Completeness of systems of complex exponentials and the Lambert 𝑊 functions

We study some of the properties of the solution system { e i λ n t } \{e^{i\lambda _nt}\} of the delay-differential equation y ′ ( t ) = a y ( t − 1 ) y’(t) = ay(t-1) . We first establish some general results on the stability of the completeness of exponential systems in L 2 L^2 and then show that the solution system above is always complete, but is not an unconditional basis in L 2 ( − 1 / 2 , 1 / 2 ) L^2(-1/2,1/2) .