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 $$-\frac{d^2}{d{x}^2}+x+q(x),x\in [0,\mathrm{\infty }),$$ together with the boundary condition ${\phi }^{\mathrm{\prime }}(0)-b\phi (0)=0$, $b\in \mathbb{R}$, where $$q\in \{p\in L_{\mathbb{R}}^2({\mathbb{R}}_{+},(1+x{)}^{r}dx):p^{\mathrm{\prime }}\in L_{\mathbb{R}}^2({\mathbb{R}}_{+},(1+x{)}^{r}dx)\}$$ with $r>1$.