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

Abstract

A self-adjoint first order system with Hermitian $\pi$-periodic potential $Q(z)$, integrable on compact sets, is considered. It is shown that all zeros of $\Delta + 2e^{-i\int_0^\pi \Im q dt}$ are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which $Q(z)$ is $\frac{\pi}{2}$-periodic. Furthermore, the zeros of $\Delta - 2e^{-i\int_0^\pi \Im q dt}$ are all double zeros if and only if the associated self-adjoint system is unitarily equivalent to one in which $Q(z) = \sigma_2 Q(z) \sigma_2$. Here $\Delta$ denotes the discriminant of the system and $\sigma_0$, $\sigma_2$ are Pauli matrices. Finally, it is shown that all instability intervals vanish if and only if $Q = r\sigma_0 + q\sigma_2$, for some real valued $\pi$-periodic functions $r$ and $q$ integrable on compact sets.