1--D Schrödinger operators with local interactions on a discrete set

Abstract

Spectral properties of 1-D Schrödinger operators ${\mathrm{H}}_{X,\alpha }:=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+{\sum }_{x_n\in X}{\alpha }_n\delta (x-x_n)$ with local point interactions on a discrete set $X=\{x_n{\}}_{n=1}^{\infty }$ are well studied when $d_{\ast }:={\inf }_{n,k\in \mathbb{N}}|x_n-x_k|>0$. Our paper is devoted to the case $d_{\ast }=0$. We consider ${\mathrm{H}}_{X,\alpha }$ 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 ${\mathrm{H}}_{X,\alpha }$ 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 ${\mathrm{H}}_{X,\alpha }$ to be self-adjoint, lower-semibounded, and discrete in the case $d_{\ast }=0$. The operators with ${\delta }^{\prime }$-type interactions are investigated too. The obtained results demonstrate that in the case $d_{\ast }=0$, as distinguished from the case $d_{\ast }>0$, the spectral properties of the operators with $\delta$ and ${\delta }^{\prime }$-type interactions are substantially different.

54 pages; several corrected typos, deleted Corollary 3.21, added references

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