  We consider the influence of a power-law deviation from the critical coupling
such that the system is critical at its surface. We develop a scaling theory
showing that such a perturbation introduces a new length scale which governs
the scaling behaviour of the density profiles as well as the finite-size
behaviour of the surface properties. Exact results are obtained for the Ising
quantum chain when the perturbation varies linearly whereas the quadratic
perturbation is mainly studied numerically. The scaling theory is well
confirmed in both cases.
