The Birman-Schwinger principle bounds the number of eigenvalues of a Schrödinger operator below certain level (in terms of an integral operator involving the potential and the resolvent of the Laplacian).

This has been used in many "real physics" articles, e. g. "Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter" by Lieb and Thirring.

On might argue that Birman-Schwinger principle does not constitute a very deep application of functional analysis, which is probably true, but nevertheless even its general rigorous formulation (and, of course, proof) requires some FA.