Contractions of product density operators of systems of identical fermions and bosons

Recurrence and explicit formulae for contractions (partial traces) of antisymmetric and symmetric products of identical trace class operators are derived. Contractions of product density operators of systems of identical fermions and bosons are proved to be asymptotically equivalent to, respectively, antisymmetric and symmetric products of density operators of a single particle, multiplied by a normalization integer. The asymptotic equivalence relation is defined in terms of the thermodynamic limit of expectation values of observables in the states represented by given density operators. For some weaker relation of asymptotic equivalence, concerning the thermodynamic limit of expectation values of product observables, normalized antisymmetric and symmetric products of density operators of a single particle are shown to be equivalent to tensor products of density operators of a single particle. This paper presents the results of a part of the author's thesis [W. Radzki, "Kummer contractions of product density matrices of systems of $n$ fermions and $n$ bosons" (Polish), MS thesis, Institute of Physics, Nicolaus Copernicus University, Toru\'{n}, 1999].Comment: 20 pages. The manuscript has been shortened. A few typos correcte

On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

Stabilization of solutions of dissipative Hamiltonian systems

AbstractWe study the stabilization of solutions of damped Hamiltonian systems. We give sufficient conditions for convergence of these solutions, decay estimate and examples of applications

Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

AbstractWe study connected branches of nonconstant 2π-periodic solutions of the Hamilton equationẋ(t)=λJ∇H(x(t)),where λ∈(0,+∞), H∈C2(Rn×Rn,R) and ∇2H(x0)=A00B for x0∈∇H−1(0). The Hessian ∇2H(x0) can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given (x0,λ0). As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system ẋ(t)=J∇H(x(t)) emanating from x0. We describe also minimal periods of trajectories near x0