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Recently the non-perturbative numerical bootstrap method is rapidlydeveloping, especially its application in nonlinear quantum mechanics,many-body physics, lattice models and matrix models. Here we use this numericalbootstrap to study the non-perturbative instanton effects in symmetricdouble-well potentials, using an efficient implementation of the algorithmproposed recently by Aikawa, Morita and Yoshimura. The ground state levelsplitting, caused by instantons, is computed for the quartic and the sexticcase, where the coupling constant $g$ characterizes the strength of instantoneffects. Inspired by well-known perturbative results that is only valid atlarge $g$ value under the dilute-gas limit, a qualitative formula is proposedfor the ground state level splitting across all values of $g$, which agreeswell 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' andthe 'strong' regimes of instanton effects. This qualitative formula isnon-perturbative and can not be obtained by loop summation alone. It is beyondthe dilute-gas approximation, and might be from a renormalization-group-likeprocedure or other undiscovered analytic methods.

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