We study the critical energy and magnetization profiles for the Ising quantum chain with a marginal extended surface perturbation of the form A/y, y being the distance from the surface (Hilhorst-van Leeuwen model). For weak local couplings, A<A_c, the model displays a continuous surface transition with A-dependent exponents, whereas, for A>A_c, there is surface order at the bulk critical point. If conformal invariance is assumed to hold with such marginal perturbations, it predicts conformal profiles with the same scaling form as for the unperturbed quantum chain, with marginal surface exponents replacing the unperturbed ones. The results of direct analytical and numerical calculations of the profiles confirm the validity of the conformal expressions in the regimes of second- and first-order surface transitions.
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