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)

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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:

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 k for some k.  For such V the algebra A is the universal enveloping algebra sl(n+1), where n is the number of independent variables.

It is clear that if a differential operator satisfies Property 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.

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