1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hilbert-Polya conjecture:QM and number theory

  1. Sep 19, 2005 #1
    Now i would like to solve RH by using quantum physics,the problem is equivalent to find an operator [tex]H=aD^{2}+V(x)[/tex] with a real or complex potential V so the eigenvalues of H are precisely the roots of the function (a is a real constant) [tex]\zeta(b+is)[/tex],for b=1/2 the Riemann functional equation:

    [tex]\zeta(1-s)=\chi(s)\zeta(s)[/tex]

    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 [tex]\phi_{n}[/tex] and [tex]\phi_{k}[/tex] being the eigenfunctions associated to the eigenvalues s and s* then we would have:

    [tex](<\phi_{n}|H|\phi_{n}>)*=(<\phi_{k}|H|\phi_{k}>) [/tex]

    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:

    [tex]E_{n}-E^{0}_{n}=\delta{E(n)}=<\psi|V|\psi>[/tex]

    with [tex]E^{0}_{n},\psi [/tex] 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:

    [tex]V(x)=\int_{-\infty}^{\infty}dnR(n,x)\delta{E(n)}[/tex] 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 [tex]\zeta(b+is)[/tex]. for every a 0<b<1
     
    Last edited: Sep 20, 2005
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Hilbert-Polya conjecture:QM and number theory
  1. Hilbert spaces in QM (Replies: 2)

  2. Hilbert Spaces & QM (Replies: 1)

  3. Group theory and QM (Replies: 4)

Loading...