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

Abstract

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