A generalized virial theorem and the balance of kinetic and potential energies in the semiclassical limit

We obtain two-sided bounds on kinetic and potential energies of a bound state of a quantum particle in the semiclassical limit, as the Planck constant $\mathrm{\hslash }\text{ri}0$. Proofs of these results rely on the generalized virial theorem obtained in the paper as well as on a decay of eigenfunctions in the classically forbidden region.