Summary: The threshold behaviour of negative eigenvalues for Schrödinger operators of the type \[H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot)\] is considered. The potentials \(U\) and \(V\) are real-valued bounded functions of compact support, \( \lambda\) is a positive parameter, and positive sequence \(\alpha_\lambda\) has a finite or infinite limit as \(\lambda\to 0\). Under certain conditions on the potentials there exists a bound state of \(H_\lambda\) which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence \(\alpha_\lambda \), asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.