Algebraic Heun operator and band-time limiting

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.

27 pages

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• Entanglement of Free Fermions on Hadamard Graphs
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• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
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• Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes
• New realizations of algebras of the Askey-Wilson type in terms of Lie and quantum algebras
• Bethe ansatz diagonalization of the Heun-Racah operator
• The rational Sklyanin algebra and the Wilson and para-Racah polynomials
• Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Polynomial algebras from Lie algebra reduction chains $\mathfrak{g}\supset {\mathfrak{g}}^{\prime }$
• A new commutativity property of exceptional orthogonal polynomials
• The missing label of ${\mathfrak{su}}_3$ and its symmetry
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• Generalized quadratic commutator algebras of PBW-type
• Entanglement of free-fermion systems, signal processing and algebraic combinatorics
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• Classical Heun observables and elliptic solvability