 1129 Sergey Naboko and Sergey Simonov
 Zeroes of the spectral density of the periodic Schroedinger operator with Wignervon Neumann potential
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Feb 25, 11

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Abstract. We consider the Schroedinger operator L_α on the halfline with a periodic background potential and the Wignervon Neumann potential of Coulomb type: csin(2ωx+d)/(x+1). It is known that the continuous spectrum of the operator L_α has the same bandgap structure as the free periodic operator, whereas in each band of the absolutely continuous spectrum there exist two points (socalled critical or resonance) where the operator L_α has a subordinate solution, which can be either an eigenvalue or a `halfbound' state. The phenomenon of an embedded eigenvalue is unstable under the change of the boundary condition as well as under the local change of the potential, in other words, it is not generic. We prove that in the general case the spectral density of the operator L_α has powerlike zeroes at critical points (i.e., the absolutely continuous spectrum has pseudogaps). This phenomenon is stable in the abovementioned sense.
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