Improved energy bounds for Schr\"odinger operators

Given a potential $V$ and the associated Schr"odinger operator $-\mathrm{\Delta }+V$, we consider the problem of providing sharp upper and lower bound on the energy of the operator. It is known that if for example $V$ or $V^{-1}$ enjoys suitable summability properties, the problem has a positive answer. In this paper we show that the corresponding isoperimetric-like inequalities can be improved by means of quantitative stability estimates.