  We derive a lower bound on the location of global extrema of eigenfunctions
for a large class of non-local Schr\"odinger operators in convex domains under
Dirichlet exterior conditions, featuring the symbol of the kinetic term, the
strength of the potential, and the corresponding eigenvalue, and involving a
new universal constant. We show a number of probabilistic and spectral
geometric implications, and derive a Faber-Krahn type inequality for non-local
operators. Our study also extends to potentials with compact support, and we
establish bounds on the location of extrema relative to the boundary edge of
the support or level sets around minima of the potential.
