Borel summability of the unequal double well (Q5903179)

For \(\epsilon >0\) and \(g\in {\mathbb{C}}\), \(| \arg g| \leq \pi /4\) let \(H(p,\epsilon)=p^ 2+x^ 2(1+gx)^ 2+\epsilon g^ 2x^ 4\) be the oscillator of the potential of the unequal double well. The stability in the sense of T. Kato of any eigenvalue of H(0,\(\epsilon)\) at \(\alpha >0\), \(| g| <\alpha\); the holomorphy of eigenvalues of H(g,\(\epsilon)\) in the sector \(\{| g| <\alpha,| \arg g| \leq \pi /4\}\) and the Borel summability of the Rayleigh-Schrödinger perturbation expansion for the perturbed eigenvalues are proved.