Generalized Hermite Polynomials and Spectral Degeneracies of a Singular Sextic Oscillator

We study a quasi-exactly solvable singular sextic oscillator and its algebraic spectrum. For a distinguished range of parameters, we prove that the discriminant of the characteristic polynomial of the matrix determining the algebraic spectrum admits a natural factorization into three factors. One of these factors is the square of a generalized Hermite polynomial $H_{m,n}$, whose zeros are poles of a rational solution of the fourth Painlevé equation. Hence, the spectral degeneracies (level crossing points) corresponding to a component of the discriminant locus are in exact correspondence with the zeros of generalized Hermite polynomials, providing an exact Painlevé IV analogue of the Shapiro–Tater asymptotic correspondence originally conjectured for the quartic oscillator and Painlevé II. We also characterize the values of the parameters for which the sextic oscillator admits simultaneously two quasi-polynomial eigenfunctions with opposite exponential behaviour at infinity, and show that this phenomenon is also governed by generalized Hermite polynomials. Our result also yields a new determinantal representation of $H_{m,n}$ as the resultant of the characteristic polynomials of two complementary blocks of the matrix determining the algebraic spectrum.