Semiclassical Trajectory-Coherent Approximation in Quantum Mechanics I. High-Order Corrections to Multidimensional Time-Dependent Equations of Schrödinger Type

The knowledge of quantum phase flow induced under the Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow curves obey the quantum Hamilton's equations and play the role of characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phase-space analytic geometry based on the star-product operation can hardly be visualized. The statement "quantum trajectory belongs to a constraint submanifold" can be changed e.g. to the opposite by a unitary transformation. Some of relations among quantum objects in phase space are, however, left invariant by unitary transformations and support partly geometric relations of belonging and intersection. Quantum phase flow satisfies the star-composition law and preserves hamiltonian and constraint star-functions.

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“…The phase-space trajectories appearing in [30,31,32,33] are connected to specific quantum states like in the de Broglie -Bohm theory.…”

Section: Composition Law For Quantum Trajectories and Energy Consementioning

confidence: 99%

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Krivoruchenko

Faessler

2007

Journal of Mathematical Physics

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“…The functions belonging to the class P t , at any fixed time t ∈ R 1 are concentrated, as → 0 in the neighborhood of a point lying on the phase curve z = Z(t, 0), t ∈ R 1 [3]. Therefore, it is natural to call the functions of the class P t trajectory-concentrated functions.…”

Section: The Nonlocal Gross-pitaevskii Equation and The Cauchy Problemmentioning

confidence: 99%

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

Lisok¹

2013

SIGMA

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Abstract. We consider the symmetry properties of an integro-differential multidimensional Gross-Pitaevskii equation with a nonlocal nonlinear (cubic) term in the context of symmetry analysis using the formalism of semiclassical asymptotics. This yields a semiclassically reduced nonlocal Gross-Pitaevskii equation, which can be treated as a nearly linear equation, to determine the principal term of the semiclassical asymptotic solution. Our main result is an approach which allows one to construct a class of symmetry operators for the reduced Gross-Pitaevskii equation. These symmetry operators are determined by linear relations including intertwining operators and additional algebraic conditions. The basic ideas are illustrated with a 1D reduced Gross-Pitaevskii equation. The symmetry operators are found explicitly, and the corresponding families of exact solutions are obtained.

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“…Equations (28) are called a system in variations in vector form [27]. In general, it has n complex linear independent solutions, which can be written in the form of 2n-dimensional vector columns…”

Section: Solutions Of the Gross-pitaevskii Equationmentioning

confidence: 99%

“…Further developments of the Maslov method [25,26] (see also [27] and references therein) allow one to construct the localized solutions of the Cauchy problem in the class of trajectoryconcentrated functions.…”

Section: Introductionmentioning

confidence: 99%

Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

Борисов¹

2005

SIGMA

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Abstract. The Gross-Pitaevskii equation with a local cubic nonlinearity that describes a many-dimensional system in an external field is considered in the framework of the complex WKB-Maslov method. Analytic asymptotic solutions are constructed in semiclassical approximation in a small parameter , → 0, in the class of functions concentrated in the neighborhood of an unclosed surface associated with the phase curve that describes the evolution of surface vertex. The functions of this class are of the one-soliton form along the direction of the surface normal. The general constructions are illustrated by examples.

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Semiclassical Trajectory-Coherent Approximation in Quantum Mechanics I. High-Order Corrections to Multidimensional Time-Dependent Equations of Schrödinger Type

Cited by 79 publications

References 56 publications

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Weyl’s symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

Symmetry and Intertwining Operators for the Nonlocal Gross-Pitaevskii Equation

Transverse Evolution Operator for the Gross-Pitaevskii Equation in Semiclassical Approximation

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