from the physics and mathematics we know that
$ \frac{\sin(\sqrt u)}{\sqrt u} $
$ J_{l}(\sqrt u) $
have only real zeros
on the other hand these functions are entire, have no poles, only zeros and can be defined on the whole complex plane $ \mathbb C $.
So my question is: Is there a method or based on inverse spectral problem so we can say if a function has or hasn't a second Order Sturm Liouville operator in the form
$ -y''(x)+q(x)y(x)=zy(x) $
so the eigenvalues of these operator are precisely the zeros of the entire function??
I have checked the papers
http://arxiv.org/abs/0712.3238 The Schrödinger operator with Morse potential on the right half line