Direct Fisher inference of the quartic oscillator's eigenvalues

TL;DR

This paper introduces a Fisher information-based method to directly infer eigenvalues of the quartic oscillator, avoiding explicit Schrödinger equation solutions and free fitting parameters, thus providing a novel analytical approach.

Contribution

It develops a Fisher information framework that derives eigenvalues of anharmonic oscillators through a PDE, bypassing traditional variational methods with free parameters.

Findings

Successfully infers eigenvalues without solving Schrödinger's equation

Provides a parameter-free analytical method for quantum anharmonic oscillators

Establishes a Legendre transform structure linking Fisher information and quantum eigenvalues

Abstract

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.