B. Van den Bossche

Darboux transformations of the one-dimensional stationary Dirac equation

A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian, as well as the corresponding eigenfunctions, are determined through the knowledge of only two eigenfunctions of the first Dirac Hamiltonian. Moreover this operator, together with its adjoint and the two Hamiltonians, generate a quadratic deformation of the superalgebra subtending the usual supersymmetric quantum mechanics. Our developments are illustrated in the free-particle case and the generalized Coulomb interaction. In the latter case, a relativistic counterpart of shape invariance is observed.

Anomalous pion decay in effective QCD at finite temperature

Abstract The width of the process π 0 → γγ is calculated within effective QCD at finite temperature. We use two different effective models which describe chiral symmetry breaking and its restoration at finite temperature in SU(2) ƒ : (i) an effective lagrangian with a nonlocal separable interaction kernel, (ii) A- and B-scaledNJL lagrangians which implement the scale anomaly of QCD. We calculate the temperature dependences of the quark and pion masses as well as of the pion-park-antiquark coupling strength below and above the double quark mass threshold. At zero temperature we obtain a fairly good agreement with the experimental value for the decay width. The temperature dependence of the decay width is related to that of the pion mass. Γ π 0 → γγ is enhanced in the vicinity of the Mott temperature ( M π = 2 m q ) for the nonlocal as well as for the A-scaled model and suppressed for the B-scaled model. We compare our results with the SU(2) Nambu-Jona-Lasinio model which is contained as a limiting case in our approach.

Phase transition and thermodynamics of a hot and dense system in a scaled NJL model

Abstract The chiral phase transition of a hot and dense system of quarks is studied within a modified SU(3) NJL Lagrangian that implements the QCD scale anomaly. The u - and s -quark condensates can or can not feel the same chiral restoration depending on the considered region of the three-dimensional space ( T c , μ uc , μ sc ). The temperature behaviour of the pressure and the energy and entropy densities of the u - and s -quark system is investigated. At high temperature, the non-vanishing bare s -quark mass only modifies slightly the usual behaviour associated with an ideal quark gas.

SU(3) scaled effective lagrangians for a hot and strange system

Abstract We construct two effective lagrangians which reproduce the explicit flavor SU(3) symmetry breaking and the trace and axial anomalies of QCD. All analytical expressions are derived for a hot strange matter but the results are only shown for zero baryonic density. We study the coupling of the quark condensates (u and s) to the gluon condensate. We also consider the temperature behavior of the pion and of the kaon masses. We show that the results crucially depend on the choice of the two free parameters of the model: the chiral transition can be of first or of second order; the coupling between the gluon and the quark condensates can be large or not; the meson masses can increase or decrease with temperature. The introduction of the QCD trace anomaly in the SU(3) NJL model kills the predictive character of the NJL.

DARBOUX TRANSFORMATIONS FOR QUASI-EXACTLY SOLVABLE HAMILTONIANS

We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order transformations. The irreducible transformations of the first type give singular intermediate potentials and the ones of the second type give complex-valued intermediate potentials while final potentials are meaningful in all cases. These developments are illustrated on the so-called radial sextic oscillator.

The coupling of quark and gluon condensates in hot and dense systems

Abstract Two lagrangians, which implement the QCD scale anomaly, are used to study the coupling of quark and gluon condensates in dense and hot matter. The response of the system strongly depends on the assumed value of the vacuum gluon condensate. For a low value of this vacuum gluon condensate, quark and gluon condensates feel simultaneously the chiral phase transition while for a larger value, the gluon condensate keeps its vacuum value and the quark condensate is the only order parameter of the transition. We also calculate in both models the equation of state of nuclear matter.

Gluonic content of the scalar meson nonet in a scaled effective Lagrangian

Abstract A scaled SU(3) Nambu-Jona-Lasinio Lagrangian is used to compute the mass of the nine scalar mesons in the vacuum and the mass of the glueball. This chiral model provides a gluonic content to the scalar mesons as well as a quark content to the glueball. It is shown that a suitable choice of the vacuum gluon condensate allows reproduction of the experimental masses of the scalar mesons except for K 0 ∗ (1430) . This choice corresponds to a weak coupling between the gluon and quark condensates, giving tf 0 (1500) or tf J (1710) which is nearly a pure glueball.

Differential realizations of polynomial algebras in finite-dimensional spaces of monomials

Abstract We give a complete classification of at most second-order differential ladder operators preserving finite-dimensional spaces of monomials and closing under the Lie bracket to give a cubic polynomial of the diagonal generators.

A General Approach of Quasi-Exactly Solvable Schrödinger Equations

Abstract We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schrodinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lame and the screened Coulomb potentials.

A GENERAL APPROACH OF QUASI-EXACTLY SOLVABLE SCHRÖDINGER EQUATIONS WITH THREE KNOWN EIGENSTATES

We propose a general method for constructing quasi-exactly solvable potentials with three analytic eigenstates. These potentials can be real or complex functions but the spectrum is real. A comparison with other methods is also performed.