  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.
