arXiv subjects

N. Vansteenkiste

Publications and source records attributed to N. Vansteenkiste.

11 recordsLinked to original sources

arXiv2025

A unified approach for domination and packing problems in graphs

In this paper, we introduce new concepts of domination and packing functions in graphs, which generalize, respectively, the labelled dominating and packing functions defined by Lee and Chang in 2008, and Hinrichsen et al. in 2019. These generalized functions offer a unified and simpler framework for addressing many of the variations of domination and packing concepts in graphs explored in the literature. Interestingly, their associated optimization problems turn out to be equivalent, providing insight to explain the observed coincidences in computational complexity results for graph classes where both problems, the domination one and its corresponding packing variation, have been analyzed. This equivalence also allows us to solve some computational complexity open questions, for some graph classes. Furthermore, we prove that the generalized problems remain solvable in polynomial time for graphs with bounded clique-width and strongly chordal graphs.

math.CO↗

arXiv2002

Pseudosupersymmetric quantum mechanics: General case, orthosupersymmetries, reducibility, and bosonization

Pseudosupersymmetric quantum mechanics (PsSSQM), based upon the use of pseudofermions, was introduced in the context of a new Kemmer equation describing charged vector mesons interacting with an external constant magnetic field. Here we construct the complete explicit solution for its realization in terms of two superpotentials, both equal or unequal. We prove that any orthosupersymmetric quantum mechanical system has a pseudosupersymmetry and give conditions under which a pseudosupersymmetric one may be described by orthosupersymmetries of order two. We propose two new matrix realizations of PsSSQM in terms of the generators of a generalized deformed oscillator algebra (GDOA) and relate them to the cases of equal or unequal superpotentials, respectively. We demonstrate that these matrix realizations are fully reducible and that their irreducible components provide two distinct sets of bosonized operators realizing PsSSQM and corresponding to nonlinear spectra. We relate such results to some previous ones obtained for a GDOA connected with a $C_3$-extended oscillator algebra (where $C_3 = {\rm Z}_3$) in the case of linear spectra.

math-ph↗

arXiv2002

Reducibility and bosonization of parasupersymmetric and orthosupersymmetric quantum mechanics

Order-$p$ parasupersymmetric and orthosupersymmetric quantum mechanics are shown to be fully reducible when they are realized in terms of the generators of a generalized deformed oscillator algebra and a ${\rm Z}_{p+1}$-grading structure is imposed on the Fock space. The irreducible components provide $p+1$ sets of bosonized operators corresponding to both unbroken and broken cases. Such a bosonization is minimal.

math-ph↗

arXiv2000

$C_λ$-extended oscillator algebras and some of their deformations and applications to quantum mechanics

$C_λ$-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where $C_λ$ is the cyclic group of order $λ$, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras are obtained, and used to provide a complete classification of their unitary irreducible representations under the assumption that the number operator spectrum is nondegenerate. Deformed algebras admitting Casimir operators analogous to those of their undeformed counterparts are looked for, yielding three new algebraic structures. One of them includes the Brzeziński {\em et al.} deformation of the Calogero-Vasiliev algebra as a special case. In its bosonic Fock-space representation, the realization of $C_λ$-extended oscillator algebras as generalized deformed oscillator ones is shown to provide a bosonization of several variants of supersymmetric quantum mechanics: parasupersymmetric quantum mechanics of order $p = λ-1$ for any $λ$, as well as pseudosupersymmetric and orthosupersymmetric quantum mechanics of order two for $λ=3$.

math-ph↗

arXiv1999

C$_λ$-extended Oscillator Algebras: Theory and Applications to (Variants) of Supersymmetric Quantum Mechanics

C$_λ$-extended oscillator algebras, where C$_λ$ is the cyclic group of order $λ$, are introduced and realized as generalized deformed oscillator algebras. For $λ=2$, they reduce to the well-known Calogero-Vasiliev algebra. For higher $λ$ values, they are shown to provide in their bosonic Fock space representation some interesting applications to supersymmetric quantum mechanics and some variants thereof: an algebraic realization of supersymmetric quantum mechanics for cyclic shape invariant potentials of period $λ$, a bosonization of parasupersymmetric quantum mechanics of order $p = λ-1$, and, for $λ=3$, a bosonization of pseudosupersymmetric quantum mechanics and orthosupersymmetric quantum mechanics of order two.

math-ph↗

arXiv1999

Algebraic Realization of Supersymmetric Quantum Mechanics for Cyclic Shape Invariant Potentials

We study in detail the spectrum of the bosonic oscillator Hamiltonian associated with the $C_3$-extended oscillator algebra \algthree, where $C_3$ denotes a cyclic group of order three, and classify the various types of spectra in terms of the algebra parameters $α_0, α_1$. In such a classification, we identify those spectra having an infinite number of periodically spaced levels, similar to those of cyclic shape invariant potentials of period three. We prove that the hierarchy of supersymmetric Hamiltonians and supercharges, corresponding to the latter, can be realized in terms of some appropriately chosen \algthree algebras, and of Pauli spin matrices. Extension to period-$λ$ spectra in terms of $C_λ$-extended oscillator algebras is outlined.

math-ph↗

arXiv1998

C_λ-extended oscillator algebra and parasupersymmetric quantum mechanics

The C_λ-extended oscillator algebra is generated by {1,a,a^{\dagger},N,T}, where T is the generator of the cyclic group C_λ of order λ. It can be realized as a generalized deformed oscillator algebra (GDOA). Its unirreps can thus be exhibited using the representation theory of GDOAs and their carrier space show a Z_λ grading structure. Within its infinite-dimensional Fock space representation, this algebra provides a bosonization of parasupersymmetric quantum mechanics of order p= λ- 1.

math.QA↗

arXiv1998

$C_λ$-extended harmonic oscillator and (para)supersymmetric quantum mechanics

$C_λ$-extended oscillator algebras are realized as generalized deformed oscillator algebras. For $λ= 3$, the spectrum of the corresponding bosonic oscillator Hamiltonian is shown to strongly depend on the algebra parameters. A connection with cyclic shape invariant potentials is noted. A bosonization of PSSQM of order two is obtained.

quant-ph↗

arXiv1997

Representation Theory of Generalized Deformed Oscillator Algebras

The representation theory of the generalized deformed oscillator algebras (GDOA's) is developed. GDOA's are generated by the four operators ${1,a,a^†,N}$. Their commutators and Hermiticity properties are those of the boson oscillator algebra, except for $[a, a^†]_q = G(N)$, where $[a,b]_q = a b - q b a$ and $G(N)$ is a Hermitian, analytic function. The unitary irreductible representations are obtained by means of a Casimir operator $C$ and the semi-positive operator $a^† a$. They may belong to one out of four classes: bounded from below (BFB), bounded from above (BFA), finite-dimentional (FD), unbounded (UB). Some examples of these different types of unirreps are given.

q-alg↗

arXiv1996

Representation theory of deformed oscillator algebras

The representation theory of deformed oscillator algebras, defined in terms of an arbitrary function of the number operator~$N$, is developed in terms of the eigenvalues of a Casimir operator~$C$. It is shown that according to the nature of the $N$ spectrum, their unitary irreducible representations may fall into one out of four classes, some of which contain bosonic, fermionic or parafermionic Fock-space representations as special cases. The general theory is illustrated by classifying the unitary irreducible representations of the Arik-Coon, Chaturvedi-Srinivasan, and Tamm-Dancoff oscillator algebras, which may be derived from the boson one by the recursive minimal-deformation procedure of Katriel and Quesne. The effects on non-Fock-space representations of the minimal deformation and of the quommutator-commutator transformation, considered in such a procedure, are studied in detail.

q-alg↗

arXiv1995

Comment on ``Generalized $q$-oscillators and their Hopf structures''

In a recent paper (1994 {\sl J.\ Phys.\ A: Math.\ Gen.\ }{\bf 27} 5907), Oh and Singh determined a Hopf structure for a generalized $q$-oscillator algebra. We prove that under some general assumptions, the latter is, apart from some algebras isomorphic to su$_q$(2), su$_q$(1,1), or their undeformed counterparts, the only generalized deformed oscillator algebra that supports a Hopf structure. We show in addition that the latter can be equipped with a universal $\cR$-matrix, thereby making it into a quasitriangular Hopf algebra.

q-alg↗