Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue

Let $\Omega\subset \mathbb{R}^d$ be an open set of finite measure and let $\Theta$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $\Omega$ when its second eigenvalue is close to the second eigenvalue of $\Theta$. Precisely, for every integer $k \ge 1$, we provide a quantitative control of the difference $|\lambda_k(\Omega)-\lambda_k(\Theta)|$ by the variation of the second eigenvalue $C(d,k)(\lambda_2(\Omega)-\lambda_2(\Theta))^\alpha$, for a suitable exponent $\alpha$ and a positive constant $C(d,k)$ depending only on the dimension of the space and the index $k$. We are able to find such an estimate for general $k$ and arbitrary $\Omega$ with $\alpha =\alpha_d/(d+1)^2$ where $\alpha_2 = 1/2$ and $0<\alpha_d<1$ in higher dimensions. In the particular case where $\lambda_k(\Omega)\ge \lambda_k(\Theta)$, we can improve the inequality and find an estimate with the sharp exponent $\alpha = 1/2$.