in the Wu-sprung model, given a Hamiltonian in one dimension
$$ -y''(x)+f(x)y(x)=E_{n}y(x) \qquad y(0)=0=y(\infty) $$
we can define the function $ f(x) $ implicitly as
$$ f^{-1}(x)= 2\sqrt{\pi} \frac{d^{1/2}}{dx^{1/2}}n(x) $$
here $ n(x) $ is the function counting the eigenvalues $ n(x)= \sum_{E_{n}\le x} 1$
for the case of Riemann function this $ n(x)= \frac{1}{\pi}arg\xi(1/2+i \sqrt{x}) $
so the Riemann Hypotheis is the solution to an inverse problem
literature: http://arxiv.org/pdf/math/0510341v1.pdf introduction to wu sprung model
http://mathdl.maa.org/images/upload_library/22/Ford/JosephKeller.pdf
a survey on inverse problems in physics
for the Riemann zet function the 'potential ' $ f(X) $ is defined as
$$ f^{-1} (x)=\frac{4}{\sqrt{4x+1} } +\frac{1}{2\pi } \int\nolimits_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right) -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right) $$
Where it is rigorous, the relation between quantum chaos and mathematics has been that number theory is used to prove the conjectures inspired by physics, not the other way around. The ideas from physics are not, so far, specific enough to number theory to prove anything new in the direction of the Riemann hypothesis.