Title: The quantum $H_3$ integrable system

Author: Marcos A. G. Garcia (University of Minnesota)

Abstract:

The $H_3$ integrable system is a $3D$ quantum system with rational potential related to the non-rystallographic root system $H_3$. It is isospectral to the $3D$ harmonic oscillator. It is shown that the gauge-rotated $H_3$ Hamiltonian, 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 $\vec \al\ =\ (1,2,3)$. A hidden algebra of the $H_3$ model is found. It is infinite-dimensional, finitely-generated algebra of differential operators possessing the finite-dimensional representations. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model in a space of invariants isospectral to the quantum $H_3$ rational model is defined.