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It is well known that a suggestive relation exists that links Schr\"odinger'sequation (SE) to the information-optimizing principle based on Fisher'sinformation measure (FIM). The connection entails the existence of a Legendretransform structure underlying the SE. Here we show that appeal to thisstructure leads to a first order differential equation for the SE's eigenvaluesthat, in certain cases, can be used to obtain the eigenvalues withoutexplicitly solving SE. Complying with the above mentioned equation constitutesa necessary condition to be satisfied by an energy eigenvalue.  We show that the general solution is unique.

Special features of the relation between Fisher Information and Schr\"odinger eigenvalue equation