Let $d \ge 2$ be an integer and let $X=\mathcal S_{d-1}$ the unit-sphere in $\mathbb R^d$. Let $\tau_d$ be the uniform distribution on $X$. Define a function $K:X \times X \to \mathbb R$ by $K(x,y) := e^{-c\|x-y\|^\beta}$, for some scale parameter $c>0$ and $\beta \ge 1$. I'm particularly interested in the case $\beta \in \{1,2\}$. The case $\beta = 1$ corresponds to the Laplace kernel while the case $\beta=2$ corresponds to the Gaussian kernel.

Now, consider the kernel integral operator $T_K:L^2(X,\tau_d) \to L^2(X,\tau_d)$ defined by

$$ T_Kf:X \to \mathbb R,\,T_K f(x) = \int_X K(x,x')f(x')d\tau_d(x'). $$

This is a compact positive operator has a countable spectrum / eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots $ with $\lambda_n \ge 0$ for all $n$.

Question. Is there an exact analytic formula for the $\lambda_n$'s (as a function of $c$, $\beta$, $d$, and $n$) ? Same question for the trace of $T_K$, namely the sum $\sum_{n=1}^\infty \lambda_n$ (as a function of $c$, $\beta$, and $d$).

Edit: The trace equals $e$

It turns out that $\mbox{Tr}(T_K) = \mathbb E_{x \sim \tau_d} [K(x,x)] = e^{-c}=\mathbb E_{x \sim \tau_d}e^{-c\|x-x\|^\beta} = e$ for all $\beta \ge 1$.