Non-perturbative instanton effects in the quartic and the sextic double-well potential by the numerical bootstrap approach

Recently the non-perturbative numerical bootstrap method is rapidly developing, especially its application in nonlinear quantum mechanics, many-body physics, lattice models and matrix models. Here we use this numerical bootstrap to study the non-perturbative instanton effects in symmetric double-well potentials, using an efficient implementation of the algorithm proposed recently by Aikawa, Morita and Yoshimura. The ground state level splitting, caused by instantons, is computed for the quartic and the sextic case, where the coupling constant $g$ characterizes the strength of instanton effects. Inspired by well-known perturbative results that is only valid at large $g$ value under the dilute-gas limit, a qualitative formula is proposed for the ground state level splitting across all values of $g$, which agrees well with the bootstrap data. It has the expected behavior at both large value $|g|\to \infty$ and small value $|g|\to 0$, so it describes both the 'weak' and the 'strong' regimes of instanton effects. This qualitative formula is non-perturbative and can not be obtained by loop summation alone. It is beyond the dilute-gas approximation, and might be from a renormalization-group-like procedure or other undiscovered analytic methods.