Multi-instantons in quantum mechanics (QM)

In general, a linear combination of instanton solutions is not a solution of the imaginary-time equations of motion, because the equations not linear. Moreover, in quantum mechanics (QM), all solutions of the classical equations can depend only on one time collective coordinate (in this respect, in field theory, the situation is different). However, a linear combination of largely separated instantons (a multi-instanton configuration) renders the action almost stationary, because each instanton solution differs, at large distances, from a constant solution by only exponentially small corrections (in field theory this is only true if the theory is massive). A situation where multi-instantons play a role is provided by large order behaviour estimates of perturbation theory for potentials with degenerate minima. When one starts from a situation in which the minima are almost degenerate, one obtains, in the degenerate limit, a contribution of the superposition of two, infinitely separated, instantons, but with an infinite multiplicative coefficient. Indeed, in this limit, the fluctuations which tend to change the distance between the instanton and the anti-instanton induce a vanishingly small variation of the action. To correctly determine the limit, one has to introduce a second collective coordinate which describes these fluctuations. The determination, at leading order, of all many-instanton contributions has led to conjecture the exact form of the semi-classical expansion for potentials with degenerate minima, generalizing the exact Bohr-Sommerfeld quantization condition.