Lévy-Schrödinger wave packets

Abstract

We analyze the time--dependent solutions of the pseudo--differential Lévy--Schrödinger wave equation in the free case, and we compare them with the associated Lévy processes. We list the principal laws used to describe the time evolutions of both the Lévy process densities, and the Lévy--Schrödinger wave packets. To have self--adjoint generators and unitary evolutions we will consider only absolutely continuous, infinitely divisible Lévy noises with laws symmetric under change of sign of the independent variable. We then show several examples of the characteristic behavior of the Lévy--Schrödinger 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 Lévy process densities, and the usual Schrödinger wave functions.

41 pages, 13 figures; paper substantially shortened, while keeping intact examples and results; changed format from "report" to "article"; eliminated Appendices B, C, F (old names); shifted Chapters 4 and 5 (old numbers) from text to Appendices C, D (new names); introduced connection between Relativistic q.m. laws and Generalized Hyperbolic laws

References in corpus (7)

• Fractional Quantum Mechanics
• Exact Mapping of the 2+1 Dirac Oscillator onto the Jaynes-Cummings Model: Ion-Trap Experimental Proposal
• Selfdecomposability and selfsimilarity: a concise primer
• Mixtures in non stable Levy processes
• Bimodality and hysteresis in systems driven by confined Lévy flights
• One dimensional stable probability density functions for rational index $0<\alpha \le 2$
• On some results of Cufaro Petroni about Student t-processes

Cited by in corpus (5)

• Levy flights and nonlocal quantum dynamics
• Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I
• General form of the covariant field equations of arbitrary spin and the relativistic canonical quantum mechanics
• Markov processes and generalized Schroedinger equations
• A class of Lévy driven SDEs and their explicit invariant measures