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from the physics and mathematics we now 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

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