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

TL;DR

This paper proves uniform equiconvergence and pointwise convergence of spectral decompositions for 1D Dirac operators with regular boundary conditions, extending results to both strictly regular and not strictly regular cases.

Contribution

It establishes equiconvergence of spectral decompositions for Dirac operators with regular boundary conditions, including non-strictly regular cases, which was previously unaddressed.

Findings

Spectral decompositions form Riesz bases for these operators.

Uniform equiconvergence holds for functions of bounded variation.

Results apply to both strictly regular and not strictly regular boundary conditions.

Abstract

One dimensional Dirac operators $$ L_{bc}(v) y = i 1 & 0 0 & -1 \frac{dy}{dx} + v(x) y, \quad y = y_1 y_2, \quad x\in[0,\pi]$$, considered with $L^2$-potentials $ v(x) = 0 & P(x) Q(x) & 0$ and subject to regular boundary conditions ($bc$), have discrete spectrum. For strictly regular $bc,$ the spectrum of the free operator $ L_{bc}(0) $ is simple while the spectrum of $ L_{bc}(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 $[0,\pi].$ Analogous results are obtained for regular but not strictly regular $bc.$