# 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)

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 ${\displaystyle P}$ of second order with polynomial coefficients.

It turns out that in all known examples:

${\displaystyle \mathbf {1} }$ :  ${\displaystyle P}$ preserves some finite-dimensional polynomial vector space ${\displaystyle V}$.

The set of all differential operators with polynomial coefficients that preserve a fixed finite-dimensional polynomial vector space ${\displaystyle V}$ forms an associative algebra ${\displaystyle A}$.  In the most interesting case the vector space ${\displaystyle V}$ coincides with the space of all polynomials of degrees ${\displaystyle \leq k}$ for some ${\displaystyle k}$.  For such ${\displaystyle V}$ the algebra ${\displaystyle A}$ is the universal enveloping algebra ${\displaystyle \operatorname {sl} (n+1)}$, where ${\displaystyle n}$ is the number of independent variables.

It is clear that if a differential operator satisfies Property ${\displaystyle \mathbf {1} }$, we can find several eigenvalues and corresponding polynomial eigenvectors in an explicit algebraic form.

For the elliptic Calogero type models the flat metric ${\displaystyle g}$ related to the symbol of ${\displaystyle P}$ 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