Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases

Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics

When several inequivalent supercharges form a closed superalgebra in Quantum Mechanics it entails the appearance of hidden symmetries of a Super-Hamiltonian. We examine this problem in one-dimensional QM for the case of periodic potentials and potentials with finite number of bound states. After the survey of the results existing in the subject the algebraic and analytic properties of hidden-symmetry differential operators are rigorously elaborated in the Theorems and illuminated by several examples

Quantifying the non-ergodicity of scaled Brownian motion

We examine the non-ergodic properties of scaled Brownian motion, a non-stationary stochastic process with a time dependent diffusivity of the form $D(t)\simeq t^{\alpha-1}$. We compute the ergodicity breaking parameter EB in the entire range of scaling exponents $\alpha$, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths $T$ and short lag times $\Delta$ the EB parameter as function of the scaling exponent $\alpha$ has no divergence at $\alpha=1/2$ and present the asymptotes for EB in different limits. We generalise the analytical and simulations results for the time averaged and ergodic properties of scaled Brownian motion in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.Comment: 19 pages, 6 figures (IOP LaTeX

High pressure synthesis of FeO-ZnO solid solutions with rock salt structure: in situ X-ray diffraction studies

X-ray diffraction with synchrotron radiation has been used for the first time to study chemical interaction in the FeO-ZnO system at 4.8 GPa and temperatures up to 1300 K. Above 750 K, the chemical reaction between FeO and ZnO has been observed that resulted in the formation of rock salt (rs) Fe1-xZnxO solid solutions (0.3 \leq x \leq 0.85). The lattice parameters of these solid solutions have been in situ measured as a function of temperature under pressure, and corresponding thermal expansion coefficients have been calculated.Comment: 9 pages, 2 figures, 1 tabl