Abstract
We consider the Sturm-Liouville operator generated in the space L2[0,+∞) by the expression −d2/dx2 + x + aδ(x − b), 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 01 and λ 0 n−1 < λn ≤ λ 0 n , n = 2, 3,..., where {−λ 0 n } 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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Russian Text © A.S. Pechentsov, A.Yu. Popov, 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 2, pp. 168–179.
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Pechentsov, A.S., Popov, A.Y. Distribution of the Spectrum of a Singular Sturm-Liouville Operator Perturbed by the Dirac Delta Function. Diff Equat 55, 169–180 (2019). https://doi.org/10.1134/S0012266119020034
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DOI: https://doi.org/10.1134/S0012266119020034