Asymptotic behavior of the eigenvalues of an anharmonic oscillator

In this paper we study the properties of the spectrum of the boundary-value problem
$$ \varphi''+[\lambda-x^2-V(x)]\varphi=0,\quad-\infty<x<\infty. $$
Let $\lambda_k$ be the points of the spectrum of this problem, arranged in order of increasing absolute value. Our main result is
Theorem. {\it Let $V(x)$ satisfy the conditions
$$ |V(x)|\leqslant M,\quad|x|\leqslant L;\qquad|V(x)|\leqslant\frac M{|x|},\quad|x|>L. $$
Then for any $\varepsilon>0$
$$ |\lambda_k-2k-1|=o(k^{-1/2+\varepsilon})\ \text{for}\ k\to\infty. $$
}
Bibliography: 2 titles.