Periodic instantons and quantum-mechanical tunneling at high energy

The tunneling process at high energy is investigated for a one-dimensional system with the double-well potential. The path-integral method is used to calculate the transition amplitude between excited states in the two wells, as well as the level splitting of the excited states by expanding the action about a periodic instanton solution. The solution of half a period between two turning points is treated like an instanton configuration and the singularity of the Feynman kernel between turning points for the finite Euclidean time interval is smoothened with the end-point integrals. The level splitting obtained is in exact agreement with the WKB result. For weak coupling and energies far below the barrier height, the transition amplitude grows with energy exponentially. For energies approaching the barrier height anharmonic contributions must be taken into account.