arXiv subjects
Publications and source records attributed to Marek Kuś.
We prove nonintegrability of a model Hamiltonian system defined on the Lie algebra $\mathfrak{su}_3$ suitable for investigation of connections between classical and quantum characteristics of chaos.
We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, 'most classical' can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons, as well as mixed spin states decomposable into probabilistic mixtures of pure coherent states. The latter were the subject of the recent paper by Giraud et. al., who showed that in the lowest-dimensional, nontrivial case of spin 1, each such state can be decomposed into a mixture of eight pure states. Using our method we prove that in fact four pure states always suffice.
We discuss the nearest neighbor distribution of the eigenvalues for hermitian generators in the Lie algebra of a semisimple complex Lie Group along a sequence of irreducible representations. After the basic definitions a limit theorem for rays of irreducible representation is formulated. Then it is proved that only a certain kind of rescaling will give meaning full results in the general case. Finally, we give explicit formulas for the Lipkin operator in the case of SU(3).
Various problems concerning the geometry of the space $u^*(\cH)$ of Hermitian operators on a Hilbert space $\cH$ are addressed. In particular, we study the canonical Poisson and Riemann-Jordan tensors and the corresponding foliations into Kähler submanifolds. It is also shown that the space $\cD(\cH)$ of density states on an $n$-dimensional Hilbert space $\cH$ is naturally a manifold stratified space with the stratification induced by the the rank of the state. Thus the space $\cD^k(\cH)$ of rank-$k$ states, $k=1,...,n$, is a smooth manifold of (real) dimension $2nk-k^2-1$ and this stratification is maximal in the sense that every smooth curve in $\cD(\cH)$, viewed as a subset of the dual $u^*(\cH)$ to the Lie algebra of the unitary group $U(\cH)$, at every point must be tangent to the strata $\cD^k(\cH)$ it crosses. For a quantum composite system, i.e. for a Hilbert space decomposition $\cH=\cH^1\ot\cH^2$, an abstract criterion of entanglement is proved.
A method allowing to distinguish interacting from non-interacting systems based on available time series is proposed and investigated. Some facts concerning generalized Renyi dimensions that form the basis of our method are proved. We show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts. We use our method to distinguish interacting from non-interacting systems on the examples of logistic and Hénon maps. A classification of all possible interaction schemes is given.
We consider the characteristic polynomials of random unitary matrices $U$ drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The variances of these ``secular coefficients'' are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent $β$. The latter secular coefficients are related to the traces of powers of $U$ by Newton's well-known formulae. While the traces tend to have Gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large, the secular coefficients exhibit strong mutual correlations due to Newton's mixing of traces to coefficients. These results might become relevant for current efforts at combining semiclassics and random-matrix theory in quantum treatments of classically chaotic dynamics.