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

TL;DR

This paper investigates the stability of spectral properties and Bari basis conditions for boundary value problems involving 2x2 Dirac systems, establishing Lipschitz continuity of spectral mappings under potential perturbations.

Contribution

It introduces new Lipschitz stability results for spectral characteristics and Bari basis properties of Dirac systems with potential perturbations, extending previous work on transformation operators.

Findings

Spectral characteristics are Lipschitz continuous under potential perturbations.

Eigenfunction sequences form Bari bases under certain conditions.

Stability results hold for potentials in L^p spaces with p in [1,2].

Abstract

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.