A note on eigenvalues of ordinary differential operators (Q1382012)

In [\textit{C. Fefferman} and \textit{L. Seco}, Adv. Math. 95, No. 2, 145-305 (1992; Zbl 0797.34083)] it is shown that, for certain functions V, the asymptotic formula obtained by the standard WKB method for the eigenvalues \(E\) of the differential operator \[ -{d^2 \over dx^2} +V(x) \] can be improved and the general result reduces to one of the form \[ \int \bigl(E- V(x) \bigr)^{1/2}_+ dx+ih_1(E) =\pi\Bigl(k+ \textstyle {{1\over 2}} \Bigr)+ O (\Lambda^{-2}), \] where \(h_1(E)\) and \(\Lambda= \Lambda(E)\) are given explicitly, with \(h_1 (E)= O (\Lambda^{-1})\). The author proves that the error can be improved to \(O (\Lambda^{-3}) \) for a class of large, slowly varying potentials.