Borg's Periodicity Theorems for first order self-adjoint systems with complex potentials

Abstract

A self-adjoint first order system with Hermitian $π$-periodic potential $Q(z)$, integrable on compact sets, is considered. It is shown that all zeros of $Δ+ 2e^{-i\int_0^π\Im q dt}$ are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which $Q(z)$ is $\fracπ{2}$-periodic. Furthermore, the zeros of $Δ- 2e^{-i\int_0^π\Im q dt}$ are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which $Q(z) = σ_2 Q(z) σ_2$. Here $Δ$ denotes the discriminant of the system and $σ_0$, $σ_2$ are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if $Q = rσ_0 + qσ_2$, for some real valued $π$-periodic functions $r$ and $q$ integrable on compact sets.