Weakly coupled bound state of 2D Schr\"odinger operator with potential-measure

We consider a self-adjoint two-dimensional Schr"odinger operator $H_{\alpha \mu }$, which corresponds to the formal differential expression $$-\mathrm{\Delta }-\alpha \mu ,$$ where $\mu$ is a finite compactly supported positive Radon measure on ${\mathbb{R}}^2$ from the generalized Kato class and $\alpha

0$isthecouplingconstant\mathrm{.}Itwasprovenearlierthat$\sigma_{\rm ess}(H_{\alpha\mu}) = [0,+\infty)$\mathrm{.}Weshowthatforsufficientlysmall$\alpha$thecondition$\sharp\sigma_{\rm d}(H_{\alpha\mu}) = 1$ holds and that the corresponding unique eigenvalue has the asymptotic expansion $$\lambda (\alpha )=-(C_{\mu }+o(1))\exp (-\frac{4\pi }{\alpha \mu ({\mathbb{R}}^2)}),\alpha \to 0+,$$ with a certain constant $C_\mu > 0$\mathrm{.}Weobtainalsotheformulaforthecomputationof$C_\mu$. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend Simon’s results, see \cite{Si76}, to the case of potentials-measures. Also for regular potentials our results are partially new.