  We study global solutions to the classical one-phase free boundary problem
that have finite Morse index relative to the Alt-Caffarelli functional. We show
that such solutions are stable outside a compact set and characterize the index
as the maximal number of linearly independent $L^2$ integrable eigenfunctions
of the corresponding Robin eigenvalue problem, associated to negative
eigenvalues. As an application, we obtain a complete classification of global
solutions of finite Morse index in the plane. Our results are counterparts to
the minimal surface theorems of Fischer-Colbrie and Gulliver.
