Asymptotic Equation for Zeros of Hermite Polynomials from the Holstein-Primakoff Representation

Abstract

The Holstein-Primakoff representation for spin systems is used to derive expressions with solutions that are conjectured to be the zeros of Hermite polynomials $H_n(x)$ as $n\to \infty$. This establishes a correspondence between the zeros of the Hermite polynomials and the boundaries of the position basis of finite-dimensional Hilbert spaces.

3 pages, 3 figures, version 3 (eqs 6 & 7 rewritten in a more concise form; typo in eqn 5 fixed; footnote 9 with ref. 6 added)