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Differential Equations

, Volume 55, Issue 2, pp 169–180 | Cite as

Distribution of the Spectrum of a Singular Sturm-Liouville Operator Perturbed by the Dirac Delta Function

  • A. S. PechentsovEmail author
  • A. Yu. Popov
Ordinary Differential Equations
  • 9 Downloads

Abstract

We consider the Sturm-Liouville operator generated in the space L2[0,+∞) by the expression −d2/dx2 + x + (xb), where δ is the Dirac delta function, a < 0, and b > 0, and the boundary condition y(0) = 0. We prove that the eigenvalues λn of this operator satisfy the inequalities λ1 < λ 1 0 and λ n−1 0 < λnλ n 0 , n = 2, 3,..., where {−λ n 0 } is the sequence of zeros of the Airy function Ai (λ). The problem on the location of the first eigenvalue λ1 depending on the parameters a and b is solved. In particular, we obtain conditions under which λ1 is negative and provide a lower bound for λ1.

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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