论文标题
半线扰动的史塔克运营商的迪利奇问题
The Dirichlet problem for perturbed Stark operators in the half-line
论文作者
论文摘要
我们考虑扰动的stark操作员$h_qφ=-φ”+xφ+ q（x）φ$，$φ（0）= 0 $，in $ l^2（\ Mathbb {r} _+）$，其中$ q $是属于$ \ \ \ \ \ \ \ \ \ a} _ a} _r = \ eft左= \ eft \ eft的真实功能q \ in \ Mathcal {a} _r \ cap \ cap \ text {ac} [0，\ infty）：q'\ in \ nathcal {a} _r \ right \ right \} $，其中$ \ mathcal {a} _r {a} _r = _r = l^l^2（但是修复了$ \ {a_n \} _ {n = 1}^\ infty $是第一类通风函数的零，让$ω_r：\ mathbb {n} \ to \ athbb {r} $由规则$ $ω_r（n）= n^n^n^n^n^n^n^{r}定义$ r \ in（1,2）$和$ω_r（n）= n^{ - 1/3} $如果$ r \ in [2，\ infty）$。 o（n^{ - 1/3}ω_r^2（n））$和$κ_n（q）=-2π（-a_n）^{ - 1/2} \ int_0^\ int_0^\ infty \ infty \ text {ai}（a+a_n）（x+a_n）在$ \ mathfrak {a} _r $的有界子集上，我们首先表明$λ_n：\ Mathcal {a} _r \ to \ Mathbb {r} $
We consider the perturbed Stark operator $H_qφ= -φ" + xφ+ q(x)φ$, $φ(0)=0$, 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 $\left\{λ_n(q)\right\}_{n=1}^ \infty$ and $\left\{κ_n(q)\right\}_{n=1}^ \infty$ 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 $ω_r:\mathbb{N}\to\mathbb{R}$ be defined by the rule $ω_r(n) = n^{-1/3}\log^{1/2}n$ if $r\in(1,2)$ and $ω_r(n) = n^{-1/3}$ if $r\in[2,\infty)$. We prove that $λ_n(q) = -a_n + π(-a_n)^{-1/2}\int_0^\infty \text{Ai}^2(x+a_n)q(x)dx + O(n^{-1/3}ω_r^2(n))$ and $κ_n(q) = - 2π(-a_n)^{-1/2}\int_0^\infty \text{Ai}(x+a_n)\text{Ai}'(x+a_n)q(x)dx + O(ω_r^3(n))$, uniformly on bounded subsets of $\mathfrak{A}_r$. In order to obtain these asymptotic formulas, we first show that $λ_n:\mathcal{A}_r\to\mathbb{R}$ and $κ_n:\mathcal{A}_r\to\mathbb{R}$ are real analytic maps.
