On Self-Adjointness Of 1-D Schr\"odinger Operators With $\delta$-Interactions

Abstract

In the present work we consider in $L^2(\mathbb{R}_+)$ the Schr\"odinger operator $\mathrm{H_{X,\alpha}}=-\mathrm{\frac{d^2}{dx^2}}+\sum_{n=1}^{\infty}\alpha_n\delta(x-x_n)$. We investigate and complete the conditions of self-adjointness and nontriviality of deficiency indices for $\mathrm{H_{X,\alpha}}$ obtained in \cite{karpiiKost}. We generalize the conditions found earlier in the special case $d_n:=x_{n}-x_{n-1}=1/n$, $n\in \mathbb{N}$, to a wider class of sequences $\{x_n\}_{n=1}^\infty$. Namely, for $x_n=\frac{1}{n^{\gamma}\ln^\eta n}$ with $<\gamma,\eta>\in(1/2, 1)\times(-\infty,+\infty)\:\cup\:\{1\}\times(-\infty,1]$, the description of asymptotic behavior of the sequence $\{\alpha_n\}_{n=1}^{\infty}$ is obtained for $\mathrm{H_{X,\alpha}}$ either to be self-adjoint or to have nontrivial deficiency indices.