The Dirichlet problem for perturbed Stark operators in the half-line

We consider the perturbed Stark operator $$H_q\varphi = -\varphi '' + x\varphi q(x)\varphi $$ , $$\varphi (0)=0$$ in $$L^2({\mathbb R}_+)$$ where q is a real function that belongs to $${\mathfrak {A}}_r =\left\{ q\in \mathcal {A}_r\cap \text {AC}[0,\infty ): q'\in {A}_r\right\} $$\mathcal {A}_r L^2_{\mathbb R}({\mathbb R}_+,(1+x)^r dx)$$ and $$r>1$$ arbitrary but fixed. Let $$\left\{ \lambda _n(q)\right\} _{n=1}^\infty \kappa _{n=1}^ \infty be spectrum associated set of norming constants $$H_q$$ . $$\{a_n\}_{n=1}^\infty zeros Airy first kind, let $$\omega _r:{\mathbb N}\rightarrow {\mathbb R}$$ defined by rule _r(n) n^{-1/3}\log ^{1/2}n$$ if $$r\in (1,2)$$ n^{-1/3}$$ [2,\infty )$$ prove $$\lambda _n(q) -a_n \pi (-a_n)^{-1/2}\int _0^\infty {{\,\textrm{Ai}\,}}^2(x+a_n)q(x)dx O(n^{-1/3}\omega _r^2(n))$$ $$\kappa - 2\pi {{\,\textrm{Ai}\,}}(x+a_n){{\,\textrm{Ai}\,}}'(x+a_n)q(x)dx O(\omega _r^3(n))$$ uniformly on bounded subsets {A}}_r$$ In order obtain these asymptotic formulas, we show _n:\mathcal {A}_r\rightarrow are analytic maps.