Root Operators and “Evolution” Equations

Dattoli, G.; Torre, A.

doi:10.3390/math3030690

Cited by 4 publications

(7 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…in line with (94). As a conclusion, we mention that the result (19) as well as its normalized form (35) can be considered as the analytic continuation of those for the equation [78,79]…”

mentioning

confidence: 85%

See 2 more Smart Citations

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Torre

Lattanzi²,

Levi

2016

Annalen der Physik

Self Cite

View full text Add to dashboard Cite

A detailed study of the spinless (1+1)D free-particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter "flow" in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested. +∞ −∞ e −i p 2 2 m t e i px ψ 0 ( p)dp. (12) As we know, in analogy with (9), the evolution under the Schrödinger equation amounts in the p-domain to an energyrelated phase-factor, and hence to a simple quadratic-phase

show abstract

“…in line with (94). As a conclusion, we mention that the result (19) as well as its normalized form (35) can be considered as the analytic continuation of those for the equation [78,79]…”

mentioning

confidence: 85%

“…Mathematical interest in the initial conditions problem (95) is dictated by the presence of the pseudo-differential operator 1 − ∂ 2 ξ . It has been shown in [79] that Eq. (95) admits the solution…”

mentioning

confidence: 99%

See 1 more Smart Citation

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Torre

Lattanzi²,

Levi

2016

Annalen der Physik

Self Cite

View full text Add to dashboard Cite

show abstract

“…The method that was developed by Dattoli and his collaborators [20][21][22][23][24] subsequently noted Dirac's method or DM, replacing a second-order differential equation with a pair of first-order differential equations. It generalizes the classical factorization method on several levels.…”

Section: Introductionmentioning

confidence: 99%

Factorization à la Dirac Applied to Some Equations of Classical Physics

et al. 2021

View full text Add to dashboard Cite

In this paper, we present an application of Dirac’s factorization method to three types of the partial differential equations, i.e., the wave equation, the scattering equation, and the telegrapher’s equation. This method gives results that contribute to a better understanding of physical phenomena by generalizing the Euler and constituent equations. Its application to the wave equation shows that it is indeed a factorization method, since it gives d’Alembert’s solutions in a more general framework. In the case of the diffusion equation, a fractional differential equation has been established that has already been highlighted by other authors in particular cases, but by indirect methods. Dirac’s method brings several new results in the case of the telegraphers’ equation corresponding to the propagation of an acoustic wave in a dissipative fluid. On the one hand, its formalism facilitates the temporal interpretation of phenomena, in particular the density and compressibility of the fluid become temporal operators, which can be “seen” as susceptibilities of the fluid. On the other hand, a consequence of this temporal modeling is the highlighting in Euler’s equation of a term similar to the one that was introduced by Boussinesq and Basset in the equation of the motion of a solid sphere in a unsteady fluid.

show abstract

“…3.1 (The Zassenhaus formula gap). The framework that we have considered here to describe formally the solutions of the discretized Dirac equation(27) may be seen as a multivector extension of the framework obtained in terms of Pauli matrices by Datolli and his collaborators on the papers[10,11]. The major dif-…”

mentioning

confidence: 99%

Relativistic Wave Equations on the Lattice: An Operational Perspective

Faustino

2019

Trends in Mathematics

View full text Add to dashboard Cite

Dedicated to Professor Wolfgang Sprößig on occasion of his 70th birthday.Abstract. This paper presents an operational framework for the computation of the discretized solutions for relativistic equations of Klein-Gordon and Dirac type. The proposed method relies on the construction of an evolutiontype operador from the knowledge of the Exponential Generating Function (EGF), carrying a degree lowering operator Lt = L(∂t). We also use certain operational properties of the discrete Fourier transform over the n−dimensional Brioullin zone Q h = − π h , π h n -a toroidal Fourier transform in disguise -to describe the discrete counterparts of the continuum wave propagators,∆ − m 2 respectively, as discrete convolution operators. In this way, a huge class of discretized time-evolution problems of differential-difference and difference-difference type may be studied in the spirit of hypercomplex variables.

show abstract

Root Operators and “Evolution” Equations

Cited by 4 publications

References 40 publications

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Factorization à la Dirac Applied to Some Equations of Classical Physics

Relativistic Wave Equations on the Lattice: An Operational Perspective

Contact Info

Product

Resources

About