On perturbations of the spectrum of one-dimensional PT-symmetric periodic Schrodinger operator

TL;DR

This paper analyzes how small PT-symmetric perturbations affect the spectrum of one-dimensional periodic Schrödinger operators, revealing geometric structures like elliptical lacunae and their relation to spectral branch points.

Contribution

It provides a leading-order perturbation analysis of the spectrum and zeroes of Bloch functions for PT-symmetric periodic Schrödinger operators, highlighting geometric spectral features.

Findings

Lacunae of the Bloch spectrum are elliptical.

Focal points of these ellipses coincide with spectral branch points.

Spectrum and Bloch functions are characterized in the perturbative regime.

Abstract

For PT-symmetric periodic Schrodinger 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 the 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.