Asymptotics of small eigenvalues of Hankel matrices generated by a semiclassical Gaussian weight

Zhu, Mengkun; Chen, Yuting; Chen, Yang

doi:10.1090/proc/17209

Asymptotics of small eigenvalues of Hankel matrices generated by a semiclassical Gaussian weight
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by Mengkun Zhu, Yuting Chen and Yang Chen;

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/17209

Published electronically: April 14, 2025

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Abstract:

In this paper, we study the asymptotic behavior of the smallest eigenvalue $\lambda _{N}$, of the Hankel matrix $\mathcal {H}_{N}≔\left (\mu _{m+n}\right )_{m,n=0}^{N}$ generated by the weight $w(x)≔{\mathrm {e}}^{-tx^{2}}(1+x^{2})^{\alpha },\, x\in \mathbb {R},\, t\geq 0,\, {\alpha \in \mathbb {R}}$. An asymptotic expression of the polynomials $\mathcal {P}_{N}(z)$ with the weight is established as $N\rightarrow \infty$. Based on the orthonormal polynomials $\mathcal {P}_{N}(z)$, we obtain the specific asymptotic formulas of $\lambda _{N}$.

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Asymptotics of small eigenvalues of Hankel matrices generated by a semiclassical Gaussian weight
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Abstract: