Nodal deficiency of Neumann eigenfunctions on a symmetric dumbbell domain

We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the dumbbell, together with a one-dimensional Sturm–Liouville solution in the neck. In this limit, the corresponding eigenvalues become degenerate, with multiplicity two. The nodal deficiency, defined as the difference between the eigenvalue index and the nodal domain count, is known by the Courant nodal domain theorem to be non-negative. We show that, for small neck widths, the nodal deficiencies of the dumbbell eigenfunctions are no smaller than the nodal deficiencies of the limiting eigenfunctions in the ends, and we provide conditions under which equality is achieved. As a consequence, we establish a criterion for identifying eigenfunctions of zero nodal deficiency for the dumbbell domain.