On \(L^2(-\infty,\infty)\) consider the one-dimensional Schrödinger operator \[ H(g)=-\frac12\;\frac{d^2}{dx^2}+\frac12\;x^2+\frac{g^{m-1}x^{2m}}{1+\alpha gx^2}, \] where \(m\) is a positive integer, \(\alpha >0\) and \(g\geq0\). The case \(m=1,2,3\) occurs in physical models of laser theory and quantum field theory. In this paper, the dependence of the eigenvalues on \(g\) for sufficiently small \(g\) is investigated. Using the idea and methods of [G. Auberson, Commun.Math.Phys., 84, 531–546 (1982; Zbl 0508.34042)] established for the cases \(m=2\), it is shown that for all \(m\geq3\) the \(j\)-th eigenvalue \(E_j(g)\) depend analytically on the complexified parameter \(g\) in sectors \(S_j\{g|0<|g|\leq r_j,\, |\arg g|<\theta<\pi\}\), and an asymptotic expansion of \(E_ j(g)\) is given.