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 $P$ of second order with polynomial coefficients.

It turns out that in all known examples:

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

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

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

For the elliptic Calogero type models the flat metric $g$ related to the symbol of 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.