N. Debergh

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.

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.

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.

Realizations of the Lie superalgebra q(2) and applications.

The Lie superalgebra q(2) and its class of irreducible representations Vp of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of Vp 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 Vp 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.

POLYNOMIAL DEFORMATIONS OF sl(2, ℝ) IN A THREE-DIMENSIONAL INVARIANT SUBSPACE OF MONOMIALS

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 operators preserving a finite-dimensional subspace of monomials. We concentrate on three-dimensional spaces.

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 with two distinct irreps of the deformed algebra. Their invariant subspaces correspond to even and odd numbers of particle-hole excitations.

QUASI-EXACTLY-SOLVABLE QUANTUM LATTICE SOLITONS

We extend the exactly solvable Hamiltonian describing

On realizations of `nonlinear' Lie algebras by differential operators

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