On coupling constant thresholds in one dimension. (English) Zbl 1468.34118
Summary: The threshold behaviour of negative eigenvalues for Schrödinger operators of the type
\[H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda\alpha_\lambda V(\alpha_\lambda \cdot)\]
is considered. The potentials \(U\) and \(V\) are real-valued bounded functions of compact support, \( \lambda\) is a positive parameter, and positive sequence \(\alpha_\lambda\) has a finite or infinite limit as \(\lambda\to 0\). Under certain conditions on the potentials there exists a bound state of \(H_\lambda\) which is absorbed at the bottom of the continuous spectrum. For several cases of the limiting behaviour of sequence \(\alpha_\lambda \), asymptotic formulas for the bound states are proved and the first order terms are computed explicitly.
MSC:
34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |