It is well known that a suggestive connection links Schrodingers equation (SE) and the information-optimizing principle based on Fishers 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 SEs 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 Schrodingers equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.
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