Notes on the elliptic Ruijsenaars operators (Q1282055)

The properties of elliptic Ruijsenaars operators, \[ (H_i(k)f)(z)= \sum_{\substack{ I\subset (1,\dots, N)\\ |I|=1}} \Biggl(\pi{\theta_1(z_m- z_n- N)\over \theta_1(z_m- n_n)}\Biggr)\times f\Biggl(z+ k\Biggl( \sum_{m\in I} \varepsilon_m- i\sum_{m=1} \varepsilon_m/N\Biggr)\Biggr) \] are investigated. It is also shown that these operators can be extended to nonnegative selfadjoint operators on the Hilbert space without essential selfadjointness except in trigonometric limits.