Let $\Omega \subset \mathbb{R}^n$ be an open set with same volume as the unit ball $B$ and let $\lambda_k(\Omega)$ be the $k$-th eigenvalue of the Laplacian with Dirichlet condition of $\Omega$. Suppose $\lambda_1(\Omega)$ is close to $\lambda_1(B)$, how close is $\lambda_k(\Omega)$ to $\lambda_k(B)$? We establish quantitative bounds with sharp exponents depending on the multiplicity of $\lambda_k(B)$ through the study of a perturbed shape optimization problem, in particular we prove the persistence of the ball as minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue. This is a joint work with Dorin Bucur, Jimmy Lamboley and Raphaël Prunier.