Dirichlet spectrum and Green function

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, $lambda_{1}(Omega)=lim_{kto infty} Vert G^k(f)Vert_{2}/Vert G^{k+1}(f)Vert_{2}$ 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.