  In the first part of this article we obtain an identity relating the radial
spectrum of rotationally invariant geodesic balls and an isoperimetric quotient
$\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower
estimates for the series $\sum \lambda_{i}^{-2}(\Omega)$ where $\Omega$ is an
extrinsic ball of a proper minimal surface of $\mathbb{R}^{3}$. In the second
part we show that the first eigenvalue of bounded domains is given by iteration
of the Green operator and taking the limit, $\lambda_{1}(\Omega)=\lim_{k\to
\infty} \Vert G^k(f)\Vert_{2}/\Vert G^{k+1}(f)\Vert_{2}$ for any function
$f>0$. In the third part we obtain explicitly the $L^{1}(\Omega, \mu)$-momentum
spectrum of a bounded domain $\Omega$ in terms of its Green operator. In
particular, we obtain the first eigenvalue of a weighted bounded domain in
terms of the $L^{1}(\Omega, \mu)$-momentum spectrum, extending the work of
Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant
balls.
