  In the framework of a Varying Speed of Light theory, we study the eigenvalues
associated with the Wheeler-DeWitt equation representing the vacuum expectation
values associated with the cosmological constant. We find that the
Wheeler-DeWitt equation for the Friedmann-Lema\^{\i}tre-Robertson-Walker metric
is completely equivalent to a Sturm-Liouville problem provided that the related
eigenvalue and the cosmological constant be identified. The explicit
calculation is performed with the help of a variational procedure with trial
wave functionals related to the Bessel function of the second kind $K_{\nu
}\left( x\right) $. We find the existence of a family of eigenvalues associated
to a negative power of the scale. Furthermore, we show that at the inflationary
scale such a family of eigenvalues does not appear.
