Slowly decaying Wigner--von Neumann type potentials

Details Slowly decaying Wigner--von Neumann type potentials Slowly decaying Wigner--von Neumann type potentials

2012-01-23

arxiv.org/abs/1201.4840v1

Mathematics

Spectral Theory

Scientific paper

We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials $\sin(ax)/x^b$ with $b>0$. We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.

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