Dvofotonski raspad stanja sličnih 1s2s 1S0 u teškim atomskim sustavima

In He-like systems the direct decay of the 1s2s 1S0 state to the 1s2 1S0 ground state is "forbidden". The transition 1s2s 1S0® 1s2 1S0 by two photons sensitively probes the structure of the complete atomic system. In particular, the shape of the two-photon spectrum is sensitive to it and also reveals for heavy atomic numbers details of relativistic effects in strong central fields. A brief survey on this field of research is given with special emphasis on high nuclear charge Z.U sustavima sličnim He je izravan raspad stanja 1s2s 1S0 u osnovno stanje 1s2 1S0 “zabranjen”. Prijelaz 1s2s 1S0 → 1s2 1S0 emisijom dvaju fotona je osjetljiva proba strukture cijelog atomskog sustava. Oblik dvofotonskog spektra je posebno osjetljiv i otkriva detalje relativističkih učinaka u jakim središnjim poljima teških atoma. Daje se kratak pregled ovog polja s posebnim naglaskom na sustave visokog Z

Dvofotonski raspad stanja sličnih 1s2s 1S0 u teškim atomskim sustavima

In He-like systems the direct decay of the 1s2s 1S0 state to the 1s2 1S0 ground state is "forbidden". The transition 1s2s 1S0® 1s2 1S0 by two photons sensitively probes the structure of the complete atomic system. In particular, the shape of the two-photon spectrum is sensitive to it and also reveals for heavy atomic numbers details of relativistic effects in strong central fields. A brief survey on this field of research is given with special emphasis on high nuclear charge Z.U sustavima sličnim He je izravan raspad stanja 1s2s 1S0 u osnovno stanje 1s2 1S0 “zabranjen”. Prijelaz 1s2s 1S0 → 1s2 1S0 emisijom dvaju fotona je osjetljiva proba strukture cijelog atomskog sustava. Oblik dvofotonskog spektra je posebno osjetljiv i otkriva detalje relativističkih učinaka u jakim središnjim poljima teških atoma. Daje se kratak pregled ovog polja s posebnim naglaskom na sustave visokog Z

Analysis of unbounded operators and random motion

We study infinite weighted graphs with view to \textquotedblleft limits at infinity,\textquotedblright or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means \textquotedblleft very\textquotedblright large) networks of resistors, or in statistical mechanics models for classical or quantum systems. But more generally our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If $X$ is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on $X$ evaluated on pairs of points in $X$. And the Hilbert norm-squared in $\mathcal{H}(X)$ will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian, or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space $\mathcal{H}(X)$ which measure quantitative notions of limits at infinity in $X$, one generalizes finite-energy harmonic functions in $\mathcal{H}(X)$, and the other a deficiency index of a natural operator in $\mathcal{H}(X)$ associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of \textquotedblleft boundaries\textquotedblright in more standard random walk models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure

Improved Lieb-Oxford exchange-correlation inequality with gradient correction

We prove a Lieb-Oxford-type inequality on the indirect part of the Coulomb energy of a general many-particle quantum state, with a lower constant than the original statement but involving an additional gradient correction. The result is similar to a recent inequality of Benguria, Bley and Loss, except that the correction term is purely local, which is more usual in density functional theory. In an appendix, we discuss the connection between the indirect energy and the classical Jellium energy for constant densities. We show that they differ by an explicit shift due to the long range of the Coulomb potential.Comment: Final version to appear in Physical Review A. Compared to the very first version, this one contains an appendix discussing the link with the Jellium proble

Effects of Zeeman spin splitting on the modular symmetry in the quantum Hall effect

Magnetic-field-induced phase transitions in the integer quantum Hall effect are studied under the formation of paired Landau bands arising from Zeeman spin splitting. By investigating features of modular symmetry, we showed that modifications to the particle-hole transformation should be considered under the coupling between the paired Landau bands. Our study indicates that such a transformation should be modified either when the Zeeman gap is much smaller than the cyclotron gap, or when these two gaps are comparable.Comment: 8 pages, 4 figure

Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment

In this paper, we show that the concept of sigma-convergence associated to stochastic processes can tackle the homogenization of stochastic partial differential equations. In this regard, the homogenization problem for a stochastic nonlinear partial differential equation is studied. Using some deep compactness results such as the Prokhorov and Skorokhod theorems, we prove that the sequence of solutions of this problem converges in probability towards the solution of an equation of the same type. To proceed with, we use a suitable version of sigma-convergence method, the sigma-convergence for stochastic processes, which takes into account both the deterministic and random behaviours of the solutions of the problem. We apply the homogenization result to some concrete physical situations such as the periodicity, the almost periodicity, the weak almost periodicity, and others.Comment: To appear in: Stochastic Analysis and Application

Quantum oscillator on complex projective space (Lobachewski space) in constant magnetic field and the issue of generic boundary conditions

We perform a 1-parameter family of self-adjoint extensions characterized by the parameter $\omega_0$. This allows us to get generic boundary conditions for the quantum oscillator on $N$ dimensional complex projective space($\mathbb{C}P^N$) and on its non-compact version i.e., Lobachewski space($\mathcal L_N$) in presence of constant magnetic field. As a result, we get a family of energy spectrums for the oscillator. In our formulation the already known result of this oscillator is also belong to the family. We have also obtained energy spectrum which preserve all the symmetry (full hidden symmetry and rotational symmetry) of the oscillator. The method of self-adjoint extensions have been discussed for conic oscillator in presence of constant magnetic field also.Comment: Accepted in Journal of Physics

Higher order Schrodinger and Hartree-Fock equations

The domain of validity of the higher-order Schrodinger equations is analyzed for harmonic-oscillator and Coulomb potentials as typical examples. Then the Cauchy theory for higher-order Hartree-Fock equations with bounded and Coulomb potentials is developed. Finally, the existence of associated ground states for the odd-order equations is proved. This renders these quantum equations relevant for physics.Comment: 19 pages, to appear in J. Math. Phy