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      Comments on the Properties of Mittag-Leffler Function

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          Abstract

          The properties of Mittag-Leffler function is reviewed within the framework of an umbral formalism. We take advantage from the formal equivalence with the exponential function to define the relevant semigroup properties. We analyse the relevant role in the solution of Schr\"odinger type and heat-type fractional partial differential equations and explore the problem of operatorial ordering finding appropriate rules when non-commuting operators are involved. We discuss the coherent states associated with the fractional Sch\"odinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

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          Most cited references 6

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          On the exponential solution of differential equations for a linear operator

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            The Radiation Theories of Tomonaga, Schwinger, and Feynman

             F Dyson (1949)
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              Fractional Schrodinger equation

               N. Laskin (2002)
              Properties of the fractional Schrodinger equation have been studied. We have proven the hermiticity of fractional Hamilton operator and established the parity conservation law for the fractional quantum mechanics. As physical applications of the fractional Schrodinger equation we have found the energy spectrum for a hydrogen-like atom - fractional ''Bohr atom'' and the energy spectrum of fractional oscillator in the semiclassical approximation. A new equation for the fractional probability current density has been developed and discussed. We also discuss the relationships between the fractional and the standard Schrodinger equations.
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                Author and article information

                Journal
                2017-07-04
                Article
                1707.01135

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                16 pages, 9 figures
                math-ph math.MP

                Mathematical physics, Mathematical & Computational physics

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