  We have measured the critical current density, superconducting coherence
length, and superconducting transition temperature on single-domain,
epitaxially-grown Nb(110)/Au(111)/Nb(110) trilayers, all showing non-monotonic
behavior as a function of the thickness of the Au layer. To reveal the
underlying physical mechanism, we apply the first-principles-based microscopic
theory of inhomogeneous relativistic superconductivity which predicts
quantitatively the superconducting properties. As a result of the confinement
in the growth direction, quantum-well states appear in the Au layer which
yields a complex structure of Andreev states in the superconducting state. It
is shown how the quantum-well states modify the quasiparticle spectrum and the
Josephson supercurrent. Our study reveals the coexistence of the two
superconducting phases in the Au layer, the usual intraband s-wave phase and an
additional non-uniform phase stemming from interband pairing (without magnetic
field). These results indicate the rich interplay between quantum-size and
proximity effects, suggesting the possibility of modifying superconducting
transport properties by exploiting thickness-dependent quantum size effect
