now I am diving into scattering theory, specifically the inverse scattering transform and I am applying it to reconstruct an arbitrary function used as the potential of the schrodinger operator (see https://arxiv.org/abs/1007.0938 [1]). Now I think the $\psi^2 _m (x)$ form a frame for they to be able to span the potential $V(x)$ in the operator $-\frac{d^2}{dx^2} + V(x)$ and using the theory in [1] I reached this integral.
$$ \int_{-\infty} ^{\infty} (\psi_m (x) \psi_k (x))^2 dx $$
and the only information I have is the normalization condition, that is
$$ \int_{-\infty} ^{\infty} \psi^2 _m (x) dx = 1 $$
and orthogonality
$$ \int_{-\infty} ^{\infty} \psi_m(x) \psi_k(x) dx = 0 $$ $$ m \neq k $$
So I was wondering if there is an inequality that I could use in this case or what can be said about this integral, all of this is to try and prove the parseval relation with this eigenfunctions an $V(x)$.