In Courant, Hilbert "Methods of Mathematical Physics Vol. 1", there's a statement about the Courant's nodal theorem in more than one-dimensional spaces:
In the case of eigenvalue problems of partial differential equations, arbitrary values of $n$ may exist for which the nodes of the eigenfunctions $u_n$ subdivide the entire fundamental domain into only two subdomains.
Then the authors go on to suggest an example: the equation $\Delta u+\lambda u=0$ in the square $x\in[0,\pi], y\in[0,\pi]$ with an eigenfunction
$$\sin(2rx)\sin(y)+\mu\sin(x)\sin(2ry),$$
corresponding to the eigenvalue $\lambda=4r^2+1$ (and $\mu>0$ is supposed to be sufficiently close to 1).
But this case seems to be a bit of "cheating" because here the solution is explicitly a linear combination of two degenerate eigenfunctions, each of which has the same number of nodes $(2r-1)$, not limited to 2.
So I wonder: if we take a differential operator with non-degenerate spectrum, can we find such an eigenfunction that there'd be fewer nodes than $n$? Or can Courant's nodal theorem actually extend to such operators?