Cruz-Sampedro J., Herbst I., Martinez-Avendano R.
Perturbations of the Wigner-von Neumann Potential
Leaving the Embedded Eigenvalue Fixed
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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.