Distribution of the spectrum of a singular positive 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 l_a,b:= −d²/dx² +x+aδ(x−b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ_n of this operator satisfy the inequalities λ₁⁰ < λ₁ < λ₂⁰ and λ_n⁰ ≤ λ_n < λ_n+1⁰, n = 2, 3,..., where {−λ_n⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.