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N. Debergh

Publications and source records attributed to N. Debergh.

At least 19 recordsLinked to original sources
arXiv2022

On spacetime algebra and its relations with negative masses

We consider four subsets of the complexified spacetime algebra, namely the real even part, the real odd part, the imaginary even part and the imaginary odd part. This naturally leads to the four connected components of the Lorentz group, supplemented each time by an additional symmetry. We then examine how these four parts impact the Dirac equation and show that four types of matter arise with positive and negative masses as well as positive and negative charges.

physics.gen-ph↗
arXiv2005

On supersymmetries in nonrelativistic quantum mechanics

One-dimensional nonrelativistic systems are studied when time-independent potential interactions are involved. Their supersymmetries are determined and their closed subsets generating kinematical invariance Lie superalgebras are pointed out. The study of even supersymmetries is particularly enlightened through the already known symmetries of the corresponding Schrödinger equation. Three tables collect the even, odd, and total supersymmetries as well as the invariance (super)algebras.

math-ph↗
arXiv2005

Quasi exactly solvable quantum lattice solitons

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 properties of the spectrum of this new Hamiltonian are studied as function of the new coupling constant. This Hamiltonian as well as the original one are also related to adequate Lie structures.

quant-ph↗
arXiv2002

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.

quant-ph↗
arXiv2002

A General Approach of Quasi-Exactly Solvable Schroedinger Equations

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 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.

quant-ph↗
arXiv2001

Darboux transformations and hidden quadratic supersymmetry 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 on the free particle case and the generalized Coulomb interaction. In the latter case, a relativistic counterpart of shape-invariance is observed.

quant-ph↗
arXiv2001

Realizations of the Lie superalgebra q(2) and applications

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 explicitly, and from here two realizations of q(2) are determined. The q(2) generators are realized as differential operators in one variable x, and the basis vectors of V_p as 2-arrays of polynomials in x. Following such realizations, it is observed that the Hamiltonian of certain physical models can be written in terms of the q(2) generators. In particular, the models given here as an example are the sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each of these, it is shown how the q(2) realization of the Hamiltonian is helpful in determining the spectrum.

math-ph↗
arXiv2001

On the exact solutions of the Lipkin-Meshkov-Glick model

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 within this formalism. The Hamiltonian matrix of each $j$ multiplet can be split into two submatrices associated to two distinct irreps of the deformed algebra. Their invariant subspaces correspond to even and odd numbers of particle-hole excitations.

quant-ph↗
arXiv2001

Non-Hermitian oscillator-like Hamiltonians and $λ$-coherent states revisited

Previous $λ$-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 respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed $λ$-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.

hep-th↗
arXiv2000

On oscillatorlike Hamiltonians and squeezing

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 developments in squeezing. Coherence is also discussed.

quant-ph↗
arXiv2000

On the relation between polynomial deformations of sl(2,R) and quasi-exactly solvability

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 models. This method using the finite-dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on the second harmonic generation.

hep-th↗
arXiv1998

On realizations of nonlinear Lie algebras by differential operators

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 infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are studied in the context of these ``nonlinear'' Lie algebras and some examples dealing with quantum optics are pointed out.

hep-th↗
arXiv1996

On a Deformation of $sl(2)$ with Paragrassmannian Variables

We propose a new structure ${\cal U}^{r}_{\displaystyle{q}}(sl(2)) $. This is realized by multiplying $δ$ ($q=e^δ$, $δ\in \CC$) by $θ$, where $θ$ is a real nilpotent -paragrassmannian- variable of order $r$ ($θ^{r+1}=0$) that we call the order of deformation, the limit $r\rightarrow \infty$ giving back the standard ${\cal U}_{\displaystyle {q}}(sl(2))$. In particular we show that, for $r=1$, there exists a new ${\cal R}$-matrix associated with $sl(2)$. We also proof that the restriction of the values of the parameters of deformation give nonlinear algebras as particular cases.

q-alg↗
arXiv1996

On Nonlinear Angular Momentum Theories, Their Representations and Associated Hopf Structures

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 general context. The linear $sl(2)$-case as well as its $q$-deformation are easily recovered as specific examples. Two other physically interesting applications corresponding to the so-called Higgs and quadratic algebras are also considered. We show that these two nonlinear algebras can be equipped with a Hopf structure.

q-alg↗