  The cosmological constant induced by quantum fluctuation of the graviton on a
given background is considered as a tool for building a spectrum of different
geometries. In particular, we apply the method to the Schwarzschild background
with positive and negative mass parameter. In this way, we put on the same
level of comparison the related naked singularity (-M) and the positive mass
wormhole. We use the Wheeler-De Witt equation as a basic equation to perform
such an analysis regarded as a Sturm-Liouville problem . The cosmological
constant is considered as the associated eigenvalue. The used method to study
such a problem is a variational approach with Gaussian trial wave functionals.
We approximate the equation to one loop in a Schwarzschild background. A zeta
function regularization is involved to handle with divergences. The
regularization is closely related to the subtraction procedure appearing in the
computation of Casimir energy in a curved background. A renormalization
procedure is introduced to remove the infinities together with a
renormalization group equation.
