Uniform Basis Property of Root Vector Systems of Dirac Operators

Abstract

On the segment [0, π], we study the one-dimensional Dirac operator ℒ with boundary conditions U regular in the Birkhoff sense and with a complex-valued summable potential P = (pij(x)), i, j = 1, 2.We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator ℒ, assuming that boundary conditions U and the number \( \int_0^{\pi}\left({p}_1(x)-{p}_4(x)\right) dx \) are fixed and the potential P takes values from the ball B(0, R) of radius R in the space Lϰ, ϰ > 1. We succeed to select the system of root functions such that it consists of eigenfunctions of the operator ℒ apart from a finite set of root vectors such that their number can be estimated uniformly with respect to the ball ‖P‖ϰ ≤ R.