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.
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