On coupling constant thresholds in one dimension

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