Schrödinger operator on the axis with potentials depending on two parameters

A Schrödinger operator on the axis is considered; its localized potential is the sum of a small potential and certain potentials with contracting supports, which can increase unboundedly when their supports are contracted. Sufficient conditions are presented for the absence (or existence) of eigenvalues for such an operator. In the case where eigenvalues exist, their asymptotic expansion is constructed.