Bethe ansatz for the Ruijsenaars model of BC1- type

We consider one-dimensional elliptic Ruijsenaars model of type BC1. We show that when all coupling constants are integers, it has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the A1-case

Locus configurations and ∨\vee-systems

We present a new family of the locus configurations which is not related to $\vee$-systems thus giving the answer to one of the questions raised in \cite{V1} about the relation between the generalised quantum Calogero-Moser systems and special solutions of the generalised WDVV equations. As a by-product we have new examples of the hyperbolic equations satisfying the Huygens' principle in the narrow Hadamard's sense. Another result is new multiparameter families of $\vee$-systems which gives new solutions of the generalised WDVV equation.Comment: 12 page

Recollement of Deformed Preprojective Algebras and the Calogero-Moser Correspondence

The aim of this paper is to clarify the relation between the following objects: $(a)$ rank 1 projective modules (ideals) over the first Weyl algebra A_1(\C); $(b)$ simple modules over deformed preprojective algebras $\Pi_{\lambda}(Q)$ introduced by Crawley-Boevey and Holland; and $(c)$ simple modules over the rational Cherednik algebras $H_{0,c}(S_n)$ associated to symmetric groups. The isomorphism classes of each type of these objects can be parametrized geometrically by the same space (namely, the Calogero-Moser algebraic varieties); however, no natural functors between the corresponding module categories seem to be known. We construct such functors by translating our earlier results on \A-modules over $A_1$ to a more familiar setting of representation theory. In the last section we extend our construction to the case of Kleinian singularities \C^2/\Gamma , where $\Gamma$ is a finite cyclic subgroup of \SL(2, \C) .Comment: 16 pp., LaTex, to appear in Moscow Math. J.(2007

A remark on rational isochronous potentials

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form $U(x) = 1/2 \omega^2 x^2$ or $U(x) = 1/8 \omega^2 x ^2 + c^2 x^{-2}.$Comment: 5 pages, contribution to a special issue of JNMP dedicated to F. Calogero, slightly revised versio

Generalized Lame operators

We introduce a class of multidimensional Schr\"odinger operators with elliptic potential which generalize the classical Lam\'e operator to higher dimensions. One natural example is the Calogero--Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lam\'e operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh--Veselov conjecture for the elliptic Calogero--Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to B_2 case, another one is a certain deformation of the A_2 case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald--Ruijsenaars type.Comment: 38 pages, latex; in the new version a reference was adde

On algebraic integrability of the deformed elliptic Calogero--Moser problem

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

A class of Baker-Akhiezer arrangements

We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh–Veselov Baker–Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker–Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasi-invariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero–Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1

Deformations of the root systems and new solutions to generalised WDVV equations

A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (check-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the new solutions for the generalized WDVV equations.Comment: 8 page