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

Abstract

We consider the perturbed Stark operator $H_q\phi =-\phi "+x\phi +q(x)\phi$, $\phi (0)=0$, in $L^2({\mathbb{R}}_{+})$, where $q$ is a real-valued function that belongs to ${\mathfrak{A}}_r=\left\{q\in {\mathcal{A}}_r\cap \text{AC}[0,\infty ):q^{\prime }\in {\mathcal{A}}_r\right\}$, 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 ${\kappa }_n(q)=-2\pi (-a_n{)}^{-1/2}{\int }_0^{\infty }\text{Ai}(x+a_n){\text{Ai}}^{\prime }(x+a_n)q(x)dx+O({\omega }_r^3(n))$, 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.

32 pages, typos corrected, some minor correction in statement of Lemma 6.1 and in proofs in Section 6. Main results unchanged

References in corpus (3)

• The inverse problem for perturbed harmonic oscillator on the half-line with Dirichlet boundary conditions
• Asymptotics of resonances for 1d Stark operators
• Criteria for eigenvalues embedded into the absolutely continuous spectrum of perturbed Stark type operators