The low-temperature limit of transfer operators in fixed dimension

We construct the 0'th order low-temperature WKB-phase for the first eigenfunction of a transfer operator in a domain around a non-degenerate critical point for the potential. The 0'th order low-temperature phase is shown to solve the eikonal equation in the strong-coupling limit and we obtain estimates on the 0'th order phase, which are preserved in the limit. We furthermore use the IMS localization technique to study the two highest eigenvalues of the transfer operator in the case where the potential is allowed to have many non-degenerate global minima.