Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

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 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 finitelygenerated Lie algebra of differential operators, taken in a finite-dimensional representation.

A Unified Construction of Generalized Classical Polynomials Associated with Operators of Calogero–Sutherland Type

“…[14] to obtain the energy eigenvalues in (14) and (15) by purely algebraic means. We also mention that our point of view is closely related to the theory of quasi-exactly solvable Schrödinger operators; see, e.g., [4,15]. Moreover, the identities which are the key to our results are stated in Corollaries 2.2 and 2.3 below, and they were known before only in special cases: an important special case of these identities for the Sutherland model (case II) is a consequence of a well-known result on Jack polynomials which, to our knowledge, is due to Stanley (see Proposition 2.1 in [49]), and a generalisation of the latter to the deformed case and other non-deformed cases can be found in [48] and [13,39,45], respectively.…”

THE QUANTUM H3 INTEGRABLE SYSTEM

“…(1,2,3) n for any n ∈ N and therefore they preserve the flag P (1,2,3) . The second class operators (raising operators) act on the space P…”

“…It can be shown that the Hamiltonian h H 3 has a certain degeneracy -it preserves two different flags: one with minimal characteristic vector (1,2,3) and another one with characteristic vector (1,3,5). The fact that the operator h H 3 with coefficients (20) commutes with f given by (31) implies that common eigenfunctions of the operators h H 3 and f are elements of the flag of spaces P (1,3,5) .…”

“…The fact that the operator h H 3 with coefficients (20) commutes with f given by (31) implies that common eigenfunctions of the operators h H 3 and f are elements of the flag of spaces P (1,3,5) .…”

“…A goal of this Section is to show that each one of these subspaces is a representation space of an infinite-dimensional algebra of differential operators which we call h (3) and to study this algebra.…”