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The Riesz basis property with brackets for Dirac systems with summable potentials
A. M. Savchuk,
I. V. Sadovnichaya Lomonosov Moscow State University, Moscow, Russia
Abstract:
In the space
$\mathbb H=(L_2[0,\pi])^2$, we study the Dirac operator
$\mathcal L_{P,U},$ generated by the differential
expression
$\ell_P(\mathbf y)=B\mathbf y'+P\mathbf y$, where
$$
B=\begin{pmatrix}
-i&0\\
0&i
\end{pmatrix},
\qquad
P(x)=
\begin{pmatrix}
p_1(x)& p_2(x)\\
p_3(x)& p_4(x)
\end{pmatrix},
\qquad
\mathbf y(x)=
\begin{pmatrix}
y_1(x)\\
y_2(x)
\end{pmatrix},
$$
and the regular boundary conditions
$$
U(\mathbf y)=
\begin{pmatrix}
u_{11}& u_{12}\\
u_{21}& u_{22}
\end{pmatrix}
\begin{pmatrix}
y_1(0)\\
y_2(0)
\end{pmatrix}+
\begin{pmatrix}
u_{13}& u_{14}\\
u_{23}& u_{24}
\end{pmatrix}
\begin{pmatrix}
y_1(\pi)\\
y_2(\pi)
\end{pmatrix}=0.
$$
The elements of the matrix
$P$ are assumed to be complex-valued functions summable over
$[0,\pi]$. We show that the spectrum of the operator
$\mathcal L_{P,U}$ is discrete and consists of eigenvalues
$\{\lambda_n\}_{n\in\mathbb Z}$, such that
$\lambda_n=\lambda_n^0+o(1)$ as
$|n|\to\infty$, where
$\{\lambda_n^0\}_{n\in\mathbb Z}$ is the spectrum of the operator
$\mathcal L_{0,U}$ with zero potential and the same boundary conditions. If the boundary conditions are strongly regular, then the spectrum of the operator
$\mathcal L_{P,U}$ is asymptotically simple. We show that the system of eigenfunctions and associate functions of the operator
$\mathcal L_{P,U}$ forms a Riesz base in the space
$\mathbb H$ provided that the eigenfunctions are normed. If the boundary conditions are regular, but not strongly regular, then all eigenvalues of the operator
$\mathcal L_{0,U}$ are double, all eigenvalues of the operator
$\mathcal L_{P,U}$ are asymptotically double, and the system formed by the corresponding two-dimensional root subspaces of the operator
$\mathcal L_{P,U},$ is a Riesz base of subspaces (Riesz base with brackets) in the space
$\mathbb H$.
UDC:
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