Fractional Floquet theory

TL;DR

This paper introduces a fractional version of Floquet theory for fractional Schrödinger equations, enabling reduction to standard quantum mechanics and providing new tools for analyzing quantum resonances with periodic Hamiltonians.

Contribution

It proposes the fractional Floquet theorem formulated with Mittag-Leffler functions, extending Floquet theory to fractional quantum systems.

Findings

The fractional Floquet theorem is formulated using Mittag-Leffler functions.

The method reduces fractional Schrödinger equations to standard quantum mechanics.

Examples demonstrate the application to quantum resonances.

Abstract

A fractional generalization of the Floquet theorem is suggested for fractional Schr\"odinger equations (FTSE)s with the time-dependent periodic Hamiltonians. The obtained result, called the fractional Floquet theorem (fFT), is formulated in the form of the Mittag-Leffler function, which is considered as the eigenfunction of the Caputo fractional derivative. The suggested formula makes it possible to reduce the FTSE to the standard quantum mechanics with the time-dependent Hamiltonian, where the standard Floquet theorem is valid. Two examples related to quantum resonances are considered as well to support the obtained result.