## Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators    [PDF]

I. Marquette, C. Quesne
The type III Hermite \$X_m\$ exceptional orthogonal polynomial family is generalized to a double-indexed one \$X_{m_1,m_2}\$ (with \$m_1\$ even and \$m_2\$ odd such that \$m_2 > m_1\$) 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 Painlev\'e 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 \$m_2-m_1+1\$, which may alternatively be interpreted in terms of a special type of \$(m_2-m_1+2)\$th-order shape invariance property.
View original: http://arxiv.org/abs/1212.3474
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