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

We consider a self-adjoint two-dimensional Schrödinger operator [Formula: see text] , which corresponds to the formal differential expression [Formula: see text] where μ is a finite compactly supported positive Radon measure on [Formula: see text] from the generalized Kato class and [Formula: see text] is the coupling constant. It was proven earlier that [Formula: see text]. We show that for sufficiently small α the condition [Formula: see text] holds and that the corresponding unique eigenvalue has the asymptotic expansion [Formula: see text] with a certain constant [Formula: see text]. We also obtain a formula for the computation of [Formula: see text]. 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.