01-43 Cruz-Sampedro J., Herbst I., Martinez-Avendano R.

Perturbations of the Wigner-von Neumann Potential Leaving the Embedded Eigenvalue Fixed (32K, Plain TeX) Jan 29, 01

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Abstract. \magnification=1200 \def\real{{\bf R} {\bf Abstract.} We investigate the Schr\"odinger operator $H=-d^2/dx^2+(\gamma/x)\sin \alpha x+V$, acting in $ L^p(\real)$, $1\leq p<\infty$, where $\gamma \in \real \setminus\{ 0 \} $, $\alpha >0$, and $V \in L^1(\real)$. 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 $V\in L^1(\real)$, 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 $L^1(\real)$ of codimension one.

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