Estimations de l'effet tunnel pour le double puits. (Estmates of tunneling for the double wells) (Q1058059)

This paper is devoted to the Schrödinger operator \(P=-h^ 2\Delta +V\), (h small), in the case where V is a \(C^{\infty}\) function, symmetrical about a hyperplane of \({\mathbb{R}}^ n\), and admitting two non-degenerate minima (called wells). Using the WKB method and introducing a pseudo- differential calculus adapted to the WKB developments, we find estimates of the splitting for the energy levels which are O(h). The lower bound we obtain is of order \(h^{1/2}e^{-S_ 0/h}\) (where \(S_ 0\) is the Agmon distance between the wells), while the upper bound is of order \(h^{-a}e^{-S_ 0/h}\) where \(a\geq 0\) depends on the energy level we consider. We also find an upper bound involving the modified Agmon metric associated to \(V-E_ 0h\), where \(E_ 0h\) is the first term in the WKB development of the eigenvalue.