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Under an adiabatic compactification on an $S^1$, we obtain a quantum mechanics with an elliptic Lam\\'e-like potential.\n  We perform a perturbative calculation of the ground state energy in this quantum mechanics to large orders obtaining an asymptotic series. Using the Borel-Pad\\'e technique, we determine the expected locations of branch cuts in the Borel plane of the perturbative series and show that they match the values of the uniton actions. 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