We study three solvable two-dimensional systems perturbed by a point interaction centered at the
origin. The unperturbed systems are the isotropic harmonic oscillator, a square pyramidal
potential and a combination thereof. We study the spectrum of the perturbed systems. We show
that, while most eigenvalues are not affected by the point perturbation, a few of them are strongly
perturbed. We show that for some values of one parameter, these perturbed eigenvalues may take
lower values than the immediately lower eigenvalue, so that level crossings occur. These level
crossings are studied in some detail.
