Analysis technique for exceptional points in open quantum systems and QPT analogy for the appearance of irreversibility

We propose an analysis technique for the exceptional points (EPs) occurring in the discrete spectrum of open quantum systems (OQS), using a semi-infinite chain coupled to an endpoint impurity as a prototype. We outline our method to locate the EPs in OQS, further obtaining an eigenvalue expansion in the vicinity of the EPs that gives rise to characteristic exponents. We also report the precise number of EPs occurring in an OQS with a continuum described by a quadratic dispersion curve. In particular, the number of EPs occurring in a bare discrete Hamiltonian of dimension $n_\textrm{D}$ is given by $n_\textrm{D} (n_\textrm{D} - 1)$; if this discrete Hamiltonian is then coupled to continuum (or continua) to form an OQS, the interaction with the continuum generally produces an enlarged discrete solution space that includes a greater number of EPs, specifically $2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) [2^{n_\textrm{C}} (n_\textrm{C} + n_\textrm{D}) - 1]$, in which $n_\textrm{C}$ is the number of (non-degenerate) continua to which the discrete sector is attached. Finally, we offer a heuristic quantum phase transition analogy for the emergence of the resonance (giving rise to irreversibility via exponential decay) in which the decay width plays the role of the order parameter; the associated critical exponent is then determined by the above eigenvalue expansion.Comment: 16 pages, 7 figure

A Numerical Study of the Hydrodynamic Stable Concentration Boundary Layers in a Membrane System Under Microgravitational Conditions

On the basis of the classic formula of the concentration Rayleigh number and the Kedem–Katchalsky equation for diffusive membrane transport, we derived the equations of sixteenth order which show the dependence of the thicknesses of the concentration boundary layers on the difference of the solution concentrations, the concentration Rayleigh number, the solute permeability coefficient of the membrane and the diffusion coefficients in the solution, the kinematic viscosity of the solution, the density of solutions, the temperature and gravitational acceleration. The obtained equation has numerical solutions in the first, third and fourth quadrant of a co-ordinate system. However, only two solutions from the first quadrant of the co-ordinate system have physical meaning. Confining ourselves to the set of solutions with physical meaning only, the thicknesses of concentration boundary layers for different parameters occurring in the obtained equation were calculated numerically