arXiv subjects
Publications and source records attributed to 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.
We extend Hardy's uncertainty principle for a square integrable function and its Fourier transform to the multidimensional case using a symplectic diagonalization. We use this extension to show that Hardy's uncertainty principle is equivalent to a statement on the Wigner distribution of the function. We give a geometric interpretation of our results in terms of the notion of symplectic capacity of an ellipsoid. Furthermore, we show that Hardy's uncertainty principle is valid for a general Lagrangian frame of the phase space. Finally, we discuss an extension of Hardy's theorem for the Wigner distribution for exponentials with convex exponents.
We discuss the relation between density matrices and the uncertainty principle; this allows us to justify and explain a recent statement by Man'ko et al. We thereafter use Hardy's uncertainty principle to prove a new result for Wigner distributions dominated by a Gaussian and we relate this result to the coarse-graining of phase-space by "quantum blobs".
We express the condition for a phase space Gaussian to be the Wigner distribution of a mixed quantum state in terms of the symplectic capacity of the associated Wigner ellipsoid. Our results are motivated by Hardy's formulation of the uncertainty principle for a function and its Fourier transform. As a consequence we are able to state a more general form of Hardy's theorem.
Around 1980 Connes extended the notions of geometry to the non-commutative setting. Since then {\it non-commutative geometry} has turned into a very active area of mathematical research. As a first non-trivial example of a non-commutative manifold Connes discussed subalgebras of rotation algebras, the so-called {\it non-commutative tori}. In the last two decades researchers have unrevealed the relevance of non-commutative tori in a variety of mathematical and physical fields. In a recent paper we have pointed out that non-commutative tori appear very naturally in Gabor analysis. In the present paper we show that Janssen's result on good window classes in Gabor analysis has already been proved in a completely different context and in a very disguised form by Connes in 1980. Our treatment relies on non-commutative analogs of Wiener's lemma for certain subalgebras of rotation algebras by Gröchenig and Leinert.
We point out a connection between Gabor analysis and noncommutative analysis. Especially, the strong Morita equivalence of noncommutative tori appears as underlying setting for Gabor analysis, since the construction of equivalence bimodules for noncommutative tori has a natural formulation in the notions of Gabor analysis. As an application we show that Feichtinger's algebra is such an equivalence bimodule. Furthermore, we present Connes's construction of projective modules for noncommutative tori and the relevance of a generalization of Wiener's lemma for twisted convolution by Gröchenig and Leinert. Finally we indicate an approach to the biorthogonality relation of Wexler-Raz on the existence of dual atoms of a Gabor frame operator based on results about Morita equivalence.