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Factorization of Laplacian

  1. Jun 28, 2009 #1
    given the Laplacian for a certain metric 'g'

    [tex] \Delta f = \partial _i \partial ^{i}f + (\partial ^{i}f) \partial _ i log |g|^{1/2} [/tex]

    where a sum over 'i' dummy variable is assumed

    the idea is , could we factorize this Hamiltonian a second order differential operator into two first order differential operator as

    [tex] 1/2 + iL [/tex] and [tex] 1/2 - iL [/tex] so [tex] \Delta f= (1+2+iL)(1/2-iL)f [/tex] ?

    the idea taken from mathematics is the following:

    accroding to selberg trace [tex] \lambda =s(1-s) [/tex] where s is a complex zero and lambda is the eigenvalue of the Laplacian of a surface.

    in terms of Riemann Hypothesis : if we can find such factorization eigenvalues of [tex] 1/2 + iL [/tex] would be precisely the zeros of Riemann zeta function , providing that [tex] 1/4+ \gamma ^2 [/tex] are eigenvalues of a certain surface.
     
  2. jcsd
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