Algebraic quantum Hamiltonians on the plane

TL;DR

This paper classifies second order differential operators with polynomial coefficients that preserve polynomial spaces, focusing on flat metrics and potential operators, and derives explicit forms for certain elliptic Calogero-Moser Hamiltonians and Inosemtsev models.

Contribution

It provides a classification of algebraic quantum Hamiltonians on the plane with polynomial coefficients and explicit forms for specific elliptic models.

Findings

Derived polynomial forms for elliptic $A_2$ and $G_2$ Calogero-Moser Hamiltonians.

Classified second order differential operators preserving polynomial spaces with flat metrics.

Obtained explicit forms for elliptic Inosemtsev model.

Abstract

We consider second order differential operators $P$ with polynomial coefficients that preserve the vector space $V_k$ of polynomials of degrees not greater then $k$. We assume that the metric associated with the symbol of $P$ is flat and that the operator $P$ is potential. In the case of two independent variables we obtain some classification results and find polynomial forms for the elliptic $A_2$ and $G_2$ Calogero-Moser Hamiltonians and for the elliptic Inosemtsev model.