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 Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 3, Pages 151–182 (Mi izv1969)

Asymptotic behaviour of the positive spectrum of a family of periodic Sturm–Liouville problems under continuous passage from a definite problem to an indefinite one

D. A. Popov

A. N. Belozersky Institute of Physico-Chemical Biology, M. V. Lomonosov Moscow State University

Abstract: We consider the problem of the spectrum of a parameter-dependent family of periodic Sturm–Liouville problems for the equation $u"+\lambda^2(g(x)-a)u=0$, where $a\in\mathbb R$ is the parameter of the family and $\lambda$ is the spectral parameter. It is assumed that $g\colon\mathbb R\to\mathbb R$ is a sufficiently smooth $2\pi$-periodic function with one simple maximum $g(x_{\max})= a_1>0$ and one simple minimum $g(x_{\min})=a_2>0$ over a period, and that the functions $g(x-x_{\min})$ and $g(x-x_{\max})$ are even. Under these assumptions, the first two asymptotic terms are calculated explicitly for the positive eigenvalues on the whole interval $0\le a<a_1$, including the neighbourhoods of the points $a=a_1$ and $a=a_2$. For $\lambda\gg1$, it is shown that the spectrum consists of two branches $\lambda=\lambda_{\pm}(a,p)$, indexed by the signs $\pm$ and by an integer $p\in\mathbb Z^+$, $p\gg1$. A unified interpolation formula is derived to describe the asymptotic behaviour of the spectrum branches in the passage from the definite (classical) problem with $a<a_2$ to the indefinite problem with $a>a_2$.

Keywords: definite and indefinite Sturm–Liouville problems, asymptotic behaviour of the spectrum, turning points.

DOI: https://doi.org/10.4213/im1969

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English version:
Izvestiya: Mathematics, 2009, 73:3, 579–610

Bibliographic databases:

UDC: 517.927.25
MSC: 11F72, 34B24, 34C10, 34D05, 34E05, 34E20, 34E99, 34L10, 34L15, 35P05, 41A60, 46C20, 46N50, 46N99, 47A10, 47A15, 47B50, 58J50, 81Q50

Citation: D. A. Popov, “Asymptotic behaviour of the positive spectrum of a family of periodic Sturm–Liouville problems under continuous passage from a definite problem to an indefinite one”, Izv. RAN. Ser. Mat., 73:3 (2009), 151–182; Izv. Math., 73:3 (2009), 579–610

Citation in format AMSBIB
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• https://doi.org/10.4213/im1969
• http://mi.mathnet.ru/eng/izv/v73/i3/p151

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This publication is cited in the following articles:
1. D. A. Popov, “On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains”, Izv. Math., 75:5 (2011), 1007–1045
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