2012/12/31 by I. Marquette, C. Quesne · 1 citation
Mathematics · Physics and Astronomy · #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1088/1751-8113/46/15/155201
published as J. Phys. A: Math. Theor. 46 (2013) 155201 · 22 pages, no figure, published version
arxiv created 2013/03/29 · arxiv updated 2015/06/12
The type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one Xm1,m2 (with m1 even and m2 odd such that m2 > m1) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order m2-m1+1, which may alternatively be interpreted in terms of a special type of (m2-m1+2)th-order shape invariance property.