In this second paper on quantum fluctuations near the classical instanton configurations, see {em Phys. Rev. D bf 92}, 025046 (2015) and arXiv:1501.03993, we focus on another well studied quantum-mechanical problem, the one-dimensional Sine-Gordon po
tential (the Mathieu potential). Using only the tools from quantum field theory, the Feynman diagrams in the instanton background, we calculate the tunneling amplitude (the instanton density) to the three-loop order. The result confirms (to seven significant figures) the one recently recalculated by G. V. Dunne and M. {U}nsal, {it Phys. Rev. bf D 89}, 105009 (2014) from the resurgence perspective. As in the double well potential case, we found that the largest contribution is given by the diagrams originating from the Jacobian. We again observe that in the three-loop case individual Feynman diagrams contain irrational contributions, while their sum does not.
This paper deals with quantum fluctuations near the classical instanton configuration. Feynman diagrams in the instanton background are used for the calculation of the tunneling amplitude (the instanton density) in the three-loop order for quartic do
uble-well potential. The result for the three-loop contribution coincides in six significant figures with one given long ago by J.~Zinn-Justin. Unlike the two-loop contribution where all involved Feynman integrals are rational numbers, in the three-loop case Feynman diagrams can contain irrational contributions.
It is shown that the molecular ion ${rm H}^{++}_3$ does not exist in a form of the equilateral triangle. To this end, a compact variational method is presented which is based on a linear superposition of six specially tailored trial functions contain
ing non-linear parameters. Careful optimization of a total of fifteen parameters gives consistently lower variational results for the electronic energy than can be obtained with standard methods of quantum chemistry even with huge basis sets as large as mcc-pV7Z.
