A note on the PT-invariant periodic potential V(x)=4 cos^2 x + 4 i V_0 sin 2x

TL;DR

This paper demonstrates that a PT-symmetric periodic potential can be transformed into a Hermitian form for certain parameters and identifies a second critical point where the spectrum ceases to be real, revealing new physical insights.

Contribution

It uncovers a second critical point in the PT-symmetric potential, extending understanding of spectral phase transitions beyond the known threshold.

Findings

Potential can be mapped to Hermitian form for V_0<0.5

Existence of a second critical point at V_0^c ~ 0.888437

Eigenvalues become complex beyond the second critical point

Abstract

It is shown that the PT symmetric Hamiltonian with the periodic potential V(x) = 4 cos^2 x + 4 i V_0 sin 2x can be mapped into a Hermitian Hamiltonian for $V_0<0.5$, by a similarity transformation. It is also shown that there exist a second critical point of the potential V(x), apart from the known critical point $V_0=0.5$, for $V_0^c ~ .888437$ after which no part of the eigenvalues and the band structure remains real. Relevant physical consequence of this finding has been pointed out.