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

Abstract

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