In classical quantum mechanics (and specifically in the introductury texts on this topic) while calculating expectation values of certain operators in the Schrödinger approach we often have to do partial integrations such as:
$$\int_{\mathbb{R}} \overline{\psi(x)} \psi''(x) \mathrm{d}x = - \int_{\mathbb{R}} \overline{\psi'(x)} \psi'(x) \mathrm{d}x$$
or more general
$$\int_{\mathbb{R}} \overline{A\psi(x)} (B\psi)'(x) \mathrm{d}x = - \int_{\mathbb{R}} \overline{(A\psi)'(x)} B\psi(x) \mathrm{d}x$$
for certain operators $A$ and $B$.

The dropping of the boundary term which usually features partial integration is most often wrongly justified by the false belief that
$\psi$ is square integrable implies that it is decaying to zero at infinity.

(even in cases that $\psi$ ís square integrable and does decay to zero, $\psi' \psi$ does not have to satisfy any of both conditions).

Question: is there some "natural" condition or observation on the properties of $\psi$ (or the set of functions where it comes from) by which we can generically justify the dropping of the boundary terms in the above partial integrations?

With "natural" I mean: can we give this property a physical interpretation within the q.m. framework?

(generalisation to other domains than just $\mathbb{R}$ is of course also welcome)