Calogero model revisited, commuting Hamiltonians, Hurwitz numbers

Abstract

The generalized Mironov-Morozov-Natanson (MMN) equation includes a set of commuting operators, which can be considered as Hamiltonians for the quantum Calogero-Sutherland problem with a special value of the coupling constant (free fermion point). These Hamiltonians can be considered as the center of the enveloping algebra of the group $GL_N(C)$. Another commuting series of Hamiltonians is presented, parametrized by an arbitrary matrix $A\in GL_N$. These Hamiltonians are related to the Hurwitz numbers in the same way as in the case of the MMN equation and generate a generalized variant of the Calogero-Surtheland model.