The Heun operator as a Hamiltonian

Abstract

IIt is shown that the celebrated Heun operator $H_e=-(a_0{x}^3+a_1{x}^2+a_{2}x)\frac{d^2}{d{x}^2}+(b_0{x}^2+b_{1}x+b_2)\frac{d}{dx}+c_{0}x$ is the Hamiltonian of the $sl(2,R)$-quantum Euler-Arnold top of spin $\nu$ in a constant magnetic field. For $a_0\ne 0$ it is canonically-equivalent to $B{C}_1(A_1)-$ 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 $\nu$ 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.

11 pages, typos corrected, Refs.[2,9,10,13,16], text with Eq.(20) and Conclusions added, to be published at J Phys A (Letters)

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