  In this paper we clarify the relationship between inhomogeneous quantum spin
chains and classical integrable many-body systems. It provides an alternative
(to the nested Bethe ansatz) method for computation of spectra of the spin
chains. Namely, the spectrum of the quantum transfer matrix for the
inhomogeneous ${\mathfrak g}{\mathfrak l}_n$-invariant XXX spin chain on $N$
sites with twisted boundary conditions can be found in terms of velocities of
particles in the rational $N$-body Ruijsenaars-Schneider model. The possible
values of the velocities are to be found from intersection points of two
Lagrangian submanifolds in the phase space of the classical model. One of them
is the Lagrangian hyperplane corresponding to fixed coordinates of all $N$
particles and the other one is an $N$-dimensional Lagrangian submanifold
obtained by fixing levels of $N$ classical Hamiltonians in involution. The
latter are determined by eigenvalues of the twist matrix. To support this
picture, we give a direct proof that the eigenvalues of the Lax matrix for the
classical Ruijsenaars-Schneider model, where velocities of particles are
substituted by eigenvalues of the spin chain Hamiltonians, calculated through
the Bethe equations, coincide with eigenvalues of the twist matrix, with
certain multiplicities. We also prove a similar statement for the ${\mathfrak
g}{\mathfrak l}_n$ Gaudin model with $N$ marked points (on the quantum side)
and the Calogero-Moser system with $N$ particles (on the classical side). The
realization of the results obtained in terms of branes and supersymmetric gauge
theories is also discussed.
