Eigenvalues of Schrödinger operators near thresholds: two term approximation

We consider one dimensional Schrodinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with a nonlinear dependence on parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. Potentials $U$ and $V_\lambda$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_\lambda$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $\lambda\to 0$. We also construct two-term asymptotic formulas for the threshold eigenvalues.