  I have made an ample study of one dimensional quantum oscillators, ranging
from logarithmic to exponential potentials. I have found that the eigenvalues
of the hamiltonian of the oscillator with the limiting (approachissimo)
harmonic potential (~ p(x)2) maps the zeros of the Riemann function height up
in the Riemann line. This is the potential created by the field of J(x) that is
the Riemann generator of the prime number counting function, p(x), that in turn
can be defined by an integral transformation of the Riemann zeta function. This
plays the role of the spring strength of the quantum limiting harmonic
oscillator. The number theory meaning of this result is that the roots height
up of the zeta function are the eigenvalues of a Hamiltonian whose potential is
the number of primes squared up to a given x. Therefore this may prove the
never published Hilbert-Polya conjecture. The conjecture is true but does not
imply the truth of the Riemann hypothesis. We can have complex conjugated zeros
off the Riemman line and map them with another hermitic operator and a general
expression is given for that. The zeros off the line affect the fluctuation of
the eigenvalues but not their mean values.
