One application is in scattering theory (which can be applied to lots of fields, such as quantum mechanics and general relativity). The relevant theorem is the analytic Fredholm theorem.
The analytic Fredholm theorem (rather, a simpler version of it) tells you that if you have a family of Fredholm operators which are holomorphic (over an open, connected set), and the family is invertible at a point, then the family of inverse operators form a meromorphic family, and the poles have finite rank.
This theorem is used to obtain meromorphic continuations of resolvents in PDE theory. The study of the resolvent is of fundamental importance in scattering theory. In particular, poles of the meromorphic continuation of the resolvent are called resonances, which carry important information about decay rates of solutions. This is a very expansive and deep subject, and I won't do it justice without writing an essay. See http://math.mit.edu/~dyatlov/res/res_final.pdf for more.