On coupling constant thresholds in one dimension - Golovaty, Yu.D. (Carpathian Mathematical Publications, 2021)
The threshold behaviour of negative eigenvalues for Schr"odinger operators of the type \[ H_λ=-d^2dx^2+U+ λα_λ V(α_λ ·) \] is considered. The potentials U and V are real-valued bounded functions of compact support, λ is a positive parameter, and positive sequence α_λ has a finite or infinite limit as λ→ 0. Under certain conditions on the potentials there exists a bound state of H_λ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence α_λ, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.