Direct Fisher inference of the quartic oscillator's eigenvalues
  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. 
S. P. Flego, A. Plastino, A. R. Plastino
quant-ph