This paper is concerned with the self-adjoint Dirac-type systems on the interval \([0,b)\), where \(0 < b \leq +\infty\), which have the following form \[ -iJy'(x,z) + V(x) y(x,z) = z y(x,z),\tag{1} \] where \[ y = \left(\begin{matrix} y_1, \ldots, y_m \end{matrix}\right)^\top,\quad J = \left(\begin{matrix} I_{m_1} & 0 \\ 0 & I_{m_2} \end{matrix}\right),\quad V = \left(\begin{matrix} 0 & v \\ v^* & 0 \end{matrix}\right) \] and \(m = m_1 + m_2\), \(m_1, m_2 \in \mathbb{N}\), \(I_{m_k}\), for \(k = 1,2\), is the \(m_k \times m_k\) identity matrix, \(v\) is a \(m_1 \times m_2\) matrix-valued function. The potential \(v\) is assumed summable on \([0,b)\), when \(b < +\infty\), and it is assumed locally summable on \([0,+\infty)\), when \(b = +\infty\).
The main goal of this paper is to show that the potential of system (1) with additional local smoothness conditions is uniquely determined by asymptotics of a Weyl-Titchmarsh function \(M(x,z)\), given by \[ M(x,z) = \psi_{21}(x,z) \psi_{11}(x,z)^{-1},\quad z \in \mathbb{C}_+, \tag{2} \] where \(\psi_{11}\) and \(\psi_{21}\) are \(m_1 \times m_1\) and \(m_2 \times m_1\) matrix-valued functions and \[ \psi(x,z) = \left(\begin{matrix} \psi_{11}(x,z) \\ \psi_{21}(x,z) \end{matrix}\right) \] is the Weyl solution of system (1), that is \(\psi(x,z)\) is square-integrable and satisfies some boundary condition at \(+\infty\), when \(b = +\infty\), and \(\psi\) satisfies some boundary condition at \(b\), when \(b < +\infty\).
The first result of the paper slightly generalizes the well-known local uniqueness theorem for system (1). Let \(c \in \mathbb{R}\) and let \[ \Gamma = \{z : \operatorname{Re}(z) = c \operatorname{Im}(z),\, \operatorname{Im}(z) > 0\}. \] Theorem 3.3. Let \(\tilde{M}(0,z)\) be the Weyl-Titchmarsh function of system (1) with some potential \(\tilde{v}\), given by (2). Suppose that \(a \in (0,b)\). If \[ M(0,z) - \tilde{M}(0,z) = O\left(e^{2iaz}\right) \] as \(|z| \to +\infty\), \(z \in \Gamma\), then \[ v = \tilde{v}\quad \text{a.e. on \([0,a]\)}.\tag{3} \] Conversely, if (3) holds, then \[ M(0,z) - \tilde{M}(0,z) = o\left(e^{2iaz}\right)\tag{4} \] as \(|z| \to +\infty\), \(z \in \mathbb{C}_+\).
Next result is the main result of the paper.
Theorem 4.2. Under the assumptions of Theorem 3.3, if \(v, \tilde{v} \in C^n([a, a + \varepsilon))^{m_1 \times m_2}\) for some \(\varepsilon > 0\), then \[ M(0,z) - \tilde{M}(0,z) = o\left(z^{-(n+1)}e^{2iaz}\right) \] as \(|z| \to +\infty\), \(z \in \Gamma\), if and only if \[ v = \tilde{v}\quad \text{a.e. on \([0,a]\)} \] and \(v^{(p)}(a+) = \tilde{v}^{(p)}(a+)\) for every \(p = 0,1,\ldots,n\), where all \(v^{(p)}(a+)\) denote the \(p\)th right-hand derivative of \(v\) at the point \(a\).