1--D Schr\"odinger operators with local interactions on a discrete set

TL;DR

This paper investigates the spectral properties of one-dimensional Schrödinger operators with local point interactions on a discrete set where the points can accumulate, using extension theory and boundary triplets to establish conditions for self-adjointness, discreteness, and boundedness.

Contribution

It extends the analysis of Schrödinger operators with point interactions to the case where the interaction points accumulate, linking spectral properties to Jacobi matrices and analyzing both delta and delta-prime interactions.

Findings

Spectral properties are characterized via Jacobi matrices.

Necessary and sufficient conditions for self-adjointness, discreteness, and boundedness are established.

Spectral behaviors differ significantly between delta and delta-prime interactions when points accumulate.

Abstract

Spectral properties of 1-D Schr\"odinger 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 $d_*:=\inf_{n,k\in\N}|x_n-x_k|>0$. Our paper is devoted to the case $d_*=0$. 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 $d_*=0$. The operators with $\delta'$-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 $\delta$ and $\delta'$-type interactions are substantially different.