The smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight

TL;DR

This paper derives the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by a singularly perturbed Laguerre weight, using asymptotic expressions of associated orthonormal polynomials.

Contribution

It provides the first asymptotic analysis of the smallest eigenvalue for Hankel matrices with a singularly perturbed Laguerre weight.

Findings

Asymptotic expression for orthonormal polynomials $\\mathcal{P}_N(z)$ as $N\rightarrow\infty$

Asymptotic behavior of the smallest eigenvalue $\lambda_N$ of the Hankel matrix

Insights into spectral properties of Hankel matrices with singular perturbations

Abstract

An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre weight $w_{\alpha}(x;t)=x^{\alpha}{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~\alpha>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $\lambda_{N}$, of the Hankel matrix generated by the weight $w_{\alpha}(x;t)$.