Related papers: Algebraic spectral relations for elliptic quantum …
Explicit algebraic relations between the quantum integrals of the elliptic Calogero--Moser quantum problems related to the root systems ${\bf A_2}$ and ${\bf B_2}$ are found.
Algebraic integrability of the elliptic Calogero-Moser quantum problem related to the deformed root systems A_2(2) is proved. Explicit formulae for integrals are found.
Algebraic integrability of the elliptic Calogero--Moser quantum problem related to the deformed root systems $\pbf{A_{2}(2)}$ is proved. Explicit formulae for integrals are found.
Calogero-Moser systems are classical and quantum integrable multi-particle dynamics defined for any root system $\Delta$. The {\em quantum} Calogero systems having $1/q^2$ potential and a confining $q^2$ potential and the Sutherland systems…
Calogero-Moser systems can be generalized for any root system (including the non-crystallographic cases). The algebraic linearization of the generalized Calogero-Moser systems and of their quadratic (resp. quartic) perturbations are…
We discuss elliptic quantum Calogero-Moser-Sutherland models, including their relativistic generalizations due to Ruijsenaars and van Diejen, and the relations of these models to classes of special functions developed and explored in recent…
The rings of quantum integrals of the generalized Calogero-Moser systems related to the deformed root systems ${\cal A}_n(m)$ and ${\cal C}_n(m,l)$ with integer multiplicities and corresponding algebras of quasi-invariants are investigated.…
We consider the generalised Calogero-Moser-Sutherland quantum integrable system associated to the configuration of vectors $AG_2$, which is a union of the root systems $A_2$ and $G_2$. We establish the existence of and construct a suitably…
The relation between quantum systems associated to root systems and radial parts of Laplace operators on symmetric spaces is established. From this it follows the complete integrability of some quantum systems.
The deformed quantum Calogero-Moser-Sutherland problems related to the root systems of the contragredient Lie superalgebras are introduced. The construction is based on the notion of the generalized root systems suggested by V. Serganova.…
The integrability of the deformed quantum elliptic Calogero-Moser problem introduced by Chalykh, Feigin and Veselov is proven. Explicit recursive formulae for the integrals are found. For integer values of the parameter this implies the…
The space of m-harmonic polynomials related to a Coxeter group G and a multiplicity function m on its root system is defined as the joint kernel of the properly gauged invariant integrals of the corresponding generalised quantum…
Classical Calogero-Moser models with rational potential are known to be superintegrable. That is, on top of the r involutive conserved quantities necessary for the integrability of a system with r degrees of freedom, they possess an…
We exhibit the elliptic Calogero-Moser system as a Hitchin system of G-principal Higgs pairs. The group G, though naturally associated to any root system, is not semi-simple. We then interpret the Lax pairs with spectral parameter of [dP1]…
We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases.…
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find…
We discuss integrable many-body systems in one dimension of Calogero-Moser-Sutherland type, both classical and quantum as well as nonrelativistic and relativistic. In particular, we consider fundamental properties such as integrability, the…
Hidden symmetry of the quantum Calogero-Moser system with the inverse-square potential is explicitly demonstrated in algebraic sense. We find the underlying algebra explaining the super-integrability phenomenon for this system. Applications…
A model describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is partially solved in the quantum case in a totally algebraic way. As an example, the ground state and…
We develop a systematic procedure for constructing quantum many-body problems whose spectrum can be partially or totally computed by purely algebraic means. The exactly-solvable models include rational and hyperbolic potentials related to…