On coupling constant thresholds in one dimension

Keywords:
1D Schrödinger operator, coupling constant threshold, negative eigenvalue, zero-energy resonance, half-bound state, δ′-potential, point interactionAbstract
The threshold behaviour of negative eigenvalues for Schr\"{o}dinger operators of the type Hλ=−d2dx2+U+λαλV(αλ⋅) is considered. The potentials U and V are real-valued bounded functions of compact support, λ is a positive parameter, and positive sequence αλ has a finite or infinite limit as λ→0. Under certain conditions on the potentials there exists a bound state of Hλ which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence αλ, asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.