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Eigenvalues and Eigenfunctions of a Schr\"{o}dinger Operator Associated with a FiniteCombination of Dirac-Delta Functions and CH Peakons

KeyWord:Schr\"{o}dinger operator; Boundary conditions; Soliton; Peakon solution; Cammassa-Holm equation
 Author Name Affiliation Shouzhong Fu School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, Guangdong 526061, China Zhijun Qiao School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edingburg, TX 78539, USA Zhong Wang Zhongshan Polytechnic College, Zhongshan, Guangdong 528400, China
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In this paper, we first study the Schr\"{o}dinger operators with the following weighted function $\sum\limits_{i=1}^n p_i \delta(x - a_i)$, which is actually a finite linear combination of Dirac-Delta functions, and then discuss the same operator equipped with the same kind of potential function. With the aid of the boundary conditions, all possible eigenvalues and eigenfunctions of the self-adjoint Schr\"{o}dinger operator are investigated. Furthermore, as a practical application, the spectrum distribution of such a Dirac-Delta type Schr\"{o}dinger operator either weighted or potential is well applied to the remarkable integrable Camassa-Holm (CH) equation.