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The confinement of a superconductor in a thin film changes its Fermi-leveldensity of states and is expected to change its critical temperature $T_c$.Previous calculations have reported large discontinuities of $T_c$ when thechemical potential coincides with a subband edge. By solving the BCS gapequation exactly, we show that such discontinuities are artifacts and that$T_c$ is a continuous function of the film thickness. We also find that $T_c$is reduced in thin films compared with the bulk if the confinement potential islower than a critical value, while for stronger confinement $T_c$ increaseswith decreasing film thickness, reaches a maximum, and eventually drops tozero. Our numerical results are supported by several exact solutions. Wefinally interpret experimental data for ultrathin lead thin films in terms of athickness-dependent effective mass.

Rise and fall of shape resonances in thin films of BCS superconductors