On a formula for characteristic determinant of a boundary value problem for $n \times n$ Dirac type system

Consider the following $n \times n$ Dirac type equation:
\begin{equation} \label{eq:Dir-type,eq-n} -i y' - i Q(x) y = \lambda B(x) y, \quad y = {\rm col}(y_1, \ldots, y_n), \quad x \in [0,\ell], \end{equation}
on a finite interval $[0,\ell]$ subject to the general two-point boundary conditions
\begin{equation} \label{eq:BC-s} C y(0) + D y(\ell) = 0, \quad C, D \in \mathbb{C}^{n \times n}, \quad \text{rank} (CC^* + DD^*) =n. \end{equation}
Here $Q = (Q_{jk})_{j,k=1}^n$ is an integrable potential matrix and $B = {\rm diag}(\beta_1, \ldots, \beta_n) = B^*$ is a diagonal integrable matrix “weight”. If $n=2m$ and $B(\cdot) = {\rm diag}(-I_m, I_m)$, this equation turns into the classical $n\times n$ Dirac equation.
In this talk we discuss the spectral properties of the boundary value problems (BVP) \eqref{eq:Dir-type,eq-n}–\eqref{eq:BC-s}. The key role in our investigation is playing the following representation of the characteristic determinant $\Delta_Q(\cdot)$ of the BVP \eqref{eq:Dir-type,eq-n}–\eqref{eq:BC-s}:
\begin{equation} \label{eq:Ch-det-repres-n} \Delta_Q(\lambda) = \Delta_0(\lambda) + \int_{b_-}^{b_+} g(u) e^{i \lambda u} \,du, \quad g \in L^1[b_-, b_+]. \end{equation}
Here $\Delta_0(\cdot)$ is the characteristic determinant of the unperturbed BVP ($Q=0$) and $b_{\pm}$ are explicitly expressed via entries of the matrix function $B(\cdot)$. Formula \eqref{eq:Ch-det-repres-n} is valid under certain assumptions on $\{\beta_k(\cdot)\}_1^n$ and $Q$. In particular, it holds for $B = B^* = const$. In this case the implication $Q \in L^p[b_-, b_+]\otimes \Bbb C^{n\times n} \Longrightarrow g \in L^p[b_-, b_+]$ holds. In the case of $B = {\rm diag}(b_1, b_2) = {\rm const}$ formula \eqref{eq:Ch-det-repres-n} is obtained in [1].
In turn, under certain assumption on $\{\beta_k(\cdot)\}_1^n$ formula \eqref{eq:Ch-det-repres-n} yields asymptotic behavior of the spectrum in the case of regular boundary conditions. Namely, we show that $\lambda_m = \lambda_m^0 + o(1)$ as $m \to \infty$, where $\{\lambda_m\}_{m \in \mathbb{Z}}$ and $\{\lambda_m^0\}_{m \in \mathbb{Z}}$ are sequences of eigenvalues of perturbed and unperturbed BVPs, respectively. It is also shown that for $Q \in L^p$, $p \in (1,2]$, (and constant $B$) the following estimate holds:
$$ \sum_{m \in \mathbb{Z}} |\lambda_m - \lambda_m^0|^{p'} + \sum_{m \in \mathbb{Z}} (1+|m|)^{p-2} |\lambda_m - \lambda_m^0|^{p} < \infty, \quad p' := p/(p-1). $$

The talk is based on a joint paper [2] with Anton Lunyov.