139 research outputs found
Umbral Methods and Harmonic Numbers
The theory of harmonic based function 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.Comment: 6 page
Root Operators and "Evolution" Equations
Root-operator factorization à la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicomponent vector. We will review the method along with its extension to root operators of degree higher than two. Also, we will show the results obtained by the Dirac-method as well as results from other methods, specifically in connection with evolution-like equations ruled by square-root operators, that we will address to as relativistic evolution equations
Comments on the Properties of Mittag-Leffler Function
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.Comment: 16 pages, 9 figure
Space Charge and Quantum Corrections in Free Electron Laser Evolution
Effects producing gain dilution in Free Electron Laser devices are well
documented. We develop here a unified point of view allowing the introduction
of space charge effects, along with the gain deterioration due to inhomogeneous
broadening contributions and discuss the relevant interplay. We outline future
developments and comment on the possibility of including in the formalism
effects of quantum mechanical nature.Comment: 10 pages, 7 figure
Preface to "Oscillator-Amplifier Free Electron Lasers an Outlook to Their Feasibility and Performances"
Free Electron Lasers (FELs) are certainly among the most interesting devices, belonging to the realm of coherent radiation sources [...
An operational point of view to the theory of multi-variable/multi-index Hermite polynomials
The use of algebraic tools of operational and umbral nature is exploited to
develop a new point of view and to extend the theory of Hermite polynomials,
with more than one variable also of complex nature. The techniques we adopt
includes multivariable/many index Hermite- Kampe-de-Feriet polynomials of order
two and higher. It will be shown that the treatment, foreseen here, simplifies
the study of the relevant properties and the associated computational
technicalities
- …