Covariance in Fractional Calculus

TL;DR

This paper introduces a covariance-based generalization of fractional calculus in multiple dimensions and applies it to approximate the ground state energy of a fractional 2D harmonic oscillator using variational methods.

Contribution

It presents a novel covariance-based framework for multi-dimensional fractional calculus and demonstrates its application to quantum harmonic oscillator energy estimation.

Findings

Successful approximation of the fractional 2D harmonic oscillator ground state energy.

New covariance-based approach extends fractional calculus to multi-dimensional spaces.

Potential for broader applications in quantum physics and fractional differential equations.

Abstract

Based on the requirement of covariance, we propose a new approach for generalizing fractional calculus in multi-dimensional space. As a first application we calculate an approximation for the ground state energy of the fractional 2-dimensional harmonic oscillator using the Ritz variational principle.