Ladder operators for solvable potentials connected with exceptional orthogonal polynomials

30th International Colloquium on Group Theoretical Methods in Physics

Quesne, C.^1

Physique Nucleaire Theorique et Physique Mathematique, Universite Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, Brussels

B-1050, Belgium^1

关键词: Bound state;    Heisenberg algebras;    Higher-order;    Integrals of motion;    Orthogonal polynomial;    Quantum potentials;    Supersymmetric quantum mechanics;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012064/pdf
DOI  :  10.1088/1742-6596/597/1/012064

来源: IOP

【 摘 要 】

Exceptional orthogonal polynomials constitute the main part of the bound-state wavefunctions of some solvable quantum potentials, which are rational extensions of well-known shape-invariant ones. The former potentials are most easily built from the latter by using higher-order supersymmetric quantum mechanics (SUSYQM) or Darboux method. They may in general belong to three different types (or a mixture of them): types I and II, which are strictly isospectral, and type III, for which k extra bound states are created below the starting potential spectrum. A well-known SUSYQM method enables one to construct ladder operators for the extended potentials by combining the supercharges with the ladder operators of the starting potential. The resulting ladder operators close a polynomial Heisenberg algebra (PHA) with the corresponding Hamiltonian. In the special case of type III extended potentials, for this PHA the k extra bound states form k singlets isolated from the higher excited states. Some alternative constructions of ladder operators are reviewed. Among them, there is one that combines the state-adding and state-deleting approaches to type III extended potentials (or so-called Darboux-Crum and Krein-Adler transformations) and mixes the k extra bound states with the higher excited states. This novel approach can be used for building integrals of motion for two-dimensional superintegrable systems constructed from rationally-extended potentials.