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Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

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Abstract

We consider the Sturm–Liouville operator generated in the space L 2[0,+∞) by the expression l a,b:= −d 2/dx 2 +x+aδ(x−b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ n of this operator satisfy the inequalities λ1 0 < λ1 < λ2 0 and λn 0 ≤ λn < λn+1 0, n = 2, 3,..., where {−λn 0} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λn as n → +∞ depending on the parameters a and b.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    A. S. Pechentsov

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  1. A. S. Pechentsov
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Correspondence to A. S. Pechentsov.

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Original Russian Text © A.S. Pechentsov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 8, pp. 1058–1063.

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Pechentsov, A.S. Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function. Diff Equat 53, 1029–1034 (2017). https://doi.org/10.1134/S0012266117080079

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  • DOI: https://doi.org/10.1134/S0012266117080079

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