Discreteness of the spectrum of vectorial Schrödinger operators with δ-interactions

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• 作者：Xiaoyun Liu ; Xiaojing Zhao ; Guoliang Shi
• 关键词：34B24 ; 34L05 ; 47e05 ; vectorial Schrödinger operators ; δ ; interactions ; self ; adjointness ; discrete spectrum
• 刊名：Boundary Value Problems
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：2016
• 期：1
• 全文大小：1,663 KB
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• 作者单位：Xiaoyun Liu (1)
  Xiaojing Zhao (2)
  Guoliang Shi (1)
  
  1. Department of Mathematics, Tianjin University, Tianjin, 300072, P.R. China
  2. Department of Mathematics, Anyang Institute of Technology, Anyang, 455000, P.R. China
  
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770

文摘

This paper deals with the vectorial Schrödinger operators with δ-interactions generated by \(L_{X,A,Q}:=-\frac{d^{2}}{dx^{2}} +Q(x)+\sum_{k=1}^{\infty}A_{k}\delta(x-x_{k})\), \(x\in[ 0,+\infty)\). First, we obtain an embedding inequality. Then using standard form methods, we prove that the operator \(\mathbf{H}_{X,A,Q}\) given in this paper is self-adjoint. Finally, a sufficient condition and a necessary condition are given for the spectrum of the operator \(\mathbf {H}_{X,A,Q}\) to be discrete. By giving additional restrictions on the symmetric potential matrix \(Q(x)\) and \(A_{k}\), we also give a necessary and sufficient condition for a special case. The conditions are analogous to Molchanov’s discreteness criteria. Keywords vectorial Schrödinger operators δ-interactions self-adjointness discrete spectrum