  Yang--Mills theory in four dimensions is studied by using the Coulomb gauge.
The Coulomb gauge Hamiltonian involves integration of matrix elements of an
operator P built from the Laplacian and from a first-order differential
operator. The operator P is studied from the point of view of spectral theory
of pseudo-differential operators on compact Riemannian manifolds, both when
self-adjointness holds and when it is not fulfilled. In both cases,
well-defined matrix elements of P are evaluated as a first step towards the
more difficult problems of quantized Yang--Mills theory.
