Solutions of the fractional Schr\"odinger equation via diagonalization - A plea for the harmonic oscillator basis part 1: the one dimensional case

TL;DR

This paper develops an analytical approach to solve the fractional Schrödinger equation using the harmonic oscillator basis, introducing new interpretations of non-local operators and identifying a novel symmetry in fractional quantum systems.

Contribution

It presents a fully analytical calculation of matrix elements for the non-local kinetic energy in the harmonic oscillator basis and explores symmetry extensions in fractional quantum mechanics.

Findings

Analytical matrix elements for fractional kinetic energy derived

Diagonalization of fractional harmonic oscillator Schrödinger equation achieved

Discovery of a new symmetry extending parity in fractional wave equations

Abstract

A covariant non-local extention if the stationary Schr\"odinger equation is presented and it's solution in terms of Heisenbergs's matrix quantum mechanics is proposed. For the special case of the Riesz fractional derivative, the calculation of corresponding matrix elements for the non-local kinetic energy term is performed fully analytically in the harmonic oscillator basis and leads to a new interpretation of non local operators in terms of generalized Glauber states. As a first application, for the fractional harmonic oscillator the potential energy matrix elements are calculated and the and the corresponding Schr\"odinger equation is diagonalized. For the special case of invariance of the non-local wave equation under Fourier-transforms a new symmetry is deduced, which may be interpreted as an extension of the standard parity-symmetry.