Double-well tunneling splitting ($E_{\pm }=E_{\mathrm{well}}\mp \hslash K{e}^{-S_I/\hslash }$)

A symmetric two-well system has nearly localized left/right states coupled by a positive quantum tunnelling magnitude $\Delta =\hslash K{e}^{-S_I/\hslash }$. Its effective Hamiltonian is $E_{\mathrm{well}}I-\Delta {\sigma }_x$, so the superposition with an even spatial wavefunction is lower, and the level splitting is $2\Delta$. The dilute instanton gas reproduces these two exponents through even/odd crossing sums.