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 $Q\to \tilde{Q}$ of different spectral characteristics of a BVP associated in $L^2([0,1];{\mathbb{C}}^2)$ with the following $2\times 2$ Dirac type equation $$L_U(Q)y=-i{B}^{-1}{y}^{\mathrm{\prime }}+Q(x)y=\lambda y,B=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(b_1,b_2),b_1<0<b_2,y=\mathrm{c}\mathrm{o}\mathrm{l}(y_1,y_2),(1)$$ with a potential matrix $Q\in L^p=L^p([0,1];\Bbb C^{2\times2})$andsubjecttoregularboundaryconditions$Uy={U_1,U_2}y=0$. 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 $Q\to K_Q^\pm$fromtheballsin$L^p$tothespecialBanachspaces$X_{\infty,p}^2,X_{1,p}^2$, 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 $Q\to {\mathrm{\Phi }}_Q$, where ${\mathrm{\Phi }}_Q$ 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 $l^{p^{\mathrm{\prime }}}$ or into $l^p(\{(1+\mathrm{\mid }n\mathrm{\mid }{)}^{p-2}\})$; we also establish its Lipshitz property on compacts. We show similar result for the mapping $Q\to F_Q-F_0$ into $l^{p^{\mathrm{\prime }}}(\mathbb{Z};C([0,1];{\mathbb{C}}^2))$, 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 $L^2([0,1];{\mathbb{C}}^2)$. Under a simple additional assumption this system forms a Bari basis if and only if BC are self-adjoint.