The problem of calculating the $\beta$-Bogoliubov coefficient in Non-Oscillating models

The calculation of the Bogoliubov coefficients is a key piece to obtain the reheating temperature of the Universe. In all cases this calculation is performed either in toy models where some derivative of the potential is discontinuous at some points or by making some approximations to the model. The result of these calculations is applied to more realistic models without checking if they really apply to this realistic model because the exact calculation of the Bogoliubov coefficients for a viable model, which is usually depicted by a smooth potential, requires a very complicated numerical calculation. Here we want to point out the difficulties that one encounters when trying to compute numerically these coefficients for a smooth potential without making approximations, which could lead to completely different results.