Semigroups of Jensen set-valued functions

A set-valued function F defined on a convex cone S in a real vector space X into the set n(Y) of all non-empty subsets of a real vector space Y is said to be the Jensen function iff $${1\over 2}\lbrack F(x)+F(x)\rbrack=F\lbrack{1\over 2}(x+y)\rbrack$$ for all x, y ∈ S. This note deals with iteration semigroups of Jensen set-valued functions on S.     

Nonlinear diffusion and Nelson-Brown movement

The nonlinear diffusion process, which can be described as the Nelson-Brown motion, is considered. The obtained equation becomes the classical linear diffusion equation for small relaxation times, and for long relaxation times it is transferred into the Schrödinger-like equation. The possible nonequilibrium stationary states are discussed.     

Symmetry enforced gauge invariance of nuclear magnetic shielding constants

E.S.R. experiments performed at 1·3 K by optical detection are reported for the photo-excited triplet state of palladiumporphin in a single crystal of n-octane, and the observation of a level anticrossing signal is described.

In the crystal the orbital degeneracy of the ³ E _u state of the free molecule is lifted by the crystal field and in n-octane the energy difference between the two orbital components |x> and |y> is found to be 58 ± 2 cm^-1. The spinorbit coupling (SOC) and the orbital Zeeman interaction couple the triplet manifolds of |x> and |y>, and for a proper understanding of the magnetic properties of these states it is necessary to work in the basis of the six spin-orbit functions deriving from the ³ E _u state of the free molecule. It is shown that either of the two triplet states can be described by an effective spin hamiltonian of the common form and expressions for the zero-field parameters D and E and the principal values of the g tensor are given. The experimental values of the parameters in the lowest triplet state are D = -24·38 ± 0·03 GHz, |E| = 320 ± 60 MHz, g _‖ = 1·677 ± 0·001 and g _⊥ = 1·989 ± 0·002. The matrix element of the SOC connecting the |x> and |y> triplet manifolds amounts to qZ = 15 ± 3 cm^-1 and the vibronic orbital angular momentum (in units of ?) in the ³ E _u state of the free molecule to qΛ = 1·5 ± 0·3. A tentative value of 0·63 for the orbital reduction factor q is obtained by comparison with a theoretical estimate of Λ. The value of q is indicative of weak Jahn-Teller coupling.     

MBPT studies of van der Waals molecules

The many-body perturbation theory is employed for the calculation of the interaction potential for the F^- … He system in the framework of the supermolecule method. A particular attention is paid to the choice of the basis set functions for the two subsystems and the related basis set superposition effects. It has been found that the main features of the interaction potential are recovered in the SCF approximation. The SCF potential has its minimum at the distance R = 6·4 a ₀ with the interaction energy of 53 cm^-1. The complete fourth-order MBPT method gives the potential minimum position and depth equal to 6·5 a ₀ and 64 cm^-1, respectively. The basis set superposition effects estimated by using the counterpoise technique are negligibly small for the SCF interaction potential, while at the correlated level their magnitude is comparable to the value of the total correlation contribution to the interaction energy. The basis set superposition effect in calculations of the electron correlation contribution to weak intermolecular interactions is found to be the major factor limiting the reliability of the corresponding theoretical data.     

Projection Evolution of Quantum States—the Delayed Choice Puzzle

Physics of Atomic Nuclei - We discuss the problem of the time variable in the nearly standard formulation of the quantum mechanics. In order to be able to describe the outcome of some of the...     

Noncommutative differential geometry with higher-order derivatives

We build a toy model of differential geometry on the real line, which includes derivatives of the second order. Such construction is possible only within the framework of noncommutative geometry. We introduce the metric and briefly discuss two simple physical models of scalar field theory and gauge theory in this geometry.Partially supported by KBN grant 2 P302 168 4.