  The aim of this article is to prove a quantitative inequality for the first
eigenvalue of a Schr\"odinger operator in the ball. More precisely, we optimize
the first eigenvalue $\lambda(V)$ of the operator $\mathcal L_v:=-\Delta-V$
with Dirichlet boundary conditions with respect to the potential $V$, under
$L^1$ and $L^\infty$ constraints on $V$. The solution has been known to be the
characteristic function of a centered ball, but this article aims at proving a
sharp growth rate of the following form: if $V^*$ is a minimizer, then
$\lambda(V)-\lambda(V^*)\geq C ||V-V^*||_{L^1(\Omega)}^2$ for some $C>0$. The
proof relies on two notions of derivatives for shape optimization: parametric
derivatives and shape derivatives. We use parametric derivatives to handle
radial competitors, and shape derivatives to deal with normal deformation of
the ball. A dichotomy is then established to extend the result to all other
potentials. We develop a new method to handle radial distributions and a
comparison principle to handle second order shape derivatives at the ball.
Finally, we add some remarks regarding the coercivity norm of the second order
shape derivative in this context.
