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arXiv:0808.0489

A new approach to the star-genvalue equation

Maurice A. de Gosson, Franz Luef

Published Aug 4, 2008• 0 citations
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Abstract

We show that the eigenvalues and eigenfunctions of the stargenvalue equation can be completely expressed in terms of the corresponding eigenvalue problem for the quantum Hamiltonian. Our method makes use of a Weyl-type representation of the star-product and of the properties of the cross-Wigner transform, which appears as an intertwining operator.

DOI
10.48550/arxiv.0808.0489

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We show that the eigenvalues and eigenfunctions of the stargenvalue equation can be completely expressed in terms of the corresponding eigenvalue problem for the quantum Hamiltonian. Our method makes use of a Weyl-type representation of the star-product and of the properties of the cross-Wigner transform, which appears as an intertwining operator.
The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1
This chapter presents Wigner’s approach to quantum mechanics, based on the Wigner function in phase space. It explains Wigner–Weyl quantization, which makes it possible to associate functions on phase space to wave functions and operators, and it develops the technology to do quantum mechanics in this formalism. This includes the star product, Moyal evolution, and star-eigenvalue equations. It also develops semiclassical methods in this formulation, and it has a section on Berry’s semiclassical formula for the Wigner function in one-dimensional systems.
The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order ℏ2\hbar^2ℏ2 (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in ℏ\hbarℏ, starting with ℏ0\hbar^0ℏ0, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.