Coherent oscillations and incoherent tunnelling in one - dimensional asymmetric double - well potential

For a model 1d asymmetric double-well potential we calculated so-called survival probability (i.e. the probability for a particle initially localised in one well to remain there). We use a semiclassical (WKB) solution of Schroedinger equation. It is shown that behaviour essentially depends on transition probability, and on dimensionless parameter which is a ratio of characteristic frequencies for low energy non-linear in-well oscillations and inter wells tunnelling. For the potential describing a finite motion (double-well) one has always a regular behaviour. For the small value of the parameter there is well defined resonance pairs of levels and the survival probability has coherent oscillations related to resonance splitting. However for the large value of the parameter no oscillations at all for the survival probability, and there is almost an exponential decay with the characteristic time determined by Fermi golden rule. In this case one may not restrict oneself to only resonance pair levels. The number of perturbed by tunnelling levels grows proportionally to the value of this parameter (by other words instead of isolated pairs there appear the resonance regions containing the sets of strongly coupled levels). In the region of intermediate values of the parameter one has a crossover between both limiting cases, namely the exponential decay with subsequent long period recurrent behaviour.Comment: 19 pages, 7 figures, Revtex, revised version. Accepted to Phys. Rev.

Fractional Dynamics of Relativistic Particle

Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected to a non-potential four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_{\mu} u^{\mu}+c^2=0, where c is a speed of light in vacuum. In the general case, the fractional dynamics of relativistic particle is described as non-Hamiltonian and dissipative. Conditions for fractional relativistic particle to be a Hamiltonian system are considered

EVALUATION OF DEVELOPMENT LEVEL OF TERRITORIES OF UKRAINE BASED ON METHODS OF MULTICRITERIA OPTIMIZATION

З використанням методів багатокритеріальної оптимізації запропоновані підходи до розв’язання задачі оцінювання рівня розвитку та виявлення депресивних територій України за набором статистичних показників, визначених в Законі України ”Про стимулювання розвитку регіонів”.С использованием методов многокритериальной оптимизации предложены подходы для решения задачи оценивания уровня и обнаружения депрессивных территорий по набору статистических показателей, определенных в Законе Украины ”О стимулировании развития регионов”.Using methods of multicriteria optimization the approaches to solve the problem of development level evaluation and detection of depressive territories by statistical indices given in Ukrainian Law “On stimulation of development of regions” are proposed