Classical and quantum exact solutions for a FRW multi-scalar field cosmology with an exponential potential driven inflation

A flat Fiedmann-Robertson-Walker (FRW) multi-scalar field cosmology is studied with a particular potential of the form $\rm V(\phi,\sigma)=V_0 e^{-\lambda_1 \phi-\lambda_2 \sigma}$, which emerges as a relation between the time derivatives of the scalars field momenta. Classically, by employing the Hamiltonian formalism of two scalar fields $\rm(\phi,\sigma)$ with standard kinetic energy, exact solutions are found for the Einstein-Klein-Gordon (EKG) system for different scenarios specified by the parameter $\rm\lambda^2=\lambda_1^2+\lambda_2^2$, as well as the e-folding function $\rm N_{e}$ which is also computed. For the quantum scheme of this model, the corresponding Wheeler-DeWitt (WDW) equation is solved by applying an appropriate change of variables.