Spectral measures, supports, compact operators

Question

Let $H$ be a Hilbert space, $K:H\rightarrow H$ a compact self-adjoint operator. The spectral measure of $K$ wrt $v\in H$ is uniquely determined by $$\langle K^n v,v \rangle=\int_{\mathbb{R}} x^n d\mu_v(x)$$ for all $n\in\mathbb{N}$. It seems to me that the atoms are strongly related to the eigenvalues, but what is the exact relationship?