Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics
Abstract
The knowledge of quantum phase flow induced under 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 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 firstorder ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phasespace analytic geometry based on the starproduct 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 the 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 starcomposition law and preserves Hamiltonian and constraint star functions.
 Authors:
 Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow (Russian Federation) and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D72076 Tuebingen (Germany)
 (Germany)
 Publication Date:
 OSTI Identifier:
 20929701
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2735816; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CANONICAL DIMENSION; DIFFERENTIAL EQUATIONS; GEOMETRY; HAMILTONIANS; HILBERT SPACE; MATHEMATICAL EVOLUTION; PHASE SPACE; QUANTUM MECHANICS; SEMICLASSICAL APPROXIMATION; TRANSFORMATIONS
Citation Formats
Krivoruchenko, M. I., Faessler, Amand, and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D72076 Tuebingen. Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics. United States: N. p., 2007.
Web. doi:10.1063/1.2735816.
Krivoruchenko, M. I., Faessler, Amand, & Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D72076 Tuebingen. Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics. United States. doi:10.1063/1.2735816.
Krivoruchenko, M. I., Faessler, Amand, and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D72076 Tuebingen. Tue .
"Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics". United States.
doi:10.1063/1.2735816.
@article{osti_20929701,
title = {Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics},
author = {Krivoruchenko, M. I. and Faessler, Amand and Institut fuer Theoretische Physik, Tuebingen Universitaet, Auf der Morgenstelle 14, D72076 Tuebingen},
abstractNote = {The knowledge of quantum phase flow induced under 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 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 firstorder ordinary differential equations for quantum trajectories and generalized Jacobi fields. Classical and quantum constraint systems are discussed. The phasespace analytic geometry based on the starproduct 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 the 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 starcomposition law and preserves Hamiltonian and constraint star functions.},
doi = {10.1063/1.2735816},
journal = {Journal of Mathematical Physics},
number = 5,
volume = 48,
place = {United States},
year = {Tue May 15 00:00:00 EDT 2007},
month = {Tue May 15 00:00:00 EDT 2007}
}

Weyl's spectral asymptotics formula for nonelliptic operators of the Dirichlet problem
One considers the asymptotic behavior of the spectrum of the Dirichlet problem for a class of operators with constant coefficients, including in it the hypoelliptic operators. For this class one obtains the classical Weyl formula of spectral asymptotics. The residual of the distribution function of the spectrum is estimated in terms of the measure of the interior boundary layer of the level surface of the operator symbol. These results can be carried over also to the vectorial case. One considers separately the class of operators whose quadratic form corresponds to the differential norm of the Sobolev type. For the indicatedmore »