The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrödinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrödinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrödinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.

中文翻译：

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二维周期薛定ding算子在能量本征能级上可积分

本文第一部分的主要目的是表明具有零负势的二维周期性Schrödinger算子的费米曲线是光滑的M曲线，其点在零能级下参数化Schrödinger方程的Bloch解。此外，还表明，Bloch解的极点位于反全同对合的固定椭圆上，因此，除了一个椭圆外，每个椭圆都恰好包含一个极。拓扑类型是稳定的，直到在变形参数的某个值处零级成为Schrödinger算子在（反）周期函数空间上的本征级为止。本文的第二部分致力于借助Novikov-Veselov构造的推广来构造此类算子。