The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"{o}dinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schr\"{o}dinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schr\"{o}dinger evolution is analyzed in detail. The relativistic covariance of related waveComment: Latex fil

Network Model of Electronic States of Thin Films, Solid Interfaces, and Planar Defects

Green\u27s functions for a finite free boundary network are derived. Their application to models of solid interface and surface defects is made. The density of states with respect to localized electrons around interface or defect nodes is calculated. The density of states as a function of the thickness of a thin film is calculated. Periodic conditions as well as free surface boundaries are considered

Kinetics of Adsorption on Stepped Surfaces and the Determination of Surface Diffusion Constants

Nitrogen adsorption on stepped W(110) surfaces is examined to illustrate a theory of surface kinetics. Experimental findings by Besocke et al. have shown that nitrogen chemisorbs dissociatively only at the step corner sites of a W(110) surface. Thus the rate of dissociation reveals the mobility of nitrogen and its interaction with the surface. Using continuous-time-random-walk theory, we obtain the probability that molecules reach the step corner sites as a function of time. A kinetic model of nitrogen dissociation is proposed to calculate a coverage function that is in good agreement with experiment. The surface diffusion constant of nitrogen molecules is obtained and is in accordance with previous observations that nitrogen molecules are first weakly physisorbed on the W(110) terrace. Finally, the coverage functions for different step densities are predicted