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

TL;DR

This paper develops parameter-free virial-based ansätze for Schrödinger eigenfunctions with symmetric convex potentials, successfully applying them to harmonic and quartic oscillators, offering a new analytical approach.

Contribution

It introduces a novel virial theorem-based method to construct eigenfunction ansätze without adjustable parameters for symmetric convex potentials.

Findings

Exact eigenfunctions for harmonic oscillator

Accurate approximations for quartic anharmonic oscillator

Demonstrates effectiveness of virial-based ansätze

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.