On coupling constant thresholds in one dimension

Keywords: 1D Schrödinger operator, coupling constant threshold, negative eigenvalue, zero-energy resonance, half-bound state, $\delta'$-potential, point interaction
Published online: 2021-03-06


The threshold behaviour of negative eigenvalues for Schr\"{o}dinger 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.

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How to Cite
Golovaty Y. On Coupling Constant Thresholds in One Dimension. Carpathian Math. Publ. 2021, 13 (1), 22-38.