{"title":"Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with Boundary Conditions Dependent on the Eigenparameter","authors":"Xiang Zhu, Zhao Wen Zheng, Kun Li","doi":"10.1007/s10114-023-1413-6","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider the following quadratic pencil of Schrödinger operators <i>L</i>(λ) generated in <span>\\({L^2}({\\mathbb{R}^ + })\\)</span> by the equation </p><div><div><span>$$ - {y^{\\prime \\prime }} + [p(x) + 2\\lambda q(x)]y = {\\lambda ^2}y,\\,\\,\\,\\,\\,x \\in {\\mathbb{R}^ + } = [0, + \\infty )$$</span></div></div><p> with the boundary condition </p><div><div><span>$${{{y^\\prime }(0)} \\over {y(0)}} = {{{\\beta _1}\\lambda + {\\beta _0}} \\over {{\\alpha _1}\\lambda + {\\alpha _0}}},$$</span></div></div><p> where <i>p</i>(<i>x</i>)and <i>q</i>(<i>x</i>) are complex valued functions and <i>α</i><sub>0</sub>, <i>α</i><sub>1</sub>, <i>β</i><sub>0</sub>, <i>β</i><sub>1</sub> are complex numbers with <span>\\({\\alpha _0}{\\beta _1} - {\\alpha _1}{\\beta _0} \\ne 0\\)</span>. It is proved that <i>L</i>(λ) has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity, if the conditions </p><div><div><span>$$p(x),{q^\\prime }(x) \\in AC({\\mathbb{R}^ + }),\\,\\,\\,\\,\\,\\,\\,\\mathop {\\lim }\\limits_{x \\to \\infty } [|p(x)| + |q(x)| + |{q^\\prime }(x)|] = 0$$</span></div></div><p> and </p><div><div><span>$$\\mathop {\\sup }\\limits_{0 \\le x < + \\infty } \\{ {{\\rm{e}}^{\\varepsilon \\sqrt x }}[|{p^\\prime }(x)| + |{q^{\\prime \\prime }}(x)|]\\} < + \\infty $$</span></div></div><p> hold, where <i>ε</i>> 0.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-023-1413-6","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper, we consider the following quadratic pencil of Schrödinger operators L(λ) generated in \({L^2}({\mathbb{R}^ + })\) by the equation
where p(x)and q(x) are complex valued functions and α0, α1, β0, β1 are complex numbers with \({\alpha _0}{\beta _1} - {\alpha _1}{\beta _0} \ne 0\). It is proved that L(λ) has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity, if the conditions
期刊介绍:
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.