The Heun operator as a Hamiltonian

Abstract

IIt is shown that the celebrated Heun operator $H_e=-(a_0 x^3 + a_1 x^2 + a_2 x) \frac{d^2}{dx^2} + (b_0 x^2 + b_1 x + b_2)\frac{d}{dx} + c_0 x$ is the Hamiltonian of the $sl(2,R)$-quantum Euler-Arnold top of spin $ν$ in a constant magnetic field. For $a_0 \neq 0$ 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 (half)-integer value the Hamiltonian possesses a finite-dimensional invariant subspace and a number of polynomial eigenfunctions occurs. Discrete systems on uniform and exponential lattices are introduced which are canonically-equivalent to one described by the Heun operator.