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 {\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.

51 pages, LaTeX, few equations added, one reference added, typos corrected

References in corpus (8)

• Exact Solvability of the Calogero and Sutherland Models
• Hidden Algebras of the (super) Calogero and Sutherland models
• Solvability of the G_2 Integrable System
• Quasi-Exactly-Solvable Many-Body Problems
• Hidden Algebra of Three-Body Integrable Systems
• Exact solutions of an elliptic Calogero--Sutherland model
• Solvability of the $F_4$ Integrable System
• The construction of trigonometric invariants for Weyl groups and the derivation of corresponding exactly solvable Sutherland models

Cited by in corpus (15)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Integrable families of hard-core particles with unequal masses in a one-dimensional harmonic trap
• A unified construction of generalised classical polynomials associated with operators of Calogero-Sutherland type
• Particular Integrability and (Quasi)-exact-solvability
• The quantum $H_4$ integrable system
• The quantum $H_3$ integrable system
• From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model
• Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariable Polynomials. II. $E_7$ case
• Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariate Polynomials. III. $E_8$ case
• Tremblay-Turbiner-Winternitz (TTW) system at integer index $k$: polynomial algebras of integrals
• The Lie algebra of the lowest transitively differential group of degree three
• $F_4$ Quantum Integrable, rational and trigonometric models: space-of-orbits view
• Calogero-like model without rearrangement symmetry
• Dirac Equation with Space Contributions Embedded in a Quantum-Corrected Gravitational Field
• Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals