The Heun operator as a Hamiltonian

Abstract

It is shown that the celebrated Heun operator + is the Hamiltonian of the -quantum Euler–Arnold top of spin ν in a constant magnetic field. For it is canonically equivalent to Calogero–Moser–Sutherland quantum models; if , ten known one-dimensional quasi-exactly-solvable problems are reproduced, and if, in addition, , then four well-known one-dimensional quantal exactly-solvable problems are reproduced. If spin ν of the top takes a (half)-integer value the Hamiltonian possesses a finite-dimensional invariant subspace and a number of polynomial eigenfunctions occur. Discrete systems on uniform and exponential lattices are introduced which are canonically equivalent to the one described by the Heun operator.