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

We consider the Sturm-Liouville operator generated in the space L (2)[0,+a) by the expression l (a,b):= -d (2)/dx (2) +x+a delta(x-b) and the boundary condition y(0) = 0. We prove that the eigenvalues lambda (n) of this operator satisfy the inequalities lambda(1) (0) < lambda(1) < lambda(2) (0) and lambda(n) (0) ae<currency> lambda(n) < lambda(n+1) (0), n = 2, 3,..., where {-lambda(n) (0)} is the sequence of zeros of the Airy function Ai (lambda). We find the asymptotics of lambda(n) as n -> +a depending on the parameters a and b.