Abstract
We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting with the projection operators in problems of band-time limiting and it gives a way to adapt a construction first used by Perline in a purely finite setup to quite a few other situations. We also extend his algebraic construction to cover some purely finite cases where his proposal fails to give a useful commuting operator.
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