The Heun operator as a Hamiltonian

It is shown that the celebrated Heun operator H e = − ( a 0 x 3 + a 1 x 2 + a 2 x ) d 2 d x 2 + ( b 0 x 2 + b 1 x + b 2 ) d d x + c 0 x is the Hamiltonian of the sl ( 2 , R ) -quantum Euler–Arnold top of spin ν in a constant magnetic field. For a 0 ≠ 0 it is canonically equivalent to BC 1 ( A 1 ) − Calogero–Moser–Sutherland quantum models; if a 0 = 0 , ten known one-dimensional quasi-exactly-solvable problems are reproduced, and if, in addition, b 0 = c 0 = 0 , 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.