Virial-ans\"atze for the Schr\"odinger Equation with a symmetric strictly convex potential

Abstract

Considering symmetric strictly convex potentials, a local relationship is inferred from the virial theorem, based on which a real log-concave function can be constructed. Using this as a weight function and in such a way that the virial theorem can still be verified, parameter-free ans\"atze for the eigenfunctions of the associated Schr\"odinger equation are built. To illustrate the process, the technique is successfully tested against the harmonic oscillator, in which it leads to the exact eigenfunctions, and against the quartic anharmonic oscillator, which is considered the paradigmatic testing ground for new approaches to the Schr\"odinger equation.