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

We consider a self-adjoint two-dimensional Schrödinger operator Hαμ, which corresponds to the formal differential expression−Δ−αμ, where μ is a finite compactly supported positive Radon measure on R2 from the generalized Kato class and α>0 is the coupling constant. It was proven earlier that σess(Hαμ)=[0,+∞). We show that for sufficiently small α the condition ♯σd(Hαμ)=1 holds and that the corresponding unique eigenvalue has the asymptotic expansionλ(α)=−(Cμ+o(1))exp⁡(−4παμ(R2)),α→0+, with a certain constant Cμ>0. We also obtain a formula for the computation of Cμ. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend the results of Simon [41] to the case of potentials-measures. Also for regular potentials our results are partially new.