  We examine the spectrum of a family of Sturm--Liouville operators with
regularly spaced delta function potentials parametrized by increasing strength.
The limiting behavior of the eigenvalues under this spectral flow was described
in a previor result of the last two authors with Berkolaiko, where it was used
to study the nodal deficiency of Laplacian eigenfunctions. Here we consider the
eigenfunctions of these operators. In particular, we give explicit formulas for
the limiting eigenfunctions, and also characterize the eigenfunctions and
eigenvalues for all values for the spectral flow parameter (not just in the
limit). We also develop spectrally accurate numerical tools for comparison and
visualization.
