Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

Abstract

One dimensional Dirac operators L-bc(nu) y = i ((1)(0) -(0)(1)) dy/dx +nu(x)y. y = ((y2) (y1)). x is an element of [0, pi], considered with L-2-potentials nu(x) = ((0)(P(x)) (P(x))(0)) and subject to regular boundary conditions (bc), have discrete spectrum. For strictly regular be, the spectrum of the free operator L-bc(0) is simple while the spectrum of L-bc(nu) is eventually simple, and the corresponding normalized root function systems are Riesz bases. For expansions of functions of bounded variation about these Riesz bases, we prove the uniform equiconvergence property and point-wise convergence on the closed interval [0, pi]. Analogous results are obtained for regular but not strictly regular bc.