  In this paper we study a novel spin chain with nearest-neighbors interactions
depending on the sites coordinates, which in some sense is intermediate between
the Heisenberg chain and the spin chains of Haldane-Shastry type. We show that
when the number of spins is sufficiently large both the density of sites and
the strength of the interaction between consecutive spins follow the Gaussian
law. We develop an extension of the standard freezing trick argument that
enables us to exactly compute a certain number of eigenvalues and their
corresponding eigenfunctions. The eigenvalues thus computed are all integers,
and in fact our numerical studies evidence that these are the only integer
eigenvalues of the chain under consideration. This fact suggests that this
chain can be regarded as a finite-dimensional analog of the class of
quasi-exactly solvable Schroedinger operators, which has been extensively
studied in the last two decades. We have applied the method of moments to study
some statistical properties of the chain's spectrum, showing in particular that
the density of eigenvalues follows a Wigner-like law. Finally, we emphasize
that, unlike the original freezing trick, the extension thereof developed in
this paper can be applied to spin chains whose associated dynamical spin model
is only quasi-exactly solvable.
