Periodic and constant solutions of matrix Riccati differential equations: n — 2. Proc. Roy. Sur 1’equation differentielle matricielle de type Riccati. Bull. Math. The qualitative study of second order linear equations originated in the classic paper . for a history of the Riccati transformation. Differentielle. (Q(t),’)’. VESSIOT, E.: “Sur quelques equations diffeYentielles ordinaires du second ordre .” Annales de (3) “Sur l’equation differentielle de Riccati du second ordre.
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The second author greatly appreciates the support by University of Puerto Rico and Universitat de Barcelona during the academic visit to the latter in Spring where this project was born.
This is different to construct the explicit propagators know- ing apriory the solutions of the Riccatti or characteristic equation, which can open other possibilities to study propagator with special functions as characteristic equations, equafion example, Heun equation.
In this section we find solutions to 4. Symbolic Computation, 23— Enter the email address you signed up with and we’ll email you a reset link. The solutions obtained throgh such towers are called Liouvillian. Toy models inspired by integrable Riccati equations.
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Views Read Edit View history. It follows the construction of the propagators related with these integrability conditions. Email address subscribed successfully. Ince, Ordinary differential equations, Dover, New York, The oscil- lator 1 – 2 might be introduced for the first time by Takahasi  in order to describe the process of degenerate parametric amplification in quantum optics see also [19, 20, 22, 23, 30, 31, 37].
From Proposition ricxati, it is recovered the well known result in differential Galois theory see for example : Here we follow the version of Kovacic Algorithm given in [1, 3, 17].
Orszag, Quantum OpticsSpringer, Heidelberg, We never store sensitive information about our customers in cookies. Translation of the original Latin into English by Ian Bruce. Suslov, Models of damped oscillators in quantum mechanics, Journal of Physical Mathematics 1S 16 pages. Propagators and Green Functions.
A 48, — P. This procedure is called Hamiltonian Algebrization, which is an isogaloisian transformation, i.
In this paper a Galoisian approach to build propagators through Riccati equations is presented. In fact, they present a non-periodic solution that allows us to write the propagator explicitly; the fact of the solution being non-periodic is fundamental.
Help Center Find new research papers in: Hannay, Angle variable holonomy in adiabatic excursion of an integrable Hamil- tonian, J.
The starting point is the knowledge of the integrability, in the Picard-Vessiot sense, of the characteristic equation, or equivalently the existence of an al- gebraic solution over its differential field of its associated Riccati equation. The aim of this paper is to establish a Galoisian approach to the tech- niques given by Suslov et. Key words and phrases. Click here to sign up. Angelow, Light propagation in nonlinear waveguide and classical two-dimensional oscillator, Physica A —  R.
To study Liouvillian solutions for linear second order differential equations, as well the integrability of their associated Riccati equations, we use Kovacic algorithm see  and differentiekle algebrization procedure see [1, 3].
Suslov, Dynamical invariants for variable quadratic Hamiltonians, Physica Scripta 81 5, 11 pp ; see also Preprint arXiv: The fact that in quan- tum electrodynamics the electromagnetic field can be represented as a set of forced harmonic oscillators makes quadratic Hamiltonians of special interest [7, 8, 11, 14, 35, 39]. Theorem 9 Galoisian approach to LSE. Galoisian Approach equxtion Propagators In this section we apply the Picard-Vessiot theory in the context of propagators. Kovacic Algorithm developed by J.
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The substitution that is needed to solve this Bernoulli equation is. If Pn does not exist, then case 2 cannot hold. Cookies are little nuggets of information that web servers store on your computer to make it easier for them to keep track of your browsing session. Although there are a plenty of papers concerning to explicit solutions and harmonic oscilla- tor see we recall that differential Galois theory can provide the Liou- villian solutions of characteristic equations without previous knowledge of such equations.
Takahasi, Information theory of quantum-mechanical channels Advances in Com- munication Systems: Differential Galois group, Green functions, propagators, Ric- cati equation. For example, at loot. Theory and Applications A. In this case the ODEs are in the complex domain and differentiation is with respect to a complex variable.
The main application of this Ga- loisian approach, for xe a main result of the paper, is the construction of the propagator for the so called degenerate parametric oscillator: A activation email has been sent to you.
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