Geodesic
The asymptotic expansion of the spectrum of a Sturm–Liouville operator
Sbornik. Mathematics, Tome 38 (1981) no. 1, pp. 127-141 
Kh. Kh. Murtazin; T. G. Amangil'din. The asymptotic expansion of the spectrum of a Sturm–Liouville operator. Sbornik. Mathematics, Tome 38 (1981) no. 1, pp. 127-141. http://geodesic.mathdoc.fr/item/SM_1981_38_1_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper studies the properties of the spectrum of the problem \begin{gather*} -y''(x)+q(x)y(x)=\lambda y(x),\qquad x > 0,\\ y(0)=0,\quad y(x)\in L_2[0,\infty), \end{gather*} under the assumption that $q(x)$ grows like a power of $x$ at $\infty$, while allowing that $q(x)$ may have a nonintegrable singularity at $0$. A result which lets one write down the first few terms of an asymptotic series for the eigenvalues is obtained. Bibliography: 8 titles.

[1] M. Giertz, “On the solutions in $L_2(-\infty,\infty)$ of $y''+(\lambda-q(x))y=0$ when $q$ is rapidly increasing”, Proc. London Math. Sec. Ser. 3, 14:53 (1964), 53–73 | DOI | MR | Zbl

[2] V. I. Smirnov, Kurs vysshei matematiki, t. 3 (ch. 2), izd. 9-e, izd-vo “Nauka”, Moskva, 1974 | MR

[3] A. G. Alenitsyn, “Asimptotika spektra operatora Shturma–Liuvillya v sluchae predelnogo kruga”, Diff. uravneniya, 12:3 (1976), 428–437 | MR | Zbl

[4] V. M. Babich, V. S. Buldyrev, Asimptoticheskie metody v zadachakh difraktsii korotkikh voln. Metod etalonnykh zadach, izd-vo “Nauka”, Moskva, 1972 | MR

[5] G. B. Dvait, Tablitsy integralov i drugie matematicheskie formuly, izd-vo “Nauka”, Moskva, 1977

[6] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, t. 1, IL, Moskva, 1960

[7] L. A. Sakhnovich, “O spektre angarmonicheskogo ostsillyatora”, Izv. AN SSSR, seriya matem., 28 (1964), 1345–1362 | MR

[8] N. M. Kostenko, “Asimptotika sobstvennykh chisel angarmonicheskogo ostsillyatora”, Matem. sb., 81(123) (1970), 163–175 | MR | Zbl