Borel summability of the unequal double well (Q5903569)

The best known example of a non-Borel summable perturbation series is represented by the Rayleigh-Schrödinger perturbation expansion (RSPE) of the standard double well oscillator \(H(g)=p^ 2+x^ 2+x^ 2(1+gx)^ 2\) in \(L^ 2(\mathbb R)\), \(g\in \mathbb R\). This fact is of course due to the instability of the eigenvalues as \(g\to 0\), i.e., to their asymptotic degeneracy as \(g\to 0\). However there are examples, such as that of \textit{I. W. Herbst} and \textit{B. Simon} [Phys. Lett. B 78, 304--306 (1978)] \(K(g)=p^ 2+x(1+gx)^ 2-2gx-1,\) in which there is stability but no Borel summability to the eigenvalues. Hence, also on account of recent investigations on Borel summability in four-dimensional field theories, it could be interesting to relate the lack of summability to some other more subtle physical mechanism of well-defined meaning also in a more general context. To this end, T. Spencer has suggested considering the following `unequal' double well oscillator: \((1)\quad H(g,\epsilon)=p^ 2+x^ 2(1+gx)^ 2+\epsilon g^ 2x^ 4,\) which in the limit \(g\to 0\) has an infinite action instanton for any \(\varepsilon\geq 0\). This model could in addition be of interest in itself: as a matter of fact, in some sense it represents the slightest modification of the nonsummable example, and it is natural to ask to what extent the nonsummability is `accidental', i.e., how sensitive is its dependence on the choice of the parameters in \(H(g)\)? Furthermore it can be easily proved through the Hunziker-Vock technique [\textit{W.Hunziker} and \textit{E. Vock}, Commun. Math. Phys. 83, 281--302 (1982; Zbl 0528.35023)] that any value \(E\) of \(H(0,\varepsilon)\equiv H(0)=p^ 2-x^ 2\) is stable for \(g\in\mathbb R\) small as an eigenvalue of H(g,\(\epsilon)\), \(\epsilon >0\), because the second minimum of \(V(g,\varepsilon)\equiv x^ 2(1+gx)^ 2+\varepsilon g^ 2x^ 4\) tends to \(+\infty\) as \(g\to 0\), \(\varepsilon >0\). In this note we prove that, for \(\varepsilon >0\), any eigenvalue \(E\) is actually stable as an eigenvalue \(E(g,\varepsilon)\) of \(H(g,\epsilon)\) for \(g\) complex, \(| g|\) suitably small, \(| \arg g| \leq \pi /4,\) and that the RSPE near \(E\) is Borel summable to \(E(g,\varepsilon)\) for \(g\) positive and small.