Asymptotics of the spectral data of perturbed Stark operators in the half-line with mixed boundary conditions

We obtain sharp asymptotic formulas for the eigenvalues and norming constants of Sturm-Liouville operators associated with the differential expression \[ -d^2/dx^2 + x + q(x), x∈ [0,∞), \] together with the boundary condition $φ'(0) - bφ(0) =0$, b, where \[ q∈ \{ p∈ L^2R(R_+,(1+x)^r dx) : p'∈ L^2R(R_+,(1+x)^r dx)\} \] with $r>1$.