On supersymmetries in nonrelativistic quantum mechanics (2005)
Beckers, J., Debergh, N., Nikitin, A. G.
One-dimensional nonrelativistic systems are studied when time-independent potential interactions are involved. Their supersymmetries are determined and their closed subsets generating kinematical...
Quasi exactly solvable quantum lattice solitons (2004)
Brihaye, Y., Debergh, N., Nininahazwe, A.
We extend the exactly solvable Hamiltonian describing $f$ quantum oscillators considered recently by J. Dorignac et al. by means of a new interaction which we choose as quasi exactly solvable. The...
Polynomial deformations of sl(2,R) in a three-dimensional invariant subspace of monomials (2003)
Debergh, N., Ndimubandi, J., Bossche, B. Van Den
New finite-dimensional representations of specific polynomial deformations of sl(2,R) are constructed. The corresponding generators can be, in particular, realized through linear differential...
Debergh, N., Ndimubandi, J., Bossche, B. Van Den
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...
Darboux transformations for quasi-exactly solvable Hamiltonians (2002)
Debergh, N., Samsonov, Boris F., Bossche, B. Van Den
We construct new quasi-exactly solvable one-dimensional potentials through Darboux transformations. Three directions are investigated: Reducible and two types of irreducible second-order...
A General Approach of Quasi-Exactly Solvable Schroedinger Equations (2002)
Debergh, N., Ndimubandi, J., Bossche, B. Van Den
We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schroedinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter...
Debergh, N., Pecheritsin, A. A., Samsonov, B. F., Bossche, B. Van Den
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...
On the exact solutions of the Lipkin-Meshkov-Glick model (2001)
We present the many-particle Hamiltonian model of Lipkin, Meshkov and Glick in the context of deformed polynomial algebras and show that its exact solutions can be easily and naturally obtained...
Realizations of the Lie superalgebra q(2) and applications (2001)
Debergh, N., Van Der Jeugt, J.
The Lie superalgebra q(2) and its class of irreducible representations V_p of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of V_p is given...
On a Lie algebraic approach of quasi-exactly solvable potentials with two known eigenstates (2001)
Brihaye, Y., Debergh, N., Ndimubandi, J.
We compare two recent approaches of quasi-exactly solvable Schr\" odinger equations, the first one being related to finite-dimensional representations of $sl(2,R)$ while the second one is based on...
Non-Hermitian oscillator-like Hamiltonians and $\lambda$-coherent states revisited (2001)
Beckers, J., Cariñena, J. F., Debergh, N., Marmo, G.
Previous $\lambda$-deformed {\it non-Hermitian} Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with...
On oscillatorlike Hamiltonians and squeezing (2000)
Beckers, J., Debergh, N., Szafraniec, F. H.
Generalizing a recent proposal leading to one-parameter families of Hamiltonians and to new sets of squeezed states, we construct larger classes of physically admissible Hamiltonians permitting new...
On the relation between polynomial deformations of sl(2,R) and quasi-exactly solvability (2000)
A general method based on the polynomial deformations of the Lie algebra sl(2,R) is proposed in order to exhibit the quasi-exactly solvability of specific Hamiltonians implied by quantum physical...
On oscillatorlike developments and further improvements in squeezing (1999)
Beckers, J., Debergh, N., Szafraniec, F. H.
A recent proposal of new sets of squeezed states is seen as a particular case of a general context admitting realistic physical Hamiltonians. Such improvements reveal themselves helpful in the study...
On realizations of nonlinear Lie algebras by differential operators (1998)
Beckers, J., Brihaye, Y., Debergh, N.
We study realizations of polynomial deformations of the sl(2,R)- Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite- and...
Parasupersymmetric Quantum Mechanics with Generalized Deformed Parafermions (1996)
Beckers, J., Debergh, N., Quesne, C.
A superposition of bosons and generalized deformed parafermions corresponding to an arbitrary paraquantization order $p$ is considered to provide deformations of parasupersymmetric quantum mechanics....
On Nonlinear Angular Momentum Theories, Their Representations and Associated Hopf Structures (1996)
Abdesselam, B., Beckers, J., Chakrabarti, A., Debergh, N.
Nonlinear $sl(2)$ algebras subtending generalized angular momentum theories are studied in terms of undeformed generators and bases. We construct their unitary irreducible representations in such a...
On a Deformation of $sl(2)$ with Paragrassmannian Variables (1995)
Abdesselam, B., Beckers, J., Chakrabarti, A., Debergh, N.
We propose a new structure ${\cal U}^{r}_{\displaystyle{q}}(sl(2)) $. This is realized by multiplying $\delta$ ($q=e^{\delta}$, $\delta\in \CC$) by $\theta$, where $\theta$ is a real nilpotent...