On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials

0564995 - ÚJF 2023 RIV NL eng J - Journal Article
Marquette, I. - Zelaya, Kevin
On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials.
Physica. D. Roč. 442, DEC (2022), č. článku 133529. ISSN 0167-2789. E-ISSN 1872-8022
R&D Projects: GA MŠMT EF18_053/0017163
Institutional support: RVO:61389005
Keywords : Fourth Painlevé equation * Shape-invariant Hamiltonian * Generalized Hermite polynomials * Rational potentials * Orthogonal polynomials
OECD category: Applied mathematics
Impact factor: 4, year: 2022
Method of publishing: Open access
https://doi.org/10.1016/j.physd.2022.133529

This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the -1/x and -2x hierarchies of rational solutions of the fourth Painleve equation. Such a relation unequivocally establishes the discrete spectrum structure, composed as the union of a finite-and infinite-dimensional sequence of equidistant eigenvalues separated by a gap. The two indices of the generalized Hermite polynomials define the dimension of the finite sequence and the gap. Likewise, the complete set of eigensolutions decomposes into two disjoint subsets, whose elements are written as the product of a polynomial times a weight function supported on the real line. These polynomials fulfill a second-order differential equation and are alternatively determined from a three-term recurrence relation, the initial conditions of which are also fixed in terms of generalized Hermite polynomials.
Permanent Link: https://hdl.handle.net/11104/0336562