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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be an open set of finite measure and let $\mathrm{\Theta }$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $\mathrm{\Omega }$ when its second eigenvalue is close to the second eigenvalue of $\mathrm{\Theta }$. Precisely, for every integer $k\ge 1$, we provide a quantitative control of the difference $\mathrm{\mid }{\lambda }_k(\mathrm{\Omega })-{\lambda }_k(\mathrm{\Theta })\mathrm{\mid }$ by the variation of the second eigenvalue $C(d,k)({\lambda }_2(\mathrm{\Omega })-{\lambda }_2(\mathrm{\Theta }){)}^{\alpha }$, 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 $\mathrm{\Omega }$ with $\alpha ={\alpha }_d\mathrm{/}(d+1{)}^2$ where ${\alpha }_2=1\mathrm{/}2$ and $0<{\alpha }_d<1$ in higher dimensions. In the particular case where ${\lambda }_k(\mathrm{\Omega })\ge {\lambda }_k(\mathrm{\Theta })$, we can improve the inequality and find an estimate with the sharp exponent $\alpha =1\mathrm{/}2$.