BORG'S PERIODICITY THEOREMS FOR FIRST-ORDER SELF-ADJOINT SYSTEMS WITH COMPLEX POTENTIALS

作者：Currie Sonja; Roth Thomas T; Watson Bruce A^*

来源：Proceedings of the Edinburgh Mathematical Society, 2017, 60(3): 615-633.

DOI：10.1017/S0013091516000389

• Dirac system
• inverse problems
• spectral theory

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+ 2exp(-i integral(pi)(0) Im q dt) are double zeros if and only if this self-adjoint system is unitarily equivalent to one in which Q(z) is pi/2-periodic. Furthermore, the zeros of Delta-2 exp(-i integral(pi)(0) 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.

• 出版日期2017-8