# Hilbert-Polya conjecture:QM and number theory

1. Sep 19, 2005

### eljose

Now i would like to solve RH by using quantum physics,the problem is equivalent to find an operator $$H=aD^{2}+V(x)$$ with a real or complex potential V so the eigenvalues of H are precisely the roots of the function (a is a real constant) $$\zeta(b+is)$$,for b=1/2 the Riemann functional equation:

$$\zeta(1-s)=\chi(s)\zeta(s)$$

tells us that if s is an energy of H for the case a=1/2 also s* is another energy for the H,then taking the expectation value of the Hamiltonian with $$\phi_{n}$$ and $$\phi_{k}$$ being the eigenfunctions associated to the eigenvalues s and s* then we would have:

$$(<\phi_{n}|H|\phi_{n}>)*=(<\phi_{k}|H|\phi_{k}>)$$

s=E_{n} and s*=E*_{n}=E_{k} from this last equation we get that for b=1/2 the potential is real.

The cases b<>1/2 we have complex "energies" in the form s*+(2a-1)i (just apply R Equation) so the potential for these roots must be complex and a complex potential can,t have real energies as <b> (the complex part of the potential is not 0 [H,b]=d being d a non-zero operator.)

Using perturbation theory we can calculate the potential by solving an integral equation:

$$E_{n}-E^{0}_{n}=\delta{E(n)}=<\psi|V|\psi>$$

with $$E^{0}_{n},\psi$$ the eigenvalues and eigenfunctions of H0=P^2/2m ( a free particle moving on an infinite potential well)

The last equation is an integral equation,we can solve it by Resolvent Kernel method getting finally a formula for V in the form:

$$V(x)=\int_{-\infty}^{\infty}dnR(n,x)\delta{E(n)}$$ valid for whatever b is so we could use it to prove RH is any desired form.....as we have found a Hamiltonian whose energies are the roots of $$\zeta(b+is)$$. for every a 0<b<1

Last edited: Sep 20, 2005