  The self-adjointness of the reduced Hamiltonian operators arising from the
Laplace-Beltrami operator of a complete Riemannian manifold through quantum
Hamiltonian reduction based on a compact isometry group is studied. A simple
sufficient condition is provided that guarantees the inheritance of essential
self-adjointness onto a certain class of restricted operators and allows us to
conclude the self-adjointness of the reduced Laplace-Beltrami operators in a
concise way. As a consequence, the self-adjointness of spin Calogero-Sutherland
type reductions of `free' Hamiltonians under polar actions of compact Lie
groups follows immediately.
