# Algebraic structures and deformed Schr\"{o}dinger equations from groups entropies

## TL;DR

This paper develops a generalized algebraic framework inspired by group entropy theory, leading to a G-deformed Schrödinger equation that encompasses Tsallis and Kappa statistics, with applications to quantum systems like the infinite potential well.

## Contribution

It introduces the G-algebra and G-differential calculus, unifying various statistical mechanics frameworks and deriving a generalized Schrödinger equation with novel eigenfunction properties.

## Key findings

- G-algebra generalizes real number algebra based on group entropies.
- The G-deformed Schrödinger equation includes Tsallis and Kappa cases as special instances.
- Eigenfunctions exhibit non-uniform zeros spacing described by G-algebra sums.

## Abstract

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schr\"{o}dinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the $q$-deformed (standard) Schr\"{o}dinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.