In quantum field theory, periodic instantons are finite energy solutions of Euclidean-time field equations which communicate (in the sense of quantum tunneling) between two turning points in the barrier of a potential and are therefore also known as bounces.
For completeness we add that sphalerons are the field configurations at the very top of a potential barrier.
Periodic instantons were discovered with the explicit solution of Euclidean-time field equations for double-well potentials and the cosine potential with non-vanishing energy[1] and are explicitly expressible in terms of Jacobian elliptic functions (the generalization of trigonometrical functions).
of the energies of states or wave functions related to the wells on either side of the barrier, i.e.
by the path integral method requires summation over an infinite number of widely separated pairs of periodic instantons -- this calculation is therefore said to be that in the dilute gas approximation.