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

TL;DR

This paper derives an asymptotic equation for the zeros of Hermite polynomials using the Holstein-Primakoff representation, linking polynomial zeros to quantum spin system boundaries in large dimensions.

Contribution

It introduces a novel approach connecting Hermite polynomial zeros with quantum spin representations via the Holstein-Primakoff method.

Findings

Derived an asymptotic equation for Hermite zeros

Established a correspondence between zeros and quantum spin boundaries

Provides insights into large-dimensional Hilbert space structures

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 $n \rightarrow \infty$. This establishes a correspondence between the zeros of the Hermite polynomials and the boundaries of the position basis of finite-dimensional Hilbert spaces.