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Semiclassical approximation for the non-self-adjoint Sturm-Liouville problem with the potential \(q(x) = x^{4} - a^{2}x^{2}\). (English. Russian original) Zbl 1181.34093

Math. Notes 85, No. 5, 755-759 (2009); translation from Mat. Zametki 85, No. 5, 792-796 (2009).
Summary: We study the family of differential operators \[ L(\varepsilon)y=i\varepsilon y''+(x^4-a^2x^2)y,\quad \varepsilon>0, \]
acting in the space \(L_2(\mathbb R)\), where \(\varepsilon\) is a small parameter. Since \(q(x)\to\infty\) as \(x\to\pm\infty\), the spectrum of this operator is discrete for any \(\varepsilon > 0\). The goal in this paper is to show that, for small values of \(\varepsilon\), the eigenvalues of the operator accumulate near some curves in the complex plane; we present them explicitly.

MSC:

34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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References:

[1] M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations, in Mathematical Reference Library (Nauka, Moscow, 1983) [in Russian]. · Zbl 0538.34001
[2] S. N. Tumanov and A. A. Shkalikov, Izv. Ross. Akad. Nauk, Ser. Mat. 66(4), 177 (2002) [Russian Acad. Sci. Izv. Math. 66 (4), 829 (2002)]. · doi:10.4213/im399
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