The kernel polynomial method based on Jacobi polynomials

TL;DR

This work tackles efficient computation of spectral densities for large Hermitian matrices by extending the Kernel Polynomial Method to Jacobi polynomials with tunable edge behavior. It derives an optimal positivity-preserving kernel and damping factors, generalizing Jackson damping to arbitrary Jacobi bases, and provides asymptotic scaling of the kernel resolution $Q_{\min}\sim \frac{j_{\alpha,1}^2}{(1+\alpha)N^2}$. The paper also develops explicit closed-form damping for special cases $(\alpha,\beta)=(\pm\tfrac{1}{2},\pm\tfrac{1}{2})$, and demonstrates the approach on 2D square and 3D cubic lattices, showing improved edge fidelity and compatibility with existing KPM pipelines. The results enable accurate, non-negative DOS approximations across a wide class of systems and can be integrated into standard KPM workflows for scalable spectral analysis.

Abstract

The kernel polynomial method based on Jacobi polynomials $P_n^{α,β}(x)$ is proposed. The optimal-resolution positivity-preserving kernels and the corresponding damping factors are obtained. The results provide a generalization of the Jackson damping factors for arbitrary Jacobi polynomials. For $α=\pm 1/2$, $β=\pm 1/2$ (Chebyshev polynomials of the first to fourth kinds), explicit trigonometric expressions for the damping factors are obtained. The resulting algorithm can be easily introduced into existing implementations of the kernel polynomial method.

Figures (9)

• Figure 1: The region $V$ shows $(\alpha ,\beta )$ points for which $K_N(x,y)$ is non-negative for all $N$ (see text for more details). The region $V_\infty$ shows points which don't belong to $V$ but fulfill the asymptotic non-negativity condition, in which case $K_N(x,y)$ is non-negative for $N\rightarrow \infty$ (see Section \ref{['sec:asymp_opt_kernel']}).
• Figure 2: Damping factors $g_n$ for several small values of $N$ and their asymptotic function $g(t)$ for different values of the parameters $\alpha ,\beta$.
• Figure 3: Heatmaps of the weighted kernel $\tilde{K}_N(x,y)$ for different values of $\alpha ,\beta$ for $N=15$ (top row) and $N=50$ (bottom row).
• Figure 4: Asymptotic value of the scaled kernel resolution $Q_{\rm min}$ as a function of the parameter $\alpha$ for large $N$. Symbols show the values given by Eq. \ref{['eq:sigma_exact']}.
• Figure 5: Asymptotic damping function $g(t)$ for different values of the parameter $\alpha$.