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On the asymptotic behaviour of solutions of the Dirac system and applications to the Sturm–Liouville problem with a singular potential
Journal of Spectral Theory ( IF 1.0 ) Pub Date : 2020-05-27 , DOI: 10.4171/jst/311
Alexander Gomilko 1 , Łukasz Rzepnicki 1
Affiliation  

The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval [0,1]: $$D'(x)+\begin{2bmatrix} 0 & \sigma_1(x)\\ \sigma_2(x) & 0 \end{2bmatrix} D(x)=i\mu\begin{2bmatrix} 1 & 0\\ 0 &-1 \end{2bmatrix}D(x),\quad D(0)=\begin{2bmatrix} 1 & 0\\ 0 & 1 \end{2bmatrix},$$ where $\mu\in\mathbb{C}$ is a spectral parameter, and $\sigma_j\in L_2[0,1]$, $j=1,2$. We propose a new approach for the study of asymptotic behaviour of its solutions as $\mu\to \infty$ and $| \mathrm {Im} \;\mu | \le d$. As an application, we obtain new, sharp asymptotic formulas for eigenfunctions of Sturm–Liouville operators with singular potentials.

中文翻译:

Dirac系统解的渐近行为及其对奇异势Sturm-Liouville问题的应用

本文的主要焦点是区间[0,1]上Dirac系统的以下矩阵柯西问题:$$ D'(x)+ \ begin {2bmatrix} 0&\ sigma_1(x)\\ \ sigma_2( x)&0 \ end {2bmatrix} D(x)= i \ mu \ begin {2bmatrix} 1&0 \\ 0&-1 \ end {2bmatrix} D(x),\ quad D(0)= \ begin {2bmatrix} 1&0 \\ 0&1 \ end {2bmatrix},$$,其中$ \ mu \ in \ mathbb {C} $是频谱参数,$ \ sigma_j \ in L_2 [0,1] $, $ j = 1,2 $。我们提出了一种新的方法来研究其解的渐近行为,如$ \ mu \ to \ infty $和$ |。\ mathrm {Im} \; \ mu | \ le d $。作为一种应用,我们获得了具有奇异势的Sturm-Liouville算子的本征函数的新的,尖锐的渐近公式。
更新日期:2020-05-27
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