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-valued 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 $\mathcal{A}_r = L^2(\mathbb{R}_+,(1+x)^r dx)$ 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 $\omega_r:\mathbb{N}\to\mathbb{R}$ 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 $\lambda_n(q) = -a_n + \pi (-a_n)^{-1/2}\int_0^\infty \text{Ai}^2(x+a_n)q(x)dx + O(n^{-1/3}\omega_r^2(n))$ and $\kappa_n(q) = - 2\pi (-a_n)^{-1/2}\int_0^\infty \text{Ai}(x+a_n)\text{Ai}'(x+a_n)q(x)dx + O(\omega_r^3(n))$, uniformly on bounded subsets of $\mathfrak{A}_r$. In order to obtain these asymptotic formulas, we first show that $\lambda_n:\mathcal{A}_r\to\mathbb{R}$ and $\kappa_n:\mathcal{A}_r\to\mathbb{R}$ are real analytic maps.