Schrodinger operators with slowly decaying Wigner--von Neumann type potentials

2012/01/23 by Lukic, Milivoje
#34L40 (Primary) 35J10 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1201.4840

Abstract

We consider Schrödinger 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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