2010/11/30 by MARCOS A. G. GARCÍA, ALEXANDER V. TURBINER
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Quantum Mechanics and Non-Hermitian Physics #Nonlinear Waves and Solitons
paper · doi:10.1142/s0217751x10050597
The quantum H 3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H 3 . It is shown that the gauge-rotated H 3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H 3 , is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H 3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H 3 -invariants "polynomially"-isospectral to the quantum H 3 model is defined.