Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2004-07-23
Commun.Math.Phys. 260 (2005) 17-44
Physics
High Energy Physics
High Energy Physics - Theory
51 pages, LaTeX, few equations added, one reference added, typos corrected
Scientific paper
10.1007/s00220-005-1401-y
Solvability of the rational quantum integrable systems related to exceptional root spaces $G_2, F_4$ is re-examined and for $E_{6,7,8}$ is established in the framework of a unified approach. It is shown the Hamiltonians take algebraic form being written in a certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for {\it arbitrary} values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.
Boreskov Konstantin G.
López Vieyra Juan C.
Turbiner Alexander V.
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