Geodesic
Asymptotic behavior of the eigenvalues of an anharmonic oscillator
Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 151-163

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In this paper we study the properties of the spectrum of the boundary-value problem $$ \varphi''+[\lambda-x^2-V(x)]\varphi=0,\quad-\infty\infty. $$ 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. 
@article{SM_1970_10_2_a0,
     author = {N. M. Kostenko},
     title = {Asymptotic behavior of the eigenvalues of an anharmonic oscillator},
     journal = {Sbornik. Mathematics},
     pages = {151--163},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {1970},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_10_2_a0/}
}
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N. M. Kostenko. Asymptotic behavior of the eigenvalues of an anharmonic oscillator. Sbornik. Mathematics, Tome 10 (1970) no. 2, pp. 151-163. http://geodesic.mathdoc.fr/item/SM_1970_10_2_a0/