Scattering in the PT-symmetric Coulomb potential

Scattering on the ${\cal PT}$-symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex $x$ plane from below. This trajectory reflects ${\cal PT}$ symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2L parameter, which would correspond to the angular momentum in the usual Hermitian setting. Transmission and reflection coefficients are written in closed analytic form and it is shown that similarly to other ${\cal PT}$-symmetric scattering systems the latter exhibit handedness effect. Bound-state energies are recovered from the poles of the transmission coefficients.Comment: Journal of Physics A: Mathematical and Theoretical 42 (2009) to appea

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the $\delta=0$ choice the quartic potential term is cancelled, leading to the {\it reduced} sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the quasi-exactly solvable formalism. Possible generalizations of the method are also presented

Jets and Underlying Events at LHC Energies

Jet-matter interaction remains a central question and a theoretical challenge in heavy-ion physics and might become important in high-multiplicity events in proton-proton collisions at LHC energies. Full jet measurement at LHC offer the proper tool to investigate energy loss process and fragmentation of hard parton in the medium. Since jet reconstruction will be constrained to small cone sizes, then study of the connection between jets and surrounding environment provides a further possibility to extend our exploration. We study jets at s = (14 TeV)^2 and pp collisions at s = (7 TeV)^2. We analyze the flavor components in jet-like environments. We introduce a definition for surrounding cones/belts and investigate flavor dependence and correlation of different hadron species produced in jets. Here, we focus on proton-triggered correlations. Our analysis can be extended for heavy ion collisions.Comment: 4 pages, 2 figures, Proceedings of Hot Quarks 2010, 21-26 June 2010 Las Londe Les Maures; to appear in Journal of Physics: Conference Serie

Underlying events in p+p collisions at LHC energies

General properties of hadron production are investigated in proton-proton collisions at LHC energies. We are interested in the characteristics of hadron production outside the identified jet cones. We improve earlier definitions and introduce surrounding rings/belts around the cone of identified jets. In this way even multiple jet events can be studied in details. We define the underlying event as collected hadrons from outside jet cones and outside surrounding belts, and investigate the features of these hadrons. We use a PYTHIA generated data sample of proton-proton collisions at s = (7 TeV)^2. This data sample is analysed by our new method and the widely applied CDF method. Angular correlations and momentum distributions have been studied and the obtained results are compared and discussed.Comment: 5 pages, 5 figures, to appear in the EPJ Web of Conferences, Proceedings of the International Workshop on Hot and Cold Baryonic Matter 2010 (Budapest, Hungary, 15-20 August 2010

The Nuclear Modification Factor at Large Rapidities

RHIC data on high-$p_T$ hadron production display strong suppression in a wide rapidity region, indicating strong induced energy loss for both transversally and longitudinally traveling partons. We investigate the interplay of energy loss and rapidity dependence in a perturbative QCD improved parton model, and estimate the opacity of the produced hot matter in $AuAu$ collisions at energies $\sqrt{s}=200$ $A$GeV and 63 $A$GeV at different rapidity values. Direction-dependent suppression offers the possibility to study the geometry of the hot matter.Comment: 4 pages, 2 figures, Contribution to the Poster Proceedings of the Quark Matter 2005 Conference. To be published in Nuclear Physics

PT symmetry breaking and explicit expressions for the pseudo-norm in the Scarf II potential

Closed expressions are derived for the pseudo-norm, norm and orthogonality relations for arbitrary bound states of the PT symmetric and the Hermitian Scarf II potential for the first time. The pseudo-norm is found to have indefinite sign in general. Some aspects of the spontaneous breakdown of PT symmetry are analysed.Comment: 16 pages; to appear in Phys. lett.

Representation reduction and solution space contraction in quasi-exactly solvable systems

In quasi-exactly solvable problems partial analytic solution (energy spectrum and associated wavefunctions) are obtained if some potential parameters are assigned specific values. We introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters. To obtain a larger solution space one varies the energy over a discrete set (the spectrum). A unified treatment that includes the standard as well as the new class of quasi-exactly solvable problems is presented and few examples (some of which are new) are given. The solution space is spanned by discrete square integrable basis functions in which the matrix representation of the Hamiltonian is tridiagonal. Imposing quasi-exact solvability constraints result in a complete reduction of the representation into the direct sum of a finite and infinite component. The finite is real and exactly solvable, whereas the infinite is complex and associated with zero norm states. Consequently, the whole physical space contracts to a finite dimensional subspace with normalizable states.Comment: 25 pages, 4 figures (2 in color

Underlying Event Studies for LHC Energies

Underlying event was originally defined by the CDF collaboration decades ago. Here we improve the original definition to extend our analysis for events with multiple-jets. We introduce a definition for surrounding rings/belts and based on this definition the jet- and surrounding-belt-excluded areas will provide a good underlying event definition. We inverstigate our definition via the multiplicity in the defined geometry. In parallel, mean transverse momenta of these areas also studied in proton-proton collisions at $\sqrt{s}=7$ TeV LHC energy.Comment: 6 pages and 4 figure