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arXiv · 1305.2509

On the dimensions of the oscillator algebras induced by orthogonal polynomials

G. Honnouvo·K. Thirulogasanthar
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Abstract

There is a generalized oscillator algebra associated with every class of orthogonal polynomials $\{Ψ_n(x)\}_{n=0}^{\infty}$, on the real line, satisfying a three term recurrence relation $xΨ_n(x)=b_nΨ_{n+1}(x)+b_{n-1}Ψ_{n-1}(x), Ψ_0(x)=1, b_{-1}=0$. This note presents necessary and safficient conditions on $b_n$ for such algebras to be of finite dimension. As examples, we discuss the dimensions of oscillator algebras associated with Hermite, Legendre and Gegenbauer polynomials. In addition we shall also discuss the dimensions of some generalized deformed oscillator algebras. Some remarks on the dimensions of oscillator algebras associated with multi-boson systems are also presented.

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G. Honnouvo, K. Thirulogasanthar. 2014-09-18. On the dimensions of the oscillator algebras induced by orthogonal polynomials. https://doi.org/10.1063/1.4896324

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