A complex periodic QES potential and exceptional points

TL;DR

This paper investigates a complex PT-symmetric periodic potential, demonstrating its quasi-exact solvability, the emergence of exceptional points for certain parameters, and its connection to the Mathieu equation.

Contribution

It introduces a new class of quasi-exactly solvable complex periodic potentials and analyzes the conditions for exceptional points related to the potential's parameters.

Findings

Potential is quasi-exactly solvable for specific parameters.

Exceptional points occur for odd N ≥ 3 depending on coupling strength.

Asymptotic behavior links the potential to the Mathieu equation.

Abstract

We show that the complex $\cal PT$-symmetric periodic potential $V(x) = - ({\rm i} \xi \sin 2x + N)^2$, where $\xi$ is real and $N$ is a positive integer, is quasi-exactly solvable. For odd values of $N \ge 3$, it may lead to exceptional points depending upon the strength of the coupling parameter $\xi$. The corresponding Schr\"odinger equation is also shown to go over to the Mathieu equation asymptotically. The limiting value of the exceptional points derived in our scheme is consistent with known branch-point singularities of the Mathieu equation.