Lie–Scheffers Systems

Cariñena, José F.; Ibort, Alberto; Marmo, Giuseppe; Morandi, G.

doi:10.1007/978-94-017-9220-2_9

Cited by 13 publications

(53 citation statements)

References 40 publications

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“…Similarly to standard Lie systems, the associated t-dependent Schrödinger equation related to a quantum Lie system can be solved by means of a Lie system on a Lie group [1,3]. Let us explain this fact (see [3] for applications).…”

Section: Quantum Lie Systemsmentioning

confidence: 99%

“…A Lie system is a non-autonomous system of first-order ordinary differential equations whose general solution can be written via a (generally nonlinear) autonomous function, referred to as superposition rule, a finite set of particular solutions, and some constants related to the initial conditions [1][2][3][4]. For instance, non-autonomous inhomogeneous linear systems of first-order ordinary differential equations, Bernoulli equations, Riccati equations, and matrix Riccati equations are examples of Lie systems [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

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Quantum quasi-Lie systems: properties and applications

2023

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A Lie system is a non-autonomous system of ordinary differential equations describing the integral curves of a t-dependent vector field that is equivalent to a t-dependent family of vector fields within a finite-dimensional Lie algebra of vector fields. Lie systems have been generalised in the literature to deal with t-dependent Schrödinger equations determined by a particular class of t-dependent Hamiltonian operators, the quantum Lie systems, and other systems of differential equations through the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and quantum Lie systems to cope with t-dependent Schrödinger equations associated with the here-called quantum quasi-Lie systems. To illustrate our methods, we propose and study a quantum analogue of the classical nonlinear oscillator searched by Perelomov, and we analyse a quantum one-dimensional fluid in a trapping potential along with quantum t-dependent Smorodinsky–Winternitz oscillators.

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Section: Quantum Lie Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum quasi-Lie systems: properties and applications

2023

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“…It is straightforward to verify that, due to the relation ( 5), the ODEs ( 7) possess the structure of a Lie system [5][6][7][8][9] with associated t-dependent vector field X t (6) and Vessiot-Guldberg Lie algebra isomorphic to b 2 . By the Lie-Scheffers Theorem, this guarantees that the system always admits a fundamental system of solutions, that is, a superposition rule.…”

Section: Lie-hamilton Systems From the Book Algebramentioning

confidence: 99%

“…As (7) is separable in the coordinates (x, y), the latter property is not required, and the system can be solved explicitly by quadratures, with the exact solution given by…”

Section: Lie-hamilton Systems From the Book Algebramentioning

confidence: 99%

“…In this contribution, we make use of the theory of Lie-Hamilton (LH) systems [1][2][3][4][5] in order to construct generalized time-dependent SIS epidemic models with a variable infection rate. We recall that LH systems form a particular class within Lie systems [5][6][7][8][9], due to the existence of Hamiltonian vector fields with respect to a Poisson structure.…”

Section: Introductionmentioning

confidence: 99%

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Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations

Fernández-Saiz,

Campoamor-Stursberg,

Herranz

2023

J. Phys.: Conf. Ser.

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The theory of Lie–Hamilton systems is used to construct generalized time-dependent SIS epidemic Hamiltonians with a variable infection rate from the ‘book’ Lie algebra. Although these are characterized by a set of non-autonomous nonlinear and coupled differential equations, their corresponding exact solution is explicitly found. Moreover, the quantum deformation of the book algebra is also considered, from which the corresponding deformed SIS Hamiltonians are obtained and interpreted as perturbations in terms of the quantum deformation parameter of previously known SIS systems. The exact solutions for these deformed systems are also obtained.

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Discrete matrix Riccati equations with superposition formulas

Penskoi¹,

Winternitz²

2004

Journal of Mathematical Analysis and Applications

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An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati equations. Here we shall describe discretizations of Riccati equations that preserve the superposition formulas. The approach is general enough to include q-derivatives and standard discrete derivatives.  2004 Elsevier Inc. All rights reserved.

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Lie–Scheffers Systems

Abstract: If only I knew how to get mathematicians interested in

Cited by 13 publications

References 40 publications

Quantum quasi-Lie systems: properties and applications

Quantum quasi-Lie systems: properties and applications

Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations

Discrete matrix Riccati equations with superposition formulas

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