Fourier transforms, fractional derivatives, and a little bit of quantum mechanics. (English) Zbl 1479.46053

Summary: We discuss some of the mathematical properties of the fractional derivative defined by means of Fourier transforms. We first consider its action on the set of test functions \(\mathcal{\mathcal{S}} (\mathbb{R})\), and then we extend it to its dual set, \(\mathcal{\mathcal{S}}'(\mathbb{R})\), the set of tempered distributions, provided they satisfy some mild conditions. We discuss some examples, and we show how our definition can be used in a quantum mechanical context.


46F10 Operations with distributions and generalized functions
26A33 Fractional derivatives and integrals
46N50 Applications of functional analysis in quantum physics
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