On Self-Adjointness Of 1-D Schrödinger Operators With $\delta$-Interactions

Abstract

In the present work we consider in $L^2({\mathbb{R}}_{+})$ the Schrödinger operator ${\mathrm{H}}_{\mathrm{X},\alpha }=-\frac{{\mathrm{d}}^2}{\mathrm{d}{\mathrm{x}}^2}+{\sum }_{n=1}^{\infty }{\alpha }_n\delta (x-x_n)$. We investigate and complete the conditions of self-adjointness and nontriviality of deficiency indices for ${\mathrm{H}}_{\mathrm{X},\alpha }$ 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 ${\mathrm{H}}_{\mathrm{X},\alpha }$ either to be self-adjoint or to have nontrivial deficiency indices.

To be published in Methods of Functional Analysis and Topology (2012)