Narrow Results

We study a stationary Schrödinger–Poisson system on a bounded interval of the real axis. The Schrödinger operator is defined on the bounded domain with transparent boundary conditions. This allows us to model a non-zero current through the boundary of the interval. We prove that the system always admits a solution and give explicit a priori estimates for the solutions.

We show global well-posedness in energy norm of the semi-relativistic Schrödinger–Poisson system of equations with attractive Coulomb interaction in ${ℝ}^3$ in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in ${{ℒ}}_{\underline{\lambda }}^2$. In the absence of dissipation, we show that the solution corresponding to an initial condition with negative energy blows up in finite time, which is as expected, since the nonlinearity is critical.

It turns out that a parametrization of degenerate density matrices requires a parametrization of $𝔉=U(n)/(U(k_1)\times U(k_2)\times \cdots \times U(k_m))$, $n=k_1+\cdots +k_m$ where $U(k)$ denotes the set of all unitary $k\times k$-matrices with complex entries. Unfortunately, the parametrization of this quotient space is quite involved. Our solution does not rely on Lie algebra methods directly, but succeeds through the construction of suitable sections for natural projections, by using techniques from the theory of homogeneous spaces. We mention the relation to the Lie algebra background and conclude with two concrete examples.

An overview of the Bose–Einstein condensation of correlated atoms in a trap is presented by examining the effect of interparticle correlations to one- and two-body properties of the above systems at zero temperature in the framework of the lowest order cluster expansion. Analytical expressions for the one- and two-body properties of the Bose gas are derived using Jastrow-type correlation function. In addition numerical calculations of the natural orbitals and natural occupation numbers are also carried out. Special effort is devoted for the calculation of various quantum information properties including Shannon entropy, Onicescu informational energy, Kullback–Leibler relative entropy and the recently proposed Jensen–Shannon divergence entropy. The above quantities are calculated for the trapped Bose gases by comparing the correlated and uncorrelated cases as a function of the strength of the short-range correlations. The Gross–Piatevskii equation is solved, giving the density distributions in position and momentum space, which are employed to calculate quantum information properties of the Bose gas.

Quantum splines are curves in a Hilbert space or, equivalently, in the corresponding Hilbert projective space, which generalize the notion of Riemannian cubic splines to the quantum domain. In this paper, we present a generalization of this concept to general density matrices with a Hamiltonian approach and using a geometrical formulation of quantum mechanics. Our main goal is to formulate an optimal control problem for a nonlinear system on ${𝔲}^{\ast }(n)$ which corresponds to the variational problem of quantum splines. The corresponding Hamiltonian equations and interpolation conditions are derived. The results are illustrated with some examples and the corresponding quantum splines are computed with the implementation of a suitable iterative algorithm.

We give an exact solution to the nonlinear optimization problem of approximating a Hermitian matrix by positive semi-definite matrices. Our algorithm was then used to judge whether a quantum state is entangled or not. We show that the exact approximation of a density matrices by tensor product of positive semi-definite operators is determined by the additivity property of the density matrix.