On Perturbations of the Spectrum of a One-Dimensional $\mathcal{PT}$-Symmetric Periodic Schrödinger Operator

For a $\mathcal{PT}$-symmetric periodic Schrödinger operator, which is a small perturbation of the zero potential, we calculate the spectrum and the divisor of zeroes of the Bloch function in the leading order of perturbation theory. In particular, we show that the analogs of lacunae of the Bloch spectrum are ellipses, and their focal points coincide with the branch points of the spectral curve.