The initial decay rate of a mixed state is discussed; is shown to be zero for finite energy states. In the previous paper [1] we investigated a general scheme for description of unstable systems. One of our results concerned the initial decay rate. Generalizing the earlier results of Horwitz and Marchand (see [2] and other references contained in [1]) we proved there, that the initial decay rate of any pure state of the unstable system equals to zero, if this state is so called finite energy state. Here we shall be interested in the same problem, assuming now the state of the decaying system to be in general mixed. Such assumption seems to be reasonable: firstly, from the point of view of an experiment it is too restrictive to treat only pure states of unstable systems. In particular, mixed states are generally considered in the recent studies about the influence of measuring devices on the time evolution of the unstable system [3

A BDDC (Balancing Domain Decomposition by Constraints) algorithm is developed and analyzed for a staggered discontinuous Galerkin (DG) finite element approximation of second order scalar elliptic problems. On a quite irregular subdomain partition, an optimal condition number bound is proved for two-dimensional problems. In addition, a sub-optimal but scalable condition number bound is obtained for three-dimensional problems. These bounds are shown to be independent of coefficient jumps in the subdomain partition. Numerical results are also included to show the performance of the algorithm.

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We discuss spectral and resonance properties of a Hamiltonian describing motion of an electron moving on a hybrid surface consisting on a halfline attached by its endpoint to a plane under influence of a constant magnetic field which interacts with its spin through a Rashba-type term.

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