  Inhomogeneous quantum critical systems in one spatial dimension have been
studied by using conformal field theory in static curved backgrounds. Two
interesting examples are the free fermion gas in the harmonic trap and the
inhomogeneous XX spin chain called rainbow chain. For conformal field theories
defined on static curved spacetimes characterised by a metric which is Weyl
equivalent to the flat metric, with the Weyl factor depending only on the
spatial coordinate, we study the entanglement hamiltonian and the entanglement
spectrum of an interval adjacent to the boundary of a segment where the same
boundary condition is imposed at the endpoints. A contour function for the
entanglement entropies corresponding to this configuration is also considered,
being closely related to the entanglement hamiltonian. The analytic expressions
obtained by considering the curved spacetime which characterises the rainbow
model have been checked against numerical data for the rainbow chain, finding
an excellent agreement.
