Silvia Licciardi

Evolution Equations Involving Matrices Raised to Non-Integer Exponents

The use of matrices evolution equations raised to non-integer exponents finds applications in problems involving the solution of two or three level systems ruled by Klein–Gordon type equations. We develop a fairly simple method exploiting the wealth of results developed on fractional calculus and provide an example of application.

Comments on the properties of Mittag-Leffler function

Abstract The properties of Mittag-Leffler function are 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 Schrödinger 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 Schödinger equation, analyze the relevant Poisson type probability amplitude and compare with analogous results already obtained in the literature.

Umbral Methods and Harmonic Numbers

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals.