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

We consider a self-adjoint two-dimensional Schrödinger operator H α μ , which corresponds to the formal differential expression − Δ − α μ , where μ is a finite compactly supported positive Radon measure on R 2 from the generalized Kato class and α > 0 is the coupling constant. It was proven earlier that σ ess ( H α μ ) = [ 0 , + ∞ ) . We show that for sufficiently small α the condition ♯ σ d ( H α μ ) = 1 holds and that the corresponding unique eigenvalue has the asymptotic expansion λ ( α ) = − ( C μ + o ( 1 ) ) exp ⁡ ( − 4 π α μ ( R 2 ) ) , α → 0 + , with a certain constant C μ > 0 . We also obtain a formula for the computation of C μ . 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.