Eigenvalues of Schrödinger operators near thresholds: two term approximation

Abstract: We consider one dimensional Schr\"{o}dinger operators $$H_\lambda=-\frac{d^{2}{dx^2}+U+} \lambda V_\lambda$$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The 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.