The family of metric operators, constructed by Musumbu et al (2007 J. Phys. A: Math. Theor. 40 F75), for a harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part, is re-examined in the light of an su(1,1) approach. An alternative derivation, only relying on properties of su(1,1) generators, is proposed. Being independent of the realization considered for the latter, it opens the way towards the construction of generalized non-Hermitian (not necessarily -symmetric) oscillator Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.

We construct two new exactly solvable potentials giving rise to bound-state solutions to the Schrödinger equation, which can be written in terms of the recently introduced Laguerre- or Jacobi-type X1 exceptional orthogonal polynomials. These potentials, extending either the radial oscillator or the Scarf I potential by the addition of some rational terms, turn out to be translationally shape invariant as their standard counterparts and isospectral to them.

In the context of some deformed canonical commutation relations leading to isotropic nonzero minimal uncertainties in the position coordinates, a Dirac equation is exactly solved for the first time, namely that corresponding to the Dirac oscillator. Supersymmetric quantum mechanical and shape-invariance methods are used to derive both the energy spectrum and wavefunctions in the momentum representation. As for the conventional Dirac oscillator, there are neither negative-energy states for E = −1, nor symmetry between the and cases, both features being connected with supersymmetry or, equivalently, the &ohgr; → −&ohgr; transformation. In contrast with the conventional case, however, the energy spectrum does not present any degeneracy pattern apart from that associated with the rotational symmetry. More unexpectedly, deformation leads to a difference in behaviour between the states corresponding to small, intermediate and very large j values in the sense that only for the first ones supersymmetry remains unbroken, while for the second ones no bound state exists.

In a recent paper (Del Sol Mesa A and Quesne C 2000 J. Phys. A: Math. Gen. 33 4059), we started a systematic study of the connections among different factorization types, suggested by Infeld and Hull, and their consequences for the construction of algebras. We devised a general procedure for constructing satellite algebras for all the Hamiltonians admitting a type E factorization by using the relationship between type A and E factorizations. Here we complete our analysis by showing that for Hamiltonians admitting a type F factorization, a similar method, starting from either type B or type C factorization, leads to other types of algebras. We therefore conclude that the existence of satellite algebras is a characteristic property of type E factorizable Hamiltonians. Our results are illustrated with the detailed discussion of the Coulomb problem.

The interest of quadratic algebras for position-dependent mass Schrödinger equations is highlighted by constructing spectrum generating algebras for a class of d-dimensional radial harmonic oscillators with d ≥ 2 and a specific mass choice depending on some positive parameter α. Via some minor changes, the one-dimensional oscillator on the line with the same kind of mass is included in this class. The existence of a single unitary irreducible representation belonging to the positive-discrete series type for d ≥ 2 and of two of them for d = 1 is proved. The transition to the constant-mass limit α → 0 is studied and deformed su(1,1) generators are constructed. These operators are finally used to generate all the bound-state wavefunctions by an algebraic procedure.

Recently, we introduced a new class of symmetry algebras, calledsatellite algebras, which connect with one another wavefunctionsbelonging to different potentials of a given family, andcorresponding to different energy eigenvalues. Here the role ofthe factorization method in the construction of such algebras isinvestigated. A general procedure for determining an so(2,2) orso(2,1) satellite algebra for all the Hamiltonians that admit atype E factorization is proposed. Such a procedure is based onthe known relationship between type A and E factorizations,combined with an algebraization similar to that used in theconstruction of potential algebras. It is illustrated withexamples of the generalized Morse potential, the Rosen-Morsepotential and the Kepler problem in a space of constant negativecurvature, and, in each case, the conserved quantity isidentified. It should be stressed that the method proposed isfairly general since the other factorization types may beconsidered as limiting cases of type A or E factorizations.

We comment on a recent paper by Chen et al (1998 J. Phys. A: Math. Gen. 31 6473), wherein a nonlinear deformation of su(1,1) involving two deforming functions is realized in the exactly solvable quantum-mechanical problem with Pöschl-Teller potential, and is used to derive the well known su(1,1) spectrum-generating algebra of this problem. We show that one of the defining relations of the nonlinear algebra, presented by the authors, is only valid in the limiting case of an infinite square well, and we determine the correct relation in the general case. We also use it to establish the correct link with su(1,1), as well as to provide an algebraic derivation of the eigenfunction normalization constant.

In the context of a two-parameter (伪, 尾) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters 伪, 尾 vanishes, in which case shape invariance under parameter translation occurs. In the special case where 伪 = 尾 鈮?0, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with q > 1 and its eigenvectors are therefore n-q-boson states. In the general case where 0 鈮?伪 鈮?尾 鈮?0, the eigenvectors are constructed as linear combinations of n-q-boson states by resorting to a Bargmann representation of the latter and to q-differential calculus. They are finally expressed in terms of a q-exponential and little q-Jacobi polynomials.

A harmonic oscillator Hamiltonian augmented by a non-Hermitian -symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Some quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.

We continue our previous application of supersymmetric quantum mechanical methods to eigenvalue problems in the context of some deformed canonical commutation relations leading to nonzero minimal uncertainties in position and/or momentum. Here we determine for the first time the spectrum and the eigenvectors of a one-dimensional harmonic oscillator in the presence of a uniform electric field in terms of the deforming parameters &agr;, &bgr;. We establish that whenever there is a nonzero minimal uncertainty in momentum, i.e., for &agr; ≠ 0, the correction to the harmonic oscillator eigenvalues due to the electric field is level dependent. In the opposite case, i.e., for &agr; = 0, we recover the conventional quantum mechanical picture of an overall energy-spectrum shift even when there is a nonzero minimal uncertainty in position, i.e., for &bgr; ≠ 0. Then we consider the problem of a D-dimensional harmonic oscillator in the case of isotropic nonzero minimal uncertainties in the position coordinates, depending on two parameters &bgr;, &bgr;′. We extend our methods to deal with the corresponding radial equation in the momentum representation and rederive in a simple way both the spectrum and the momentum radial wavefunctions previously found by solving the differential equation. This opens the way to solving new D-dimensional problems.

The procedure for calculating the centroid energies as well as the total and partial widths of the generalized seniority distributions of mixed configurations of identical nucleons is described. An extensive use of the quasispin formalism is made. In some particular cases, analytical formulae, some of which are new, are derived for the centroids and widths. Numerical calculations are performed in the nickel and tin isotopes. They show that although the admixtures with |Delta v|=4 should in many cases be relatively small, those with |Delta v|=2 should be very large, owing to the important contribution of the single-particle potential