Non-integrable Wavefunctions Suppose first-order perturbation yields a credible correction to the energy, but a correction to the wave function that's not square-integrable.  That can happen, I see no reason why it couldn't. Unless there is some proof that it can't happen (I haven't found any). And if it does happen, then would you believe the energy shift you calculated?
 A: I think the wiki's proof for first order correction term is rather clear.
https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)#First_order_corrections
The important sentence connect to your quesiton is "Let $V$ be a Hamiltonian representing a weak physical disturbance". The core of pertubation is the linearlization under small changes, so don't think $V$ as some usual potentional that contain singularities, if it does, it probabily meant your system is confined in a much smaller region by infinite wall.
Then, before you obtain the correction term, there is expression $(E_n^{(0)}-E_k^{(0)} )\langle k^{(0)}|n^{(1)} \rangle=\langle k^{(0)}|V|n^{(1)} \rangle$ which guarranteed the RHS is a small finite number, and each peatubated basis vector(at least the first one) is square integrable.(square integrable meant you can get a finite number, not Riemann integability.)
If you constructed a $V$ such that it's not square integrable, then you should not be using pertubation theory, because that's a "kick" not a pertubation.
A: From the basics of QM, the only wave functions which are square integrable are physical.
Therefore, the wavefunctions which are not square integrable can only appeare on math but no physical system can have such a wavefunction.
Besides the original Hamiltonian, the corrections to energies and wavefunctions only depends on the perturbations(interaction Hamiltonian). By restricting on physically acceptable wavefunctions, we can essentially restrict the allowed perturbations on the given original system. 
The great thing is that this type of restrictions on allowed interactions does naturally ocurr in the context of Quantum Field Theory (QFT). By invoking Gauge invariance on the original lagrangian. 
For example, by invoking Gauge invariance on the free Dirac lagrangian we get the quantum electrodynamics, QED, Lagrangian which only have one interaction term corresponding to fermion-photon interaction.
Also, in QFT non-physical interactions can be ruled out by dimensional analysis and renormalizability of the theory.
