  We study oscillations in the eigenfunctions for a fractional Schr\"odinger
operator on the real line. An argument in the spirit of Courant's nodal domain
theorem applies to an associated local problem in the upper half plane and
provides a bound on the number of nodal domains for the extensions of the
eigenfunctions. Using the combinatorial properties of noncrossing partitions,
we turn the nodal domain bound into an estimate for the number of sign changes
in the eigenfunctions. We discuss applications in the periodic setting and the
Steklov problem on planar domains.
