}Q(x)\, dx \int_0^x e^{-2i\lambda t}P(t)\, dt. \end{aligned}
\end{equation*}
\notag
$$
An appeal to Lemma 3.5 in [11] shows that
$$
\begin{equation}
\biggl|\int_0^\pi e^{-2i\lambda t}P(t)\, dt\biggr| \geqslant \frac{c_4e^{2\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_1}}
\end{equation}
\tag{24}
$$
$(c_4>0)$. Now, by Lemma 3.7 in [11],
$$
\begin{equation}
\biggl|\int_0^\pi e^{2i\lambda x}Q(x)\, dx\biggr| \geqslant \frac{c_5}{|{\operatorname{Im}\lambda}|^{\rho_2}}
\end{equation}
\tag{25}
$$
$(c_5>0)$. By the Hölder inequality,
$$
\begin{equation*}
\begin{aligned} \, &\biggl|\int_0^\pi e^{2i\lambda x}Q(x)\, dx\int_0^x e^{-2i\lambda t}P(t)\, dt\biggr| \\ &\qquad\leqslant\int_0^\pi |Q(x)|\, dx\int_0^x |e^{2i\lambda (x-t)}P(t)| \, dt\leqslant \|Q\|\, \|P\|<c_6. \end{aligned}
\end{equation*}
\notag
$$
From the last inequality and (24) and (25), we obtain
$$
\begin{equation}
|g_1(\pi,\lambda)|\geqslant\frac{c_7e^{2\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}} \qquad (c_7>0).
\end{equation}
\tag{26}
$$
Let us estimate the function $g_2(\pi,\lambda)$. Using (12), (13) and changing the order of integration, we obtain
$$
\begin{equation}
\begin{aligned} \, g_2(\pi,\lambda) &=\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\int_0^{t_1}e^{2i\lambda t_2}Q(t_2)g_1(t_2,\lambda)\, dt_2 \nonumber \\ &= \int_0^\pi e^{2i\lambda t_2}Q(t_2)g_1(t_2,\lambda)\, dt_2\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \end{aligned}
\end{equation}
\tag{27}
$$
and
$$
\begin{equation}
\begin{aligned} \, &g_1(t_2,\lambda) =\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\int_x^{t_2} e^{-2i\lambda t}P(t)\, dt \nonumber \\ &\qquad= \int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\, \biggl(\int_0^{t_2} e^{-2i\lambda t}P(t)\, dt -\int_0^x e^{-2i\lambda t}P(t)\, dt\biggr) \nonumber \\ &\qquad=\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\int_0^{t_2} e^{-2i\lambda t}P(t)\, dt-\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\int_0^x e^{-2i\lambda t}P(t)\, dt. \end{aligned}
\end{equation}
\tag{28}
$$
Hence
$$
\begin{equation*}
\begin{aligned} \, &g_2(\pi,\lambda) \\ &\quad= \int_0^\pi e^{2i\lambda t_2}Q(t_2)\, dt_2\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\int_0^{t_2} e^{-2i\lambda t} P(t)\, dt \int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \\ &\quad\qquad-\int_0^\pi e^{2i\lambda t_2}Q(t_2)\, dt_2\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\int_0^x e^{-2i\lambda t} P(t)\, dt\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \\ &\quad=I_1-I_2. \end{aligned}
\end{equation*}
\notag
$$
An appeal to Lemma 3.4 in [11] shows that
$$
\begin{equation}
\biggl|\int_0^{t_2} e^{-2i\lambda t}P(t)\, dt\biggr|=o(1)e^{2t_2|{\operatorname{Im}\lambda}|}.
\end{equation}
\tag{29}
$$
Using the Hölder inequality and estimates (21), (22), (29), we find that
$$
\begin{equation}
\begin{aligned} \, |I_1| &\leqslant \|Q\|\max_{0\leqslant t_2\leqslant\pi}\biggl|e^{2i\lambda t_2}\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx \int_0^{t_2} e^{-2i\lambda t}P(t)\, dt\int_{t_2}^\pi e^{-2i\lambda t_1} P(t_1)\, dt_1\biggr| \nonumber \\ &=\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}. \end{aligned}
\end{equation}
\tag{30}
$$
Consider $I_2$. It is clear that
$$
\begin{equation}
I_2=\int_0^\pi e^{2i\lambda t_2}Q(t_2)\, dt_2\, \psi(t_2,\lambda)\int_{t_2}^\pi e^{-2i\lambda t_1} P(t_1)\, dt_1,
\end{equation}
\tag{31}
$$
where
$$
\begin{equation*}
\psi(t_2,\lambda)=\int_0^{t_2} e^{2i\lambda x}Q(x)\, dx\int_0^x e^{-2i\lambda t}P(t)\, dt.
\end{equation*}
\notag
$$
From condition (19) we have
$$
\begin{equation}
\int_0^{x}Q(s) \, ds=\gamma_2x^{\rho_2}+\gamma_2x^{\rho_2}\tau(x),
\end{equation}
\tag{32}
$$
where the function $\tau(x)$ is continuous on $[0,\pi]$ and $\tau(0)=0$. Integrating by parts and using (32), we obtain
$$
\begin{equation}
\begin{aligned} \, &\psi(t_2,\lambda) =\int_0^{t_2}\biggl[ e^{2i\lambda x}\int_0^x e^{-2i\lambda t}P(t)\, dt\biggr]\, d\int_0^xQ(s)\, ds \nonumber \\ &\qquad=e^{2i\lambda t_2}\int_0^{t_2}e^{-2i\lambda t}P(t)\, dt\int_0^{t_2}Q(s)\, ds \nonumber \\ &\qquad\qquad\!-2i\lambda\int_0^{t_2} e^{2i\lambda x}\, dx\int_0^x e^{-2i\lambda t} P(t)\, dt \int_0^xQ(s)\, ds -\int_0^{t_2}P(x)\, dx\int_0^xQ(s)\, ds \nonumber \\ &\qquad=[\gamma_2t_2^{\rho_2}+\gamma_2t_2^{\rho_2}\tau_2(t_2)]e^{2i\lambda t_2}\int_0^{t_2}e^{-2i\lambda t} P(t) \, dt \nonumber \\ &\qquad\qquad-2i\lambda\int_0^{t_2} [\gamma_2x^{\rho_2}+\gamma_2x^{\rho_2}\tau_2(x)]e^{2i\lambda x}\, dx \int_0^x e^{-2i\lambda t}P(t)\, dt \nonumber \\ &\qquad\qquad-\int_0^{t_2}[\gamma_2x^{\rho_2}+\gamma_2x^{\rho_2}\tau_2(x)]P(x)\, dx. \end{aligned}
\end{equation}
\tag{33}
$$
Substituting (33) into (31), this gives
$$
\begin{equation}
\begin{aligned} \, &\int_0^\pi e^{2i\lambda t_2}Q(t_2)\, dt_2 \biggl\{[\gamma_2t_2^{\rho_2}+\gamma_2t_2^{\rho_2}\tau_2(t_2)]e^{2i\lambda t_2}\int_0^{t_2}e^{-2i\lambda t} P(t) \, dt \nonumber \\ &\qquad-2i\lambda\int_0^{t_2} [\gamma_2x^{\rho_2}+\gamma_2x^{\rho_2}\tau_2(x)]e^{2i\lambda x}\, dx \int_0^x e^{-2i\lambda t}P(t)\, dt \nonumber \\ &\qquad-\int_0^{t_2}[\gamma_2x^{\rho_2}+\gamma_2x^{\rho_2}\tau_2(x)]P(x)\, dx\biggr\}\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \nonumber \\ &=\gamma_2\biggl\{\int_0^\pi [t_2^{\rho_2}+t_2^{\rho_2}\tau_2(t_2)]e^{2i\lambda t_2}Q(t_2)\, dt_2 \nonumber \\ &\qquad\qquad\times e^{2i\lambda t_2} \int_0^{t_2}e^{-2i\lambda t}P(t)\, dt\int_{t_2}^\pi e^{-2i\lambda t_1} P(t_1)\, dt_1 \nonumber \\ &\qquad-2i\lambda \int_0^\pi e^{2i\lambda t_2}Q(t_2)\, dt_2\int_0^{t_2} [x^{\rho_2}+x^{\rho_2}\tau_2(x)]e^{2i\lambda x}\, dx \nonumber \\ &\qquad\qquad\times\int_0^x e^{-2i\lambda t}P(t)\, dt\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \nonumber \\ &\qquad-\int_0^\pi e^{2i\lambda t_2}Q(t_2) \, dt_2 \int_0^{t_2}[x^{\rho_2}+x^{\rho_2}\tau_2(x)]P(x)\, dx\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr\} \nonumber \\ &=\gamma_2\{I_{21}-I_{22}-I_{23}\}. \end{aligned}
\end{equation}
\tag{34}
$$
Employing the Hölder inequality and inequalities (10), (22), (29), we obtain
$$
\begin{equation}
\begin{aligned} \, |I_{21}| &\leqslant\|Q\|\max_{0\leqslant t_2\leqslant\pi}|t_2^{\rho_2}e^{2i\lambda t_2}|\max_{0\leqslant t_2\leqslant\pi}\biggl|e^{2i\lambda t_2} \int_0^{t_2}e^{-2i\lambda t}P(t)\, dt\biggr| \nonumber \\ &\qquad\times\max_{0\leqslant t_2\leqslant\pi}\biggl|\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr|= \frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}. \end{aligned}
\end{equation}
\tag{35}
$$
Now an appeal to the Hölder inequality, inequalities (8), $x\leqslant t_2$, Lemma 3.4 of [11], and (22) shows that
$$
\begin{equation}
\begin{aligned} \, |I_{22}| &\leqslant\|Q\|\, |\lambda|\max_{0\leqslant t_2\leqslant\pi}\biggl| e^{2i\lambda t_2}\int_0^{t_2} [x^{\rho_2}+x^{\rho_2}\tau(x)]e^{2i\lambda x}\, dx \nonumber \\ &\qquad\times\int_0^x e^{-2i\lambda t}P(t)\, dt\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr| \nonumber \\ &\leqslant\|Q\|\, |\lambda|\max_{0\leqslant t_2\leqslant\pi} \biggl|\int_0^{t_2} |x^{\rho_2}+x^{\rho_2}\tau(x)|e^{-2|{\operatorname{Im}\lambda}| x}\, dx\biggr| \nonumber \\ &\qquad\times\max_{0\leqslant t_2\leqslant\pi}\biggl|e^{2i\lambda t_2}\int_0^x e^{-2i\lambda t}P(t)\, dt\biggr| \max_{0\leqslant t_2\leqslant\pi}\biggl|\int_{t_2}^\pi e^{-2i\lambda t_1} P(t_1)\, dt_1\biggr| \nonumber \\ &=\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}. \end{aligned}
\end{equation}
\tag{36}
$$
Next, by the Hölder inequality, (9), and (22), we have
$$
\begin{equation}
\begin{aligned} \, |I_{23}| &\leqslant c_8\int_0^\pi t_2^{\rho_2}e^{-2|{\operatorname{Im}\lambda}| t_2}|Q(t_2)|\, dt_2 \int_0^{t_2}|P(x)|\, dx\biggl| \int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr| \nonumber \\ &\leqslant c_9\|P\|\max_{0\leqslant t_2\leqslant\pi} \biggl|\int_{t_2}^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr| \int_0^\pi t_2^{\rho_2}e^{-2|{\operatorname{Im}\lambda}| t_2}|Q(t_2)|\, dt_2 \nonumber \\ &=\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}. \end{aligned}
\end{equation}
\tag{37}
$$
From (34), (36) and (37) we obtain
$$
\begin{equation*}
|I_{2}|=\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}.
\end{equation*}
\notag
$$
Combining the last inequality and (30), we have
$$
\begin{equation}
g_2(\pi,\lambda)=\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}.
\end{equation}
\tag{38}
$$
Assume that $n>2$. We set
$$
\begin{equation*}
\begin{aligned} \, F_n(t_2,\lambda) &=\int_0^{t_2} e^{-2i\lambda t_3}P(t_3) \, dt_3\int_0^{t_3}e^{2i\lambda t_4} Q(t_4)\, dt_4 \\ &\qquad\times\dots\times \int_0^{t_{2n-2}}e^{-2i\lambda t_{2n-1}}P(t_{2n-1}) \, dt_{2n-1} \int_0^{t_{2n-1}}e^{2i\lambda t_{2n}}Q(t_{2n})\, dt_{2n}. \end{aligned}
\end{equation*}
\notag
$$
It is easy to see that
$$
\begin{equation}
\begin{aligned} \, g_n(\pi,\lambda) &=\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\int_0^{t_1}e^{2i\lambda t_2} Q(t_2) F_n(t_2,\lambda)\, dt_2 \nonumber \\ &=\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggl(\int_0^\pi e^{2i\lambda t_2}Q(t_2)F_n(t_2,\lambda)\, dt_2 \nonumber \\ &\qquad- \int_{t_1}^\pi e^{2i\lambda t_2}Q(t_2)F_n(t_2,\lambda)\, dt_2\biggr) \nonumber \\ &=q_n(\pi,\lambda)\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \nonumber \\ &\qquad-\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \int_{t_1}^\pi e^{2i\lambda t_2}Q(t_2)F_n(t_2,\lambda)\, dt_2, \end{aligned}
\end{equation}
\tag{39}
$$
where
$$
\begin{equation}
\begin{aligned} \, q_n(t,\lambda) &=\int_0^t e^{2i\lambda t_2}Q(t_2)F_n(t_2,\lambda)\,dt_2 \nonumber \\ &= \int_0^t e^{2i\lambda t_2}Q(t_2)\, dt_2\int_0^{t_2}e^{-2i\lambda t_3}P(t_3)\, dt_3\cdots\int_0^{t_{2n-1}} e^{2i\lambda t_{2n}}Q(t_{2n})\, dt_{2n}. \end{aligned}
\end{equation}
\tag{40}
$$
Consider the first term on the right of (39). From 3.5 in [11] we have
$$
\begin{equation*}
\biggl|\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1\biggr| \leqslant \frac{c_{10}e^{2\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_1}}.
\end{equation*}
\notag
$$
Next, by Lemma 3.7 in [11],
$$
\begin{equation*}
\sum_{n=3}^\infty |q_n(\pi,\lambda)|=\frac{o(1)}{|{\operatorname{Im}\lambda}|^{\rho_2}},
\end{equation*}
\notag
$$
and hence,
$$
\begin{equation}
\sum_{n=3}^\infty |q_n(\pi,\lambda)|\biggl|\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \biggr| =\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}.
\end{equation}
\tag{41}
$$
Let us estimate the second term in (39). Changing the order of integration and replacing $t_1=\pi-s$, we obtain
$$
\begin{equation*}
\begin{aligned} \, I_3 &=\int_0^\pi e^{-2i\lambda t_1}P(t_1)\, dt_1 \int_{t_1}^\pi e^{2i\lambda t_2} Q(t_2)F_n(t_2,\lambda) \, dt_2 \\ &= \int_0^\pi e^{2i\lambda t_2}Q(t_2)F_n(t_2,\lambda)\, dt_2\int_0^{t_2}e^{-2i\lambda t_1}P(t_1)\, dt_1 \\ &=e^{-2i\pi\lambda}\int_0^\pi e^{2i\lambda t_2}Q(t_2)F_n(t_2,\lambda)\, dt_2\int_0^{\pi-t_2}e^{2i\lambda s} \widehat P(s)\, ds, \end{aligned}
\end{equation*}
\notag
$$
where $\widehat P(s)=P(\pi-s)$. From the Hölder inequality and (23), we have
$$
\begin{equation}
|I_3|\leqslant\frac{c_{11}e^{2\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_1}}\|Q\|\max_{0\leqslant t_2\leqslant\pi}|e^{2i\lambda t_2}F_n(t_2,\lambda)|.
\end{equation}
\tag{42}
$$
Consider the function $F_n(t_2,\lambda)$ and define
$$
\begin{equation}
\begin{aligned} \, \phi(t_4,\lambda)&=e^{2i\lambda t_4}Q(t_4)\int_0^{t_{4}}e^{-2i\lambda t_5}P(t_5) \, dt_5 \nonumber \\ &\qquad\times\dots\times \int_0^{t_{2n-2}}e^{-2i\lambda t_{2n-1}} P(t_{2n-1})\, dt_{2n-1} \int_0^{t_{2n-1}}e^{2i\lambda t_{2n}}Q(t_{2n})\, dt_{2n}. \end{aligned}
\end{equation}
\tag{43}
$$
Changing the order of integration, we obtain
$$
\begin{equation}
\begin{aligned} \, F_n(t_2,\lambda) &=\int_0^{t_2} e^{-2i\lambda t_3}P(t_3)\, dt_3\int_0^{t_3}\phi(t_4,\lambda)\, dt_4 \nonumber \\ &=\int_0^{t_2}\phi(t_4,\lambda)\, dt_4\int_{t_4}^{t_2}e^{-2i\lambda t_3}P(t_3)\, dt_3, \end{aligned}
\end{equation}
\tag{44}
$$
and hence
$$
\begin{equation}
|F_n(t_2,\lambda)|\leqslant\int_0^{t_2}|\phi(t_4,\lambda)|\, dt_4\biggl|\int_{t_4}^{t_2}e^{-2i\lambda t_3} P(t_3)\, dt_3\biggr|.
\end{equation}
\tag{45}
$$
By Lemma 3.4 in [11],
$$
\begin{equation}
\biggl|\int_{t_4}^{t_2}e^{-2i\lambda t_3}P(t_3)\, dt_3\biggr| =o(1)e^{2t_2|{\operatorname{Im}\lambda}|}.
\end{equation}
\tag{46}
$$
An appeal to (45) and (46) shows that
$$
\begin{equation}
|e^{2i\lambda t_2}F_n(t_2,\lambda)|=o(1)\int_0^{t_2}|\phi(t_4,\lambda)|\, dt_4.
\end{equation}
\tag{47}
$$
Let us estimate the integral on the right-hand side of (47). From Lemma 3.1 in [11], (21), and since $t_j\geqslant t_{j+1}$, we have
$$
\begin{equation}
\begin{aligned} \, &\int_0^{t_2}|\phi(t_4,\lambda)|\, dt_4 \leqslant\|Q\|\max_{0\leqslant t_4\leqslant\pi}\biggl|e^{2i\lambda t_4} \int_0^{t_{4}}e^{-2i\lambda t_5}P(t_5)\, dt_5 \nonumber \\ &\qquad\times\dots\times \int_0^{t_{2n-2}}e^{-2i\lambda t_{2n-1}} P(t_{2n-1})\, dt_{2n-1} \int_0^{t_{2n-1}}e^{2i\lambda t_{2n}}Q(t_{2n})\, dt_{2n}\biggr| \nonumber \\ &\leqslant \|Q\|\max_{0\leqslant t_4\leqslant\pi}\biggl|\int_0^{t_{4}}|P(t_5)|\, dt_5 \int_0^{t_{2n-2}}|e^{2i\lambda(t_4-t_5+t_6-\dots-t_{2n-1})}| \, |P(t_{2n-1})| \, dt_{2n-1} \nonumber \\ &\qquad\times\dots\times \int_0^{t_{2n-1}}e^{2i\lambda t_{2n}}Q(t_{2n})\, dt_{2n}\biggr| \nonumber \\ &\leqslant \frac{c_{12}^n}{(2n-4)!\, |{\operatorname{Im}\lambda}|^{\rho_2}}. \end{aligned}
\end{equation}
\tag{48}
$$
Next, using (42), (47), (48), this gives
$$
\begin{equation}
|I_3|=\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}.
\end{equation}
\tag{49}
$$
From (39), (41), (49) we have
$$
\begin{equation}
\sum_{n=3}^\infty |g_n(\pi,\lambda)| =\frac{e^{2\pi|{\operatorname{Im}\lambda}|}o(1)}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}.
\end{equation}
\tag{50}
$$
Now (20) follows from (17), (38), (26), (50). This proves Lemma 2. Lemma 3. Suppose that where $\rho_3>0$, $\rho_4>0$. Then, in the domain $\Omega_\varepsilon^-$, where $c>0$. Proof. The required result is an easy consequence of the above analysis. Indeed, the estimate for the function $e_{22}(\,{\cdot}\,,{\cdot}\,)$ is obtained from (11)–(18) if in formula (17) for the function $e_{11}(\,{\cdot}\,,{\cdot}\,)$ one replaces $\lambda$ by $-\lambda$ and swaps the functions $P$ and $Q$. This proves the lemma. Our main result is the following theorem. Theorem 1. Suppose that $A_{14}A_{32}=A_{13}=A_{24}=0$ and one of the following two conditions (53) and (54) is satisfied: where $\rho_3>0$, $\rho_4>0$; where $\rho_1>0$, $\rho_2>0$. Then the root function system of problem (1), (2) is complete and minimal in $\mathbb{H}$. Proof. Let $|\lambda|$ be sufficiently large. If (53) is met, then by (6)
$$
\begin{equation*}
|e_{22}(\pi,\lambda)|\geqslant c_1e^{\pi|\operatorname{Im} \lambda|}
\end{equation*}
\notag
$$
$(c_1>0)$ if $\lambda\in\Omega_\varepsilon^+$. By Lemma 3,
$$
\begin{equation*}
|e_{22}(\pi,\lambda)|\geqslant\frac{c_2e^{\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_3+\rho_4}}
\end{equation*}
\notag
$$
$(c_2>0)$ if $\lambda\in\Omega_\varepsilon^-$. Now by (7) we have
$$
\begin{equation*}
|\Delta(\lambda)|\geqslant \frac{c_3e^{\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_3+\rho_4}}
\end{equation*}
\notag
$$
$(c_3>0)$ in the domain $\Omega_\varepsilon$. A similar argument shows that
$$
\begin{equation*}
|\Delta(\lambda)|\geqslant \frac{c_4e^{\pi|{\operatorname{Im}\lambda}|}}{|{\operatorname{Im}\lambda}|^{\rho_1+\rho_2}}
\end{equation*}
\notag
$$
$(c_4>0)$ if (54) is met and $\lambda\in\Omega_\varepsilon$. Now by Theorem 2.3 in [8], in both cases the root function system of problem (1), (2) is complete and minimal in $\mathbb{H}$. This proves Theorem 1. Remark 1. The case $A_{14}A_{23}=A_{13}=A_{42}=0$ was treated by Lunyov and Malamud in [12], where the more general first-order system The author wishes to express his deep gratitude to the referee.
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\Bibitem{Mak25} |
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