  It is well known that a suggestive relation exists that links Schr\"odinger's
equation (SE) to the information-optimizing principle based on Fisher's
information measure (FIM). The connection entails the existence of a Legendre
transform structure underlying the SE. Here we show that appeal to this
structure leads to a first order differential equation for the SE's eigenvalues
that, in certain cases, can be used to obtain the eigenvalues without
explicitly solving SE. Complying with the above mentioned equation constitutes
a necessary condition to be satisfied by an energy eigenvalue.
  We show that the general solution is unique.
