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Weakly coupled bound state of 2D Schrödinger operator with potential-measure

We consider a self-adjoint two-dimensional Schrödinger operator $H_{αμ}$, which corresponds to the formal differential expression \[ -Δ- αμ, \] where $μ$ is a finite compactly supported positive Radon measure on ${\mathbb R}^2$ from the generalized Kato class and $α>0$ is the coupling constant. It was proven earlier that $σ_{\rm ess}(H_{αμ}) = [0,+\infty)$. We show that for sufficiently small $α$ the condition $\sharpσ_{\rm d}(H_{αμ}) = 1$ holds and that the corresponding unique eigenvalue has the asymptotic expansion $$ λ(α) = -(C_μ+ o(1))\exp\Big(-\tfrac{4π}{αμ({\mathbb R}^2)}\Big), \qquad α\rightarrow 0+, $$ with a certain constant $C_μ> 0$. We obtain also the formula for the computation of $C_μ$. 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.