On transformation operators and Riesz basis property of root vectors system for $n \times n$ Dirac type operators. Application to the Timoshenko beam model

Abstract

The paper is concerned with the following $n\times n$ Dirac type equation$$Ly=-iB(x)^{-1}(y'+Q(x)y)=\lambda y, \quad B(x)=B(x)^*,\quad y={\rm col}(y_1,\ldots,y_n),\quad x\in[0,\ell],$$ on a finite interval $[0,\ell]$. Here $Q$ is a summable potential $n\times n$ matrix function and $B$ is an invertible self-adjoint diagonal bounded matrix function. If $n=2m$ and $B(x)={\rm diag}(-I_m,I_m)$, this equation is equivalent to Dirac equation of order $n$. We show the existence of triangular transformation operators for such equation under additional uniform separation conditions on the entries of the matrix function $B$. Here we apply this result to study direct spectral properties of the boundary value problem (BVP) associated with the above equation subject to the general boundary conditions $U(y)=Cy(0)+Dy(\ell)=0,{\rm rank}(C\ D)=n$. We apply this result to show that the deviation of the characteristic determinants of this BVP and the unperturbed BVP (with $Q=0$) is a Fourier transform of some summable function, which in turn yields asymptotic behavior of the spectrum in the case of regular boundary conditions. Namely, $\lambda_m=\lambda_m^0+o(1)$ 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 ($Q=0$) BVP, respectively. 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. The main results are applied to establish asymptotic behavior of eigenvalues and eigenvectors, and the Riesz basis property for the dynamic generator of the 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.