Coherent States and Applications in Mathematical Physics

Robert, Didier; Combescure, Monique

doi:10.1007/978-94-007-0196-0

Cited by 220 publications

(287 citation statements)

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“…In [17,35], we find the following general result (an asymptotic expansion in powers of √ ε is actually given, but we stick to the first term to ease the presentation): …”

Section: Some Previous Resultsmentioning

confidence: 91%

On semi-classical limit of nonlinear quantum scattering

Carles¹

2016

Ann. Sci. École Norm. Sup.

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ABSTRACT. We consider the nonlinear Schrödinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a complete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering theory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states.

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“…In [17,35], we find the following general result (an asymptotic expansion in powers of √ ε is actually given, but we stick to the first term to ease the presentation): …”

Section: Some Previous Resultsmentioning

confidence: 91%

On semi-classical limit of nonlinear quantum scattering

Carles¹

2016

Ann. Sci. École Norm. Sup.

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“…i.e. ∆p∆x = 2 .These properties, the their generalizations can be found in the vast literature, for example in (Sudarshan, 1963;Klauder, & Skagerstam, 1985;Zhang & Gilmore, 1990;Klauder, 1993;Ali, Antoine, Gazeau, & Mueller, 1995;Combescure, & Robert, 2012;Perelomov, 2010), and citation therein. …”

Section: Coherent States and Bosonizationmentioning

confidence: 84%

“…The interest for such states have been extended to several branches of non-linear physics for modeling multiple non-linear systems and also for systems connected with other groups (including super-groups). Coherent states can be determined for quantum groups as well as using various exponential generalizations (deformations) (Borzov & Damaskinsky, 2006;Combescure & Robert, 2012).…”

Section: Introductionmentioning

confidence: 99%

Introduction to the Coherent States Approach for Solving Non-linear Physical Problems

Torres¹,

Davila²,

Agüero³

2016

APR

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We summarize some crucial results from mathematical models that help us understand the relevance of generalized coherent states (GCS), a fundamental method used for the description of non-linear problems in physics. We mainly concentrate our review in the following models: ferromagnetism, nonlinear Schrödinger equation with external potential, nonlinear quantum oscillators, Bose -Einstein condensation (BEC) and the DNA quasi-spin model. Such models and some applications of the wide variety these involve are outlined. As variational trial states, the coherent states (CS) allow the estimation of the ground state energies and properties, yielding results which become exact in a number of nonlinear differential equations for dynamical variable of each model. Mainly we concentrate our review on two types of coherent states, The first one are based on the Heisenberg -Weyl group where it is applied the bosonization effect. The second one is based on the SU(2)/U(1) Lie group. For this case when the Hamiltonian and all physical operators are constructed by the elements of a Lie algebra, the generalized coherent states approach is directly applied without bozonization.

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“…Similarly, we write Z = (Q; P ) with Q, P ∈ C d×d for Lagrangian frames. With this denotation, the isotropy condition (1) and the normalisation condition (2) coincide with the symplecticity condition in [19] or, by taking Q = A and P = iB, with Hagedorn's original definition in [12].…”

Section: Lagrangian Frames and Ground Statesmentioning

confidence: 99%

“…In the literature, Hagedorn wave packets also coexist as generalised squeezed states; see [2]. These states are generated by applying translation and squeezing operators to the eigenstates of the standard harmonic oscillator.…”

Section: Introductionmentioning

confidence: 99%

An invariant class of wave packets for the Wigner transform

Dietert

Keller

Troppmann

2017

Journal of Mathematical Analysis and Applications

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Generalised Hagedorn wave packets appear as exact solutions of Schrödinger equations with quadratic, possibly complex, potential, and are given by a polynomial times a Gaussian. We show that the Wigner transform of generalised Hagedorn wave packets is a wave packet of the same type in phase space. The proofs build on a parametrisation via Lagrangian frames and a detailed analysis of the polynomial prefactors, including a novel Laguerre connection. Our findings directly imply the recently found tensor product structure of the Wigner transform of Hagedorn wave packets.

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Coherent States and Applications in Mathematical Physics

Cited by 220 publications

References 0 publications

On semi-classical limit of nonlinear quantum scattering

On semi-classical limit of nonlinear quantum scattering

Introduction to the Coherent States Approach for Solving Non-linear Physical Problems

An invariant class of wave packets for the Wigner transform

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