Recall that Matérn covariance function $C_\nu(d)$ is defined as

$$ C_\nu(d)=\sigma^2\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}\frac{d}{\rho}\right)^\nu K_\nu\left(\sqrt{2\nu}\frac{d}{\rho}\right), d\ge0. $$ where $\Gamma$ is Gamma function, $K_\nu$ is the modified Bessel function of the second kind, $\rho,\nu,\sigma^2$ are positive constants. My question is what does the eigenvalues of $C_\nu$ look like?

More specifically, define a symmetric positive definite kernel function $C(x,y)=C_\nu(|x-y|)$, and consider Mercer decomposition $$ C(x,y)=\sum_{i=1}^\infty \lambda_i\varphi_i(x)\varphi_i(y), x,y\in\mathbb{R}, $$ where $\lambda_i$ are positive eigenvalues and $\varphi_i$ are eigenfunctions that form an orthonormal basis of $L^2[0,1]$. Question: how fast can $\lambda_i$ decay along with $i$, say, $\lambda\sim \exp(-i^{\beta})$ for some constant $\beta$?

Note that $C_\nu(d)\to \sigma^2\exp\left(-\frac{d^2}{2\rho^2}\right)$ when $\nu\to\infty$, so the limit defines a Gaussian kernel whose eigenvalues are well understood; see About eigen-functions of the Gaussian kernel

Is there a similar result about $C_\nu$ for fixed $\nu$?