Solvability of the Hamiltonians related to exceptional root spaces: rational case

TL;DR

This paper investigates the solvability of rational quantum integrable systems linked to exceptional root spaces, demonstrating algebraic forms, invariant polynomial subspaces, and explicit eigenvalues for these Hamiltonians.

Contribution

It establishes the algebraic form and solvability of Hamiltonians related to all exceptional root spaces within a unified framework, including explicit eigenvalues and eigenfunctions.

Findings

Hamiltonians are algebraic in Weyl-invariant variables

Finite-dimensional invariant subspaces are polynomial-based and form an infinite flag

Explicit eigenvalues are computed for arbitrary coupling constants

Abstract

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.