Solvability of the \(G_2\) integrable system. (English) Zbl 0932.37052
Summary: It is shown that the three-body trigonometric \(G_2\) integrable system is exactly solvable. If the configuration space is parametrized by certain symmetric functions of the coordinates then, for arbitrary values of the coupling constants, the Hamiltonian can be expressed as a quadratic polynomial in the generators of some Lie algebra of differential operators in a finite-dimensional representation. Four infinite families of eigenstates, represented by polynomials, and the corresponding eigenvalues are described explicitly.
MSC:
| 37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |
| 70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
| 81V70 | Many-body theory; quantum Hall effect |
Keywords:
trigonometric \(G_2\) integrable system; symmetric functions; Hamiltonian; Lie algebra; eigenstatesReferences:
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