Complex periodic potentials with real band spectra

TL;DR

This paper explores complex PT-symmetric periodic potentials, revealing they can have real band spectra with unique band structure features, differing from traditional real potentials, supported by numerical and analytical methods.

Contribution

It demonstrates that certain complex PT-symmetric potentials have real band spectra and highlights qualitative differences in their band structures compared to real potentials.

Findings

Complex PT-symmetric potentials can have real band spectra.

Band edges feature only periodic wave functions, no antiperiodic ones.

Numerical and WKB analyses support these results.

Abstract

This paper demonstrates that complex PT-symmetric periodic potentials possess real band spectra. However, there are significant qualitative differences in the band structure for these potentials when compared with conventional real periodic potentials. For example, while the potentials V(x)=i\sin^{2N+1}(x), (N=0, 1, 2, ...), have infinitely many gaps, at the band edges there are periodic wave functions but no antiperiodic wave functions. Numerical analysis and higher-order WKB techniques are used to establish these results.