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

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