Generalised Freud's equation and level densities with polynomial potential

TL;DR

This paper derives generalized Freud's equations for orthogonal polynomials with polynomial weights of degree 6, 8, and 10, and uses these to compute level densities in the large N limit, advancing understanding of polynomial potential models.

Contribution

It introduces generalized Freud's equations for higher-degree polynomial potentials and applies them to derive explicit level densities as N approaches infinity.

Findings

Derived Freud's equations for d=3, 4, 5 cases.

Obtained explicit level density formulas for large N.

Connected orthogonal polynomial properties with spectral densities.

Abstract

We study orthogonal polynomials with weight $\exp[-NV(x)]$, where $V(x)=\sum_{k=1}^{d}a_{2k}x^{2k}/2k$ is a polynomial of order 2d. We derive the generalised Freud's equations for $d=3$, 4 and 5 and using this obtain $R_{\mu}=h_{\mu}/h_{\mu -1}$, where $h_{\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\mu}$, are obtained using Freud's equation and using this, explicit results of level densities as $N\rightarrow\infty$ are derived.