# Algebraic Geometry Seminar

#### 3-Body Elliptic Calogero Model: A Solution

**Speaker:**
Alexander Turbiner, Nuclear Science Institute, UNAM

**Location:**
Warren Weaver Hall 201

**Date:**
Tuesday, March 22, 2016, 3:30 p.m.

**Synopsis:**

For the 3-body elliptic Calogero Hamiltonian (two-dimensional Lamé operator with elliptic potential) we find a change of variables such that it becomes an algebraic operator (with polynomial coefficients). It is shown that the model is equivalent to the sl(3)-algebra Euler-Arnold quantum top in a constant magnetic field (in the representation (ν,0)), whose strength is defined by the coupling constant. For discrete values of the coupling constant a finite number of polynomial eigenfunctions occur: the Hamiltonian has an invariant subspace. A 3-parametric pair of commuting differential operators in two variables, one of the 2nd and another one of the 3rd degrees is explicitly constructed.