Suppose we are studying square-integrable eigenfunctions of a linear operator (e.g. an ordinary differential operator), doing perturbation theory in a small parameter.
Suppose first-order perturbation yields a credible correction to the eigenvalue, but a correction to the eigenfunction that's not square-integrable. That can happen (or can you prove that it's impossible?) Suppose it does happen; what can we say on the correction to the eigenvalue that you calculated? Is it necessarily incorrect?
Perturbation expansions are based on the assumption that the eigenvalues of $H(\lambda)$ exist and are analytic functions of $\lambda$. This can be proved for finite dimensional operators and sometimes for unbounded operators. It can be shown that the expansions for eigenvalues and eigenvectors converge (often only for $\lambda$ in a finite range).
If you get a non-square-integrable eigenvector, the assumptions must not hold.