The Wheeler-DeWitt equation as an eigenvalue problem for the cosmological constant

The Standard Cosmological Model with energy density rho, pressure p and cosmological term lambda is reconsidered. The interdependence among the cosmological equations in different equivalent forms is pointed out with lambda possibly time dependent. For constant lambda and for rho, p expressed in terms of the scalar field phi of potential V(phi), the scheme is in fact equivalent to the coupling of Einstein and scalar field equations in the Robertson-Walker metric. A Lagrangian and the corresponding Hamiltonian H, that takes the zero value, are derived directly from the equations. The Wheeler-DeWitt (WDW) equation is obtained by canonical quantization of H that is performed in two non-equivalent ways. The WDW equations are transformed into Schrodinger-like eigenvalue problems with eigenvalue lambda. The equations are separated for vanishing scalar potential. The phi-separated equation results in an eigenvalue problem in the separation constant lambda (1), that must be negative, and it is easily integrated. The R-separated equation, is again an eigenvalue problem with eigenvalue lambda. It is solved, in the flat space-time case, by preliminary fixing lambda (1), in terms of the Bessel functions of the first kind and implies that lambda can take all possible negative values. For fixed lambda (1), lambda, the wave function of the Universe vanishes in correspondence with a big-bang situation and for large R.