L\'evy-Schr\"odinger wave packets

TL;DR

This paper investigates the evolution of Lévý-Schrödinger wave packets, comparing their dynamics with Lévý processes, highlighting unique bi-modality features contrasting traditional diffusive behaviors.

Contribution

It introduces a framework for analyzing time-dependent solutions of Lévý-Schrödinger equations with symmetric, infinitely divisible Lévý noises, and explores their distinctive bi-modal evolution patterns.

Findings

Identification of bi-modality in wave packet evolution

Comparison between Lévý process densities and wave packets

Examples illustrating characteristic behaviors of the solutions

Abstract

We analyze the time--dependent solutions of the pseudo--differential L\'evy--Schr\"odinger wave equation in the free case, and we compare them with the associated L\'evy processes. We list the principal laws used to describe the time evolutions of both the L\'evy process densities, and the L\'evy--Schr\"odinger wave packets. To have self--adjoint generators and unitary evolutions we will consider only absolutely continuous, infinitely divisible L\'evy noises with laws symmetric under change of sign of the independent variable. We then show several examples of the characteristic behavior of the L\'evy--Schr\"odinger wave packets, and in particular of the bi-modality arising in their evolutions: a feature at variance with the typical diffusive uni--modality of both the L\'evy process densities, and the usual Schr\"odinger wave functions.