离散集上具有局部相互作用的1-D薛定谔算子
1--D Schrödinger operators with local interactions on a discrete set
摘要：
研究了离散集$X=\{x_n\}_{n=1}^\infty$上具有局部点相互作用的一维薛定谔算子$\mathrm{H}_{X,α}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}α_nδ(x-x_n)$的谱性质，当$d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$ .我们的论文致力于案例$d_*=0$。我们考虑$\mathrm{H}_{X,α}$的对称算子的扩张理论的框架下，应用边界三元组和相应的Weyl函数的技术。我们证明了$\mathrm{H}_{X,α}$的谱性质如自伴性、离散性和下半有界性与某些Jacobi矩阵类的相应谱性质相关。基于这种联系，我们得到了在$d_*=0$情况下算子$\mathrm{H}_{X,α}$是自伴的、下半界的和离散的充要条件。研究了具有$δ'$型相互作用的算子。结果表明，在$d_*=0$情形下，与$d_*>0$情形不同，具有$δ$和$δ'$型相互作用的算符的谱性质有很大的不同.
英文摘要：
Spectral properties of 1-D Schrödinger operators $\mathrm{H}_{X,α}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}α_nδ(x-x_n)$ with local point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty$ are well studied when $d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. We consider $\mathrm{H}_{X,α}$ in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of $\mathrm{H}_{X,α}$ like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators $\mathrm{H}_{X,α}$ to be self-adjoint, lower-semibounded, and discrete in the case $d_*=0$. The operators with $δ'$-type interactions are investigated too. The obtained results demonstrate that in the case $d_*=0$, as distinguished from the case $d_*>0$, the spectral properties of the operators with $δ$ and $δ'$-type interactions are substantially different.
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