Higher order asymptotic distribution of the eigenvalues of nondefinite Sturm–Liouville problems with one turning point

In this paper, we investigate the asymptotic behavior of the differential equation y + λr(x) − q(x) y = 0, 0 x 1, where [0, 1] contains a finite number of zeros of r(x), the so-called turning points, λ is a real parameter and the function q(x) is bounded and integrable in [0, 1]. Using a technique used previously in [B.J. Harris, S.T. Talarico, On the eigenvalues of second-order linear differential equations with fractional transition points, Math. Proc. R. Ir. Acad. Ser. A 99 (1) (1999) 29-38], we derive the higher-order asymptotic distribution of the positive eigenvalues associated with this equation for the Dirichlet problem (i.e., y(0) = y(1) = 0). This method is different from Eberhard's method [W. Eberhard, G. Freiling, The distribution of the eigenvalues for second-order eigenvalues problems in the presence of an arbitrary number of turning points, Results Math. 21 (1992) 24-41]. They have used asymptotic solution in order to obtain asymptotic distribution of the eigenvalues. In most differential equations with variable coefficient it is impossible to obtain an exact solution, so we want to obtain asymptotic distribution of the eigenvalues without solving equation. Note that in similar case, the leading term of the asymptotic distribution of positive and negative eigenvalues was derived previously by Atkinson and Mingarelli [F.V. Atkinson, A. Mingarelli, Asymptotics of the number of zeros and the eigenvalues of general weighted SturmLiouville problems, J. Reine Angew. Math. 395 (1986) 380-93].

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“…We begin by consolidating some results from [10,15] for completeness. For a complex-valued solution Ω(x, λ) of y + λx α y = 0,…”

Section: The Main Resultsmentioning

confidence: 99%

Higher-order asymptotic formula for the eigenvalues of Sturm–Liouville problem with n turning points

Saei

Akbarfam

2010

Journal of Mathematical Analysis and Applications

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“…As can be expected our techniques are completely different from those in [3]. Other papers that have appeared in this connection include [5,[8][9][10].…”

Section: Introductionmentioning

confidence: 97%

“…It was shown in [8] that the higher order asymptotic expansion of the positive eigenvalues, {u n } ∞ 1 , associated with (1) for r(t) = t α , α = 4m ± 1, is a positive odd integer satisfies the relation…”

Section: Introductionmentioning

confidence: 99%

Duality for an indefinite inverse Sturm–Liouville problem

Akbarfam

Mingarelli

2005

Journal of Mathematical Analysis and Applications

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In this paper we solve an inverse spectral problem associated with a traditional Sturm-Liouville equation with an indefinite weight function. Specifically, we reconstruct a unique positive potential from two spectral sequences, given a weight function with a simple turning point in the interior of a finite interval with fixed end boundary conditions. We show the existence of special non-linear second order differential equations satisfied by these eigenvalue functions, dubbed the dual equations, eigenvalues whose asymptotics are used in the description of the reconstruction of the unknown potential.

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“…The literature concerning the inverse problem for equations of Sturm-Liouville type is wide, for example see [3,11,13]. For eigenvalue approximation and infinite product representation of solutions in a turning point case see [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

“…It follows that the functions λ n (x) ∈ C 2 (−1, 1) if q is sufficiently smooth. In fact, from [6] it follows that the asymptotic approximations of the eigenvalues is of the form…”

Section: Introductionmentioning

confidence: 99%

The uniqueness of the solution of dual equations of an inverse indefinite Sturm–Liouville problem

Kheiri

Akbarfam

Mingarelli

2005

Journal of Mathematical Analysis and Applications

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In this paper we consider a linear second-order equation of Sturm-Liouville type:with Dirichlet boundary conditions y(−1) = y(1) = 0, where q is a positive sufficiently smooth function on [−1, 1] and λ is a real parameter. We investigate the uniqueness of the solution for the dual equations of the indefinite inverse spectral problem (I). This result is necessary for expressing inverse problem of indefinite Sturm-Liouville equation.  2005 Elsevier Inc. All rights reserved.

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Higher order asymptotic distribution of the eigenvalues of nondefinite Sturm–Liouville problems with one turning point

Cited by 9 publications

References 10 publications

Higher-order asymptotic formula for the eigenvalues of Sturm–Liouville problem with n turning points

Higher-order asymptotic formula for the eigenvalues of Sturm–Liouville problem with n turning points

Duality for an indefinite inverse Sturm–Liouville problem

The uniqueness of the solution of dual equations of an inverse indefinite Sturm–Liouville problem

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