Geodesic
Asymptotic behavior of the spectrum of an~anharmonic oscillator
Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 2, pp. 266-276

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The asymptotic expansion $$ \lambda_n=n-\frac{1}{2}+\frac{1}{2\pi\sqrt n}\biggl[\int_{-\infty}^{\infty}q(t)d(t)+o(1)\biggr], \quad n\to\infty, $$ is obtained for the spectrum of the equation $-y^{''}+[x^2/4+q(x)]y=\lambda y$, $-\infty$, of the anharmonic oscillator. The ease when the potential $v(x)$ has the form $v(x)=\alpha|x|+q(x)$ is also considered. 
@article{TMF_1981_47_2_a11,
     author = {L. A. Sakhnovich},
     title = {Asymptotic behavior of the spectrum of an~anharmonic oscillator},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {266--276},
     publisher = {mathdoc},
     volume = {47},
     number = {2},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1981_47_2_a11/}
}
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L. A. Sakhnovich. Asymptotic behavior of the spectrum of an~anharmonic oscillator. Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 2, pp. 266-276. http://geodesic.mathdoc.fr/item/TMF_1981_47_2_a11/