Abstract.
We investigate the Schrödinger operator \( H = -d^{2} / dx^{2} + (\gamma / x) \sin \alpha x + V \), acting in \( L^{p} (\mathbf{R})\), \( 1 \leq p \) < \( \infty \), where \( \gamma \in \mathbf{R} \backslash \{0\} \), \( \alpha > 0 \), and \( V \in L^{1}(\mathbf{R}) \). For \( |\gamma| \leq 2\alpha / p \) we show that H does not have positive eigenvalues. For \( |\gamma| > 2\alpha / p \) we show that the set of functions \( V \in L^{1}(\mathbf{R}) \), such that H has a positive eigenvalue embedded in the essential spectrum \( \sigma_{\rm ess}(H) = [0, \infty) \), is a smooth unbounded sub-manifold of \( L^{1}(\mathbf{R}) \) of codimension one.
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Submitted 26/01/01, accepted 18/05/01
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Cruz-Sampedro, J., Herbst, I. & Martínez-Avendaño, R. Perturbations of the Wigner-Von Neumann Potential Leaving the Embedded Eigenvalue Fixed. Ann. Henri Poincaré 3, 331–345 (2002). https://doi.org/10.1007/s00023-002-8619-4
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DOI: https://doi.org/10.1007/s00023-002-8619-4