  In the previous paper we introduced a commuting family of Baxter Q-operators
for the quantum Ruijsenaars hyperbolic system. In the present work we show that
the wave functions of the quantum system found by M. Halln\"as and S.
Ruijsenaars also diagonalize Baxter operators. Using this property we prove the
conjectured duality relation for the wave function. As a corollary, we show
that the wave function solves bispectral problems for pairs of dual Macdonald
and Baxter operators. Besides, we prove the conjectured symmetry of the wave
function with respect to spectral variables and obtain new integral
representation for it.
