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On the Embedded Eigenvalues and Dense Point Spectrum of the Stark‐Like Hamiltonians
Author(s) -
Naboko Sergei N.,
Pushnitski Alexander B.
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19971830112
Subject(s) - spectrum (functional analysis) , eigenvalues and eigenvectors , constructive , operator (biology) , mathematics , essential spectrum , sign (mathematics) , schrödinger's cat , point (geometry) , mathematical physics , section (typography) , stark effect , physics , mathematical analysis , quantum mechanics , spectral line , geometry , chemistry , biochemistry , process (computing) , repressor , computer science , transcription factor , advertising , business , gene , operating system
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.