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The existence of sharp resonant states in a single quantum well separated from its semi-infinite substrate by a barrier layer is reported here. These resonances appear as well defined peaks in the density of states. The local and total densities of states are obtained from an analytical determination of the Green functions. The expressions of the transmission and reflection phase times are also derived and compared to the density of states. The positions of the peaks in the density of states enable us to study the energy levels of resonant states as a function of the thicknesses of the barrier and well layers. Specific applications of our analytical results are given in this paper for a bilayer sandwiched between two semi-infinite GaAs and media.