# Sokolov V.V. Algebraic quantum Hamiltonians on the plane, talk at The Mini-Workshop on Integrable Equations, 17 February 2015, Independent University of Moscow (abstract)

**Speaker**: Vladimir Sokolov

**Title**: Algebraic quantum Hamiltonians on the plane
**Abstract**:

In is known that many of quantum Calogero-Moser type Hamiltonians admit a change of variables bringing them to differential operators of second order with polynomial coefficients.

It turns out that in all known examples:

: preserves some finite-dimensional polynomial vector space .

The set of all differential operators with polynomial coefficients that preserve a fixed finite-dimensional polynomial vector space forms an associative algebra . In the most interesting case the vector space coincides with the space of all polynomials of degrees for some . For such the algebra is the universal enveloping algebra , where is the number of independent variables.

It is clear that if a differential operator satisfies Property , we can find several eigenvalues and corresponding polynomial eigenvectors in an explicit algebraic form.

For the elliptic Calogero type models the flat metric related to the symbol of depends on the elliptic parameter. One of the reasons why such a metric could be interesting in itself is that families of contravariant metrics with linear dependence on a parameter are closely related to the Frobenius manifolds.

**Slides**: Sokolov V. Algebraic quantum Hamiltonians on the plane (presentation at The Mini-Workshop on Integrable Equations, 17 February 2015, Independent University of Moscow).pdf