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

Boundary value problems

Xiaojing Zhao¹  Guoliang Shi²  Xiaoyun Liu² 

[1] Department of Mathematics, Anyang Institute of Technology, Anyang, P.R. China;Department of Mathematics, Tianjin University, Tianjin, P.R. China

关键词: vectorial Schrödinger operators;    δ-interactions;    self-adjointness;    discrete spectrum;    34B24;    34L05;    47e05;   

DOI  :  10.1186/s13661-016-0536-4

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

This paper deals with the vectorial Schrödinger operators with δ-interactions generated byLX,A,Q:=−d2dx2+Q(x)+∑k=1∞Akδ(x−xk)$L_{X,A,Q}:=-\frac{d^{2}}{dx^{2}} +Q(x)+\sum_{k=1}^{\infty}A_{k}\delta(x-x_{k})$,x∈[0,+∞)$x\in[ 0,+\infty)$. First, we obtain an embedding inequality. Then using standard form methods, we prove that the operatorHX,A,Q$\mathbf{H}_{X,A,Q}$given in this paper is self-adjoint. Finally, a sufficient condition and a necessary condition are given for the spectrum of the operatorHX,A,Q$\mathbf {H}_{X,A,Q}$to be discrete. By giving additional restrictions on the symmetric potential matrixQ(x)$Q(x)$andAk$A_{k}$, we also give a necessary and sufficient condition for a special case. The conditions are analogous to Molchanov’s discreteness criteria.

【 授权许可】

CC BY