Eigenfunctions and eigenvalues of the Wigner operator

�. A. Akhundova, V. V. Dodonov, Vladimir I. Man’ko

B. W. Knight, L. Sirovich

The Wigner transform of an integral kernel on the full line generalizes the Fourier transform of a translation kernel. The eigenvalue spectra of Hermitian kernels are related to the topographic features of their Wigner transforms. Two kernels whose Wigner transforms are equivalent under the unimodular afline group have the same spectrum of eigenvalues and have eigenfunctions related by an explicit linear transformation. Any kernel whose Wigner transform has concentric ellipses as contour lines, yields an eigenvalue problem which may be solved exactly.

D. Sourias, A. D. Jannussis, A. Streclas

In this paper we study the Wigner operators of angular momentum in the phase space, which are obtained from the well-known operators of angular momentum by virtue of Wigner representation. In the sequel we construct the Lie algebra of these operators, as well as the corresponding Lie group, and we examine two representations of this group. We find also the eigenfunction and the eigenvalues of these operators.

Jens Peder Dahl

Maurice A. de Gosson, Franz Luef

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.