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

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 λ_n of this operator satisfy the inequalities λ₁ < λ₁⁰ and λ_n−1⁰ < λ_n ≤ λ_n⁰, n = 2, 3,..., where {−λ_n⁰} 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.