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

Abstract

One dimensional Dirac operators Lbc(v)y=i(100−1)dydx+v(x)y,y=(y1y2),x∈[0,π], considered with L2-potentials v(x)=(0P(x)Q(x)0) and subject to regular boundary conditions (bc), have discrete spectrum. For strictly regular bc, the spectrum of the free operator Lbc(0) is simple while the spectrum of Lbc(v) 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,π]. Analogous results are obtained for regular but not strictly regular bc.