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

TL;DR

This paper derives asymptotic formulas for the spectrum and norming constants of a perturbed Stark operator on the half-line, showing their dependence on the potential function and establishing their real analyticity.

Contribution

It provides new asymptotic formulas for spectral data of perturbed Stark operators and proves their real analyticity with respect to the potential function.

Findings

Asymptotic formulas for eigenvalues and norming constants are established.

Spectral data depend real analytically on the potential function.

Results hold uniformly on bounded subsets of the potential class.

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.