Special features of the relation between Fisher information and Schrödinger eigenvalue equation

Abstract

It is well known that a suggestive relation exists that links Schrödinger'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.

Information Theory Consequences of the Scale-Invariance of Schröedinger’s Equation

“…In [11] the present authors went way beyond the above interpretation of such PDE, provided an entirely different demonstration for Equation (13), and took it as a "prescription", a linear PDE that energy eigenvalues must necessarily comply with. A careful mathematical study of this PDE was undertaken in [11].…”

Physical implications of Fisher-information’s scaling symmetry

“…The above equation was first derived from a constrained Fisher variational problem in [23]. We are not appealing to this equation here but only to the FIM scaling symmetry instead.…”

“…The general solution for α-PDE does exist. Uniqueness is, again, proved from an analysis of the associated Cauchy problem [23].…”

“…We are not appealing to this equation here but only to the FIM scaling symmetry instead. A complete solution for α-PDE (36) is given by [23],…”

A Shannon–Tsallis transformation

“…Why? Because of the Legendre transform properties of the MaxEnt solutions [see for instance [23,24,25,26,27,28,29,30] and references therein].…”