Super-Hamiltonians for super-Macdonald polynomials

Abstract

The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables $\hat{I}{¸}_k$ are introduced, and the Hamiltonian is represented as a differential operator acting on a space of functions of both types of variables $p_k$ and $\hat{I}{¸}_k$. Eigenfunctions for this Hamiltonian are a suitable generalization of Macdonald polynomials to super-Macdonald polynomials discussed earlier in the literature. Peculiarities of the construction in comparison to the canonical bosonic case are discussed.