  It is well known that a suggestive connection links Schr\"odinger's equation
(SE) and the information-optimizing principle based on Fisher's information
measure (FIM). It has been shown that this entails the existence of a Legendre
transform structure underlying the SE. Such a structure leads to a first order
partial differential equation (PDE) for the SE's eigenvalues from which a
complete solution for them can be obtained. As an application we deal with the
quantum theory of anharmonic oscillators, a long-standing problem that has
received intense attention motivated by problems in quantum field theory and
molecular physics. By appeal to the Cramer Rao bound we are able to
Fisher-infer the particular PDE-solution that yields the eigenvalues without
explicitly solving Schr\"odinger's equation. Remarkably enough, and in contrast
with standard variational approaches, our present procedure does not involve
free fitting parameters.
