Hypergeometric type operators are shape invariant, and a factorization into a product of first order differential operators can be explicitely described in the general case. Some additional shape invariant operators depending on several parameters are defined in a natural way by starting from this general factorization. The mathematical properties of the eigenfunctions and eigenvalues of the operators thus obtained depend on the values of the parameters involved. We study the parameter dependence of orthogonality, square integrability and monotony of the eigenvalue sequence. The results obtained allow us to define certain systems of Gazeau-Klauder type coherent states and to describe some of their properties. Our systematic study recovers a number of well-known results in a natural, unified way and also leads to new findings.
We consider the dispersion of energy levels for both standard and inverted quantum harmonic oscillators in the presence of a uniform electromagnetic field. For this analysis we use a solution of the corresponding eigenproblem in terms of the Kummer functions. We find a complete description of the energy levels for a particle of mass m and electric charge q subject to the action of a harmonic oscillator and simultaneous uniform magnetic and electric fields. We also analyze the effect of spin on energy levels for an electron.
We study the approximate analytical solutions of the Dirac equation for the generalized Woods-Saxon potential with the pseudo-centrifugal term. We apply the Nikiforov-Uvarov method (which solves a second-order linear differential equation by reducing it to a generalized hypergeometric form) to spin- and pseudospin-symmetry to obtain, in closed form, the approximately analytical bound state energy eigenvalues and the corresponding upper- and lower-spinor components of two Dirac particles. The special cases κ = ±1 (s = $$ \tilde l $$ = 0, s-wave) and the non-relativistic limit can be reached easily and directly for the generalized and standard Woods-Saxon potentials. We compare the non-relativistic results with those obtained by others.
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