Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems

The paper is concerned with the stability property under perturbation $Q\to\widetilde Q$ of different spectral characteristics of a BVP associated in $L^2([0,1];\Bbb C^2)$ with the following $2\times2$ Dirac type equation $$L_U(Q)y=-iB^{-1}y'+Q(x)y=\lambda y,\quad B={\rm diag}(b_1,b_2),\quad b_1<0<b_2,\quad y={\rm col}(y_1,y_2),\quad(1)$$ with a potential matrix $Q\in L^p=L^p([0,1];\Bbb C^{2\times2})$ and subject to regular boundary conditions $Uy=\{U_1,U_2\}y=0$. Our approach to spectral stability relies on the existence of triangular transformation operators $K_Q^\pm$ for system (1) with $Q\in L^1$ established in our previous works. We prove the Lipshitz property of the mapping $Q\to K_Q^\pm$ from the balls in $L^p$ to the special Banach spaces $X_{\infty,p}^2,X_{1,p}^2$, naturally arising here, and obtain similar property for Fourier transforms of $K_Q^\pm$. These properties are of independent interest and play a crucial role in the proofs of all stability results discussed in the paper. For instance, as an immediate consequence we get the Lipshitz property of the mapping $Q\to\Phi_Q$, where $\Phi_Q$ is the fundamental matrix of the system (1). Assuming boundary conditions (BC) to be strictly regular, we show that the mapping $Q\to\sigma(L_U(Q))-\sigma(L_U(0))$ sends $L^p,p\in[1,2]$, either into $l^{p'}$ or into $l^p(\{(1+|n|)^{p-2}\})$; we also establish its Lipshitz property on compacts. We show similar result for the mapping $Q\to F_Q-F_0$ into $l^{p'}(\Bbb Z; C([0,1];\Bbb C^2))$, where $F_Q$ is a sequence of normalized eigenfunctions of $L_U(Q)$. Certain modifications of these results are proved for balls in $L^p,p\in[1,2]$. If $Q\in L^2$ we establish a criterion for the system of root vectors of $L_U(Q)$ to form a Bari basis in $L^2([0,1];\Bbb C^2)$. Under a simple additional assumption this system forms a Bari basis if and only if BC are self-adjoint.