We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill discriminant of finitely many discrete $N$-periodic Schrödinger operators (Theorem 1). Also, in the case where the spectrum is a closed interval we prove a result (Theorem 2) which is the analog of Borg's Theorem for the non-self-adjoint Jacobi case.

中文翻译：

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具有复系数的周期性雅可比算子

我们给出了具有复周期系数的雅可比算子的正反谱理论的某些结果。例如，我们证明任何首项系数为 $(-1)^N$ 的第 $N$ 次多项式都是有限多个离散 $N$-周期薛定谔算子的 Hill 判别式（定理 1）。此外，在频谱是闭区间的情况下，我们证明了一个结果（定理 2），它类似于非自伴随雅可比情况下的博格定理。