Invariance of the~generalized oscillator under a~linear transformation of the~related system of orthogonal polynomials

V. V. Borzov ; E. V. Damaskinsky

Teoretičeskaâ i matematičeskaâ fizika, Tome 190 (2017) no. 2, pp. 267-276

Résumé

We consider the families of polynomials $\mathbb P=\{P_n(x)\}_{n=0}^\infty$ and $\mathbb Q=\{Q_n(x)\}_{n=0}^\infty$ orthogonal on the real line with respect to the respective probability measures $\mu$ and $\nu$. We assume that $\{Q_n(x)\}_{n=0}^\infty$ and $\{P_n(x)\}_{n=0}^\infty$ are connected by linear relations. In the case $k=2$, we describe all pairs $(\mathbb P,\mathbb Q)$ for which the algebras $\mathfrak A_P$ and $\mathfrak A_Q$ of generalized oscillators generated by $\{Q_n(x)\}_{n=0}^\infty$ and $\{P_n(x)\}_{n=0}^\infty$ coincide. We construct generalized oscillators corresponding to pairs $(\mathbb P,\mathbb Q)$ for arbitrary $k\ge1$.

Détail

Keywords: generalized oscillator
Mots-clés : orthogonal polynomial.