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 $H_{mn}$, whose zeros are poles of a rational solution of the fourth Painlev\'e 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 Painlev\'e IV analogue of the Shapiro--Tater asymptotic correspondence originally conjectured for the quartic oscillator and Painlev\'e 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 $H_{mn}$ as the resultant of the characteristic polynomials of two complementary blocks of the matrix determining the algebraic spectrum.