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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: introduction to wu sprung model

a survey on inverse problems in physics

for the Riemann zet function the 'potential ' $ f(X) $ is defined as

$$ f^{-1} (x)=\frac{4}{\sqrt{4x+1} } +\frac{1}{2\pi } \int\nolimits_{-\sqrt{x} }^{\sqrt{x}}\frac{dr}{\sqrt{x-r^2} } \left( \frac{\Gamma '}{\Gamma } \left( \frac{1}{4} +\frac{ir}{2} \right) -\ln \pi \right) -\sum\limits_{n=1}^\infty \frac{\Lambda (n)}{\sqrt{n} } J_0 \left( \sqrt{x} \ln n\right) $$

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Also posted to (and quickly closed at) MO,… – Gerry Myerson Jan 28 '13 at 5:59
It is not accepted because it causes too many typos. HYpothesis, Hypotheis, imverse, RIemann, zet. – Gerry Myerson Jan 28 '13 at 6:01
i meant the equations, which are the ones that are really important :D – Jose Garcia Jan 28 '13 at 9:14
You corrected three of the six that I pointed out. – Gerry Myerson Jan 28 '13 at 11:39
HERE is a survey made by me about this problem and how the Riemann Weil and gutzwiller trace are analogue :) see the analogy between the Guzwiller trace and riemann weil summation formulae in QM – Jose Garcia Jan 28 '13 at 12:48

1 Answer 1

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.

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however why do not mathematician even bother solving it numerically to check that this solution is possible ?? – Jose Garcia Feb 6 '13 at 14:36
There is a lot of numerical computation about arithmetic quantum chaos, like anything else related to the Riemann hypothesis. If you know of a calculation that might be new, and can be expressed in computer code, there is no reason not to do it or ask whether it is known. – zyx Feb 6 '13 at 14:58 the operator is of the form $ H=-\frac{d^{2}}{dx^{2}}+f(x) $ and the function $ f(x) $ is given implicitly inside (1.10) as an smooth part plus corrections due to the primes and prime powers $ \sum_{n=1}^{\infty}\frac{\Lambda (n)}{\sqrt{n}}J_{0}( \sqrt{x}logp) $ – Jose Garcia Feb 7 '13 at 11:24
Operator on what Hilbert space? – zyx Feb 7 '13 at 17:16
is a problem of QM so the space is the same as the QM ... acting over integrable eigenfunctions $ \int_{-\infty}^{\infty}dx |\Psi(x)| $ – Jose Garcia Feb 7 '13 at 17:28

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