The Schrödinger-Riccati Equation

Fraga, Serafín; Vega, José Manuel García de la; Fraga, Eric S.

doi:10.1007/978-3-642-51458-6_10

Cited by 4 publications

(5 citation statements)

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“…We start from the coupled equations (5) defining the Cauchy problem whose resolution provides the exact quantum dynamics of any two-state physical scenario describable by the Hamiltonian given by equation (2). Let us introduce a new complex-valued function u(t) related to a(t) and b(t) as follows…”

Section: Towards the New Riccati Equationmentioning

confidence: 99%

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An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

et al. 2017

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A novel recipe for exactly solving in finite terms a class of special differential Riccati equations is reported. Our procedure is entirely based on a successful resolution strategy quite recently applied to quantum dynamical time-dependent SU(2) problems. The general integral of exemplary differential Riccati equations, not previously considered in the specialized literature, is explicitly determined to illustrate both mathematical usefulness and easiness of applicability of our proposed treatment. The possibility of exploiting the general integral of a given differential Riccati equation to solve an SU(2) quantum dynamical problem, is succinctly pointed out.

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Section: Towards the New Riccati Equationmentioning

confidence: 99%

“…Theoretically investigating the properties of systems in different contexts [1,2,3], such as for example classical and quantum physics [4,5,6,7,8,9,10,11,12,13,14,15,16], mathematics [17,18,19,20,21,22,23,24,25], biology [26,27], one is led to the consideration of the following non-linear non-autonomous first order differential equation…”

Section: Introductionmentioning

confidence: 99%

An example of interplay between Physics and Mathematics: Exact resolution of a new class of Riccati Equations

et al. 2017

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“…where the first equation is the special Riccati nonlinear differential equation; see, for example, [32], [34], [50], [57], [58], [72] and references therein.…”

Section: Green Function: Increasing Casementioning

confidence: 99%

The time-dependent Schroedinger equation, Riccati equation and Airy functions

Lanfear,

Suslov

2009

Preprint

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We construct the Green functions (or Feynman's propagators) for the Schrödinger equations of the form iψ t + 1 4 ψ xx ± tx 2 ψ = 0 in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is established.

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“…is the simplest nonlinear equation and only one that can be converted into linear equation. It is widely used in control system theory [1], quantum mechanics [2], fluid mechanics [3] and other fields. Since 1724, it has been intensively studied by mathematicians in a variety of ways [4].…”

Section: Introductionmentioning

confidence: 99%