Riesz basis property of root vectors system for $n \times n$ Dirac type operators

In this talk we investigate spectral properties of selfadjoint and non-selfadjoint boundary value problems (BVP) for the following first order system of ordinary differential equations
\begin{equation*} L y = -i B(x)^{-1} \bigl(y' + Q(x) y\bigr) = \lambda y , \quad B(x) = B(x)^*, \quad y= {\rm co}l(y_1, \ldots, y_n), \quad x \in [0,\ell], \end{equation*}
on a finite interval $[0,\ell]$. Here $Q \in L^1([0,\ell]; \Bbb C^{n \times n})$ is a potential matrix and $B \in L^{\infty}([0,\ell]; \Bbb R^{n \times n})$ is an invertible self-adjoint diagonal “weight” matrix.
If $n=2m$ and $B(x) = {\rm diag}(-I_m, I_m)$ this equation is equivalent to Dirac equation of order $n$.
Here we discuss the spectral properties of BVP associated with the above equation subject to general BC $U(y)=Cy(0)+Dy(\ell) = 0$, ${\rm rank}(C \ D) = n$.
As a first our main result, we mention that the deviation of the characteristic determinants $\Delta(\lambda) - \Delta_0(\lambda)$ of perturbed and unperturbed (with $Q=0$) BVPs admits the Fourier transform representation of a certain summable function explicitly expressed via kernels of the transformation operators. In turn, this representation leads to the asymptotic formula $\lambda_m = \lambda_m^0 + o(1)$ as $m \to \infty$, for the eigenvalues $\{\lambda_m\}_{m \in \Bbb Z}$ and $\{\lambda_m^0\}_{m \in \Bbb Z}$ of perturbed and unperturbed ($Q=0$) regular BVPs, respectively. In the case of $n=2$ and constant matrix $B(x) = B$ both results are obtained in [1].
Further, we prove that the system of root vectors of the above BVP constitutes a Riesz basis in a certain weighted $L^2$-space, provided that the boundary conditions are strictly regular. Along the way, we also establish completeness, uniform minimality and asymptotic behavior of root vectors. The case of constant matrix $B(x) = B$ was investigated in [1], [2], [4].
The main results are applied to establish asymptotic behavior of eigenvalues and eigenvectors, and the Riesz basis property for the dynamic generator of spatially non-homogenous damped Timoshenko beam model. We also found a new case when eigenvalues have an explicit asymptotic, which to the best of our knowledge is new even in the case of constant parameters of the model.
This is a joint work with Anton Lunyov partially published in the preprint [3].
References
[1] Lunyov A. A., Malamud M. M., "On the Riesz basis property of root vectors system for $2 \times 2$ Dirac type operators." J. Math. Anal. Appl., 441: 57–103, 2016.
[2] Lunyov A. A., Malamud M. M., “On completeness and Riesz basis property of root subspaces of boundary value problems for first order systems.” J. Spectral Theory, 5 (1):  17–70, 2015.
[3] Lunyov A. A., Malamud M. M., "On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators." arXiv:2112.07248, 2021.
[4] Lunyov A. A., Malamud M. M., "Stability of spectral characteristics of boundary value problems for $2\times 2$ Dirac type systems." J. Differ. Equat., 313: 633–742, 2022.