Related papers: Hidden Algebra of Three-Body Integrable Systems
The three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is shown to be exactly solvable. When written in appropriate variables, its…
Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable…
We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamiltonian system that preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$,…
We introduce a family of Hamiltonian systems for measurement-based quantum computation with continuous variables. The Hamiltonians (i) are quadratic, and therefore two body, (ii) are of short range, (iii) are frustration-free, and (iv)…
Taking the Hydrogen atom as an example it is shown that if the symmetry of a three-dimensional system is $O(2) \oplus Z_2$, the variables $(r, \rho, \varphi)$ allow a separation of the variable $\varphi$, and the eigenfunctions define a new…
Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are…
We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the…
The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on…
Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose…
A new (in)finite dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite dimensional representations are constructed and…
By Poissonization of Jacobi structures on real three-dimensional Lie groups $\mathbf{G}$ and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on $\mathbf{G}\otimes \mathbb{R}$.
All possible Lie bialgebra structures on the harmonic oscillator algebra are explicitly derived and it is shown that all of them are of the coboundary type. A non-standard quantum oscillator is introduced as a quantization of a triangular…
We construct integrable and superintegrable Hamiltonian systems using the realizations of four dimensional real Lie algebras as a symmetry of the system with the phase space R4 and R6. Furthermore, we construct some integrable and…
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom…
Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…
We present a method to obtain higher order integrals and polynomial algebras for two-dimensional superintegrable systems from creation and annihilation operators. All potentials with a second and a third order integrals of motion separable…
In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of Lagrangian submanifolds of $TT^*Q$…
We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential…
We propose a new definition of so called Hamiltonian forms in n-plectic geometry and show that they have a non-trivial Lie infinity-algebra structure.
In this paper the quantum integrals of the Hamiltonian of the quantum many-body problem with the interaction potential K/sinh^2(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebra gl(N) under a certain…