Quantum integrability of the deformed elliptic Calogero–Moser problem

where all but one “masses” are equal, m1 = m −1, m2 = . . . = mn = 1, m is a real parameter, p̂j = i ∂ ∂xj , j = 1, . . . , n, and ℘ is the classical Weierstrass elliptic function. The case when m is integer is a special one: in that case a stronger version of integrability (the so-called algebraic integrability) was conjectured [2]. The first results in this direction were found in [3], where it was proven in the simplest non-trivial case n = 3,m = 2. The main result of the present paper is an explicit recursive formula for the quantum integrals of the elliptic deformed CM system. This proves integrability of the system for all n and m and due to a general recent result by Chalykh, Etingof and Oblomkov [4] this also implies the algebraic integrability for integer values of the parameterm. As a by-product we have also new formulae for the integrals of the usual quantum elliptic CM problem, which was the subject of many investigations since 1970s (see in particular [5], [6], [7, 8], [9, 10, 11]). We will be using some technical tricks from these important papers. In trigonometric and rational limits we have the formulae for the quantum integrals of the corresponding deformed CM systems which are also seem to be new.