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 $\Delta +2e^{-i{\int }_0^{\pi }\Im qdt}$ 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 $\Delta -2e^{-i{\int }_0^{\pi }\Im qdt}$ 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.