Single anisotropic gap superconductivity and proximity effect in PbTaSe2

${\mathrm{PbTaSe}}_{2}$ is a noncentrosymmetric superconductor having topological Dirac surface states as shown by recent experiments. Here, we report calculations on bulk and few-layer ${\mathrm{PbTaSe}}_{2}$ using the first-principles anisotropic Migdal-Eliashberg theory, to explore the nature of the superconducting gap and the surface-bulk proximity effect in this stoichiometric system. In bulk, the multiband Pb $p$/Ta $d$ Fermi surface exhibits a single anisotropic full superconducting gap, with larger gap values appearing on in-plane orbitals that can couple effectively to the in-plane phonons. A Pb-terminated few-layer ${\mathrm{PbTaSe}}_{2}$ is found to still have a single-anisotropic-gap feature, but its superconducting gap is evidently reduced relative to the bulk case. Comparison between our theoretical results and experimental tunneling data suggests a significant proximity-enhanced superconductivity on the realistic surface of ${\mathrm{PbTaSe}}_{2}$. The present findings would apply to other same-structured $AB{\mathrm{Se}}_{2}$ ($A$ = Pb, Sn; $B$ = Ta, Nb) and are crucial for studying topological superconductors and Majorana fermions in promising topological metals with intrinsic full-gap superconductivity.