On the Embedded Eigenvalues and Dense Point Spectrum of the Stark‐Like Hamiltonians

The point spectrum lying on the essential spectrum is investigated for the one‐dimensional Schrödinger operator \documentclass{article}\pagestyle{empty}\begin{document}$ \- \frac{{d^2 }}{{dx^2 }} + q(x) $\end{document} with decaying potential q , and weakly perturbed Stark‐like operator \documentclass{article}\pagestyle{empty}\begin{document}$ - \frac{{d^2 }}{{dx^2 }} - \left| x \right|^\alpha {\rm sign }x + q(x). $\end{document} An elementary constructive technique is developed to obtain various results concerning embedded eigenvalues of Schrödinger operators. In Section 3 a constructive example of the Stark ‐ like operator with the potential q decaying slightly slowlier than o(1/|x| 1‐α/2 )and dense point spectrum on the whole real axis is presented.