arXiv subjects
Publications and source records attributed to G. Dattoli.
The parabolic functions are introduced in analogy to the circular and hyperbolic cases. We discuss the relevant properties, the geometrical interpretation and touch on possible generalizations and their link with the modular elliptic functions.
We discuss the new problems emerging in charged beam transport for SASE FEL dynamics. The optimization of the magnetic transport system for future devices requires new concepts associated with the slice emittance and the slice phase space distribution. We study the problem of electron beam slice matching and guiding in transport devices for SASE FEL emission discussing matching criteria and how the associated design of the electron transport line may affect the FEL output performances. We analyze different matching strategies by studying the relevant effect on the FEL output characteristics.
We discuss the conditions required for an optimal SASE FEL operation when bunch compression techniques are exploited to enhance the bunch peak current. We discuss the case of velocity bunching and magnetic bunch compression. With the reference to the latter technique we provide a quantitative estimate of the amount of laser heater power necessary to suppress the micro-bunching instability without creating any problem to the SASE dynamics
We analyze the dynamics of Free Electron Laser (FEL) devices, operating with a bunched beam exhibiting a longitudinal phase space correlation. We show that the presence of an energy-position correlation term is responsible for very interesting effects like an enhancement of the peak output power, a shortening of the laser pulses and an increase of the non linearly generated harmonic intensities. We conjecture that the mechanism is due to a kind of energy tapering effect associated with the correlation. We discuss the difference of the dynamics with respect to an ordinary undulator tapering and the relative advantages.
The theory of Touchard polynomials is generalized using a method based on the definition of exponential operators, which extend the notion of the shift operator. The proposed technique, along with the use of the relevant operational formalism, allows the straightforward derivation of properties of this family of polynomials and their relationship to different forms of Stirling numbers.
This report is a primer introduction on the sources of coherent syncrotron radiation from FELs compared to conventional lasers. Even though the underlying mechanisms are different, FELs can be considered a laser source to all effects. In fact FEL is based on the stimulated emission of virtual photons by a relativistic electron beam, passing through an undulator/wiggler, much like conventional lasers exploit the stimulated emission by an atomic/molecular system on which a population inversion has been realized. FEL operation is illustrated by means of the usual set of parameters (gain, saturation intensity...) used to describe the functioning of conventional lasers.
The existence of a characteristic coherence length in FEL SASE Physics determines the independent lasing of different portions, namely the slices, of the electron bunch. Each slice may be characterized by different phase space properties (not necessarily equal emittances and Twiss coefficients). This fact opens new questions on the concept of beam matching and how the various portions of the beam contribute to the performances of the output radiation, including those associated with the transverse coherence.
We review the teory of the pseudo-iperbolic functions on the basis of an algebraic point of view which employs the Eisenstein group. We frame the teory within the general context of the number decomposition and discuss the importance of these functions in the theory of the generalized Fourier transforms
The solution of the Schrodinger equation with a linear potential is considered. We use algebraic methods to obtain the explicit form of the solution for the explicitly time dependent Hamiltonian and discuss the general conditions which allow us to get solutions in terms of the Airy functions, yelding non spreading wave packets. We analyze the relevant physical meaning of these solutions and give examples of their applications. We discuss the analogy between Airy and Gauss-Weirestrass transform.
We study the properties of different type of transforms by means of operational methods and discuss the relevant interplay with many families of special functions. We consider in particular the binomial transform and its generalizations. A general method, based on the use of the Fourier transform technique, is proposed for the study of the properties of functions of operators.
We propose an operational method for the solution of differential equations involving vector products. The technique we propose is based on the use of the evolution operator, defined in such a way that the wealth of techniques developed within the context of quantum mechanics can also be exploited for classical problems. We discuss the application of the method to the solution of the Lorentz-type equations.
We introduce the Coriolis quantum states in analogy to the Landau states. We discuss their physical meaning and their role within the context of gravito-magnetic theory. We also analyse the experimental conditions under which they can be observed and their link with the Aharanov-Carmi effect.
Particular families of special functions, conceived as purely mathematical devices between the end of XVIII and the beginning of XIX centuries, have played a crucial role in the development of many aspects of modern Physics. This is indeed the case of the Euler gamma function, which has been one of the key elements paving the way to string theories, furthermore the Euler-Riemann Zeta function has played a decisive role in the development of renormalization theories. The ideas of Euler and later those of Riemann, Ramanujan and of other, less popular, mathematicians have therefore provided the mathematical apparatus ideally suited to explore, and eventually solve, problems of fundamental importance in modern Physics. The mathematical foundations of the theory of renormalization trace back to the work on divergent series by Euler and by mathematicians of two centuries ago. Feynman, Dyson, Schwinger... rediscovered most of these mathematical "curiosities" and were able to develop a new and powerful way of looking at physical phenomena.
In this brief note the operatorial methods are applied to the study of the Airy function and its generalizations.
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the exponential shift operator.
The Lamb-Bateman integral equation was introduced to study the solitary wave diffraction and its solution was written in terms of an integral transform. We prove that it is essentially the Abel integral equation and its solution can be obtained using the formalism of fractional calculus.
We consider the matrix representation of the Eisenstein numbers and in this context we discuss the theory of the Pseudo Hyperbolic Functions. We develop a geometrical interpretation and show the usefulness of the method in Physical problems related to the anomalous scattering of light by crystals
The Airy transform is an ideally suited tool to treat problem in classical and quantum optics. Even though the relevant mathematical aspects have been thoroughly investigated, the possibility it offers are wide and some aspects, as the link with special functions and polynomials, still contains unexplored aspects. In this note we will show that the so called Airy polynomials are essentially the third order Hermite polynomials. We will also prove that this identification opens the possibility of developing new conjectures on the properties of this family of polynomials.