Rational numbers in [0,1] admit no uniform distribution, but a sequence of discrete distributions with increasingly flat denominators assigns interval probabilities converging to b-a and point probabilities converging to zero.
L\'evy-Schr\"odinger wave packets
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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.
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math.PR 1years
2019 1verdicts
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Taking rational numbers at random
Rational numbers in [0,1] admit no uniform distribution, but a sequence of discrete distributions with increasingly flat denominators assigns interval probabilities converging to b-a and point probabilities converging to zero.