Exact-solvability of rational-trigonometric 
Olshanetsky-Perelomov Hamiltonians

Alexander Turbiner 
(National University of Mexico and ITEP, Moscow)

Abstract

A notion of exact-solvability for eigenvalue problem for differential 
operator based on preservation of the flag of invariant subspaces is 
introduced. It is shown that there exist a certain Weyl-invariant 
functions which if they are taken as coordinates admit eigenfuctions of 
rational-trigonometric Olshanetsky-Perelomov Hamiltonians to have a form 
of polynomials. In these coordinates the Hamiltonians appear in an 
algebraic form, namely: as differential operators with polynomial 
coefficients. All flags of invariant subspaces associated with 
$A-D-E$ rational-trigonometric Olshanetsky-Perelomov Hamiltonians are 
classified and found explicitly.