# Why the Wu-sprung model is not accepted as a solution to Riemann HYpothesis ??

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

a survey on inverse problems in physics

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

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

-
Also posted to (and quickly closed at) MO, mathoverflow.net/questions/120017/… – 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 :) vixra.org/pdf/1301.0078v2.pdf see the analogy between the Guzwiller trace and riemann weil summation formulae in QM – Jose Garcia Jan 28 '13 at 12:48

vixra.org/pdf/1301.0078v2.pdf 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
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