Exceptional points for associated Legendre functions of the second kind

TL;DR

This paper analyzes the complex structure of associated Legendre functions of the second kind, revealing the distribution of poles and exceptional points in relation to the parameter K, with implications for physical theories involving such points.

Contribution

It characterizes the pole structure and exceptional points of $Q^{-1/2-K}_{ u}( ext{cosh} ho)$ across different K values, highlighting their occurrence at real parameters.

Findings

Infinite poles for noninteger K

Finite poles for integer K, with a single pole at K=0

Exceptional points occur at real K values

Abstract

We consider the complex $\nu$ plane structure of the associated Legendre function of the second kind $Q^{-1/2-K}_{\nu}(\cosh\rho)$. We find that for any noninteger value for $K$ $Q^{-1/2-K}_{\nu}(\cosh\rho)$ has an infinite number of poles in the complex $\nu$ plane, but for any negative integer $K$ there are no poles at all. For $K=0$ or any positive integer $K$ there is only a finite number of poles, with there only being one single pole (at $\nu=0$) when $K=0$. This pattern is characteristic of the exceptional points that appear in a wide variety of physical contexts. However, unusually for theories with exceptional points, $Q^{-1/2-K}_{\nu}(\cosh\rho)$ has an infinite number of them. Other than in the $PT$-symmetry Jordan-block case, exceptional points usually occur at complex values of parameters. While not being Jordan-block exceptional points themselves, the exceptional points associated with the $Q^{-1/2-K}_{\nu}(\cosh\rho)$ nonetheless occur at real values of $K$.