Analytic structure of the associated Legendre functions of the second kind

Abstract

We consider the complex ν plane structure of the associated Legendre functions of the second kind Qν−1/2−K(cosh⁡ρ). We find that for any noninteger value of K the Qν−1/2−K(cosh⁡ρ) have an infinite number of poles in the complex ν 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 ν = 0) when K = 0. This pattern is analogous to the pattern of exceptional points that appear in a wide variety of physical contexts. However, while theories with exceptional points usually lose a finite number of degrees of freedom at the exceptional points, the Qν−1/2−K(cosh⁡ρ) lose an infinite number of poles whenever K is integer. Moreover, while theories with exceptional points usually have a finite number of such exceptional points, the Qν−1/2−K(cosh⁡ρ) possess an infinite number of points (all integer K) at which they lose degrees of freedom. 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 poles associated with the Qν−1/2−K(cosh⁡ρ) nonetheless occur at real values of K.