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

TL;DR

This paper derives bounds on kinetic and potential energies for quantum bound states in the semiclassical limit using a generalized virial theorem and eigenfunction decay properties.

Contribution

It introduces a generalized virial theorem applicable in the semiclassical limit and establishes bounds on energies based on eigenfunction decay.

Findings

Established two-sided bounds on energies as   0

Demonstrated decay of eigenfunctions in forbidden regions

Extended virial theorem to semiclassical regime

Abstract

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 $\hbar\ri 0$. 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.