Zeroes of the spectral density of the periodic Schrödinger operator with Wigner-von Neumann potential (Q2898402)

The authors consider the Schrödinger operator \(\mathcal{L}_{\alpha}\) defined on the positive half-line by the differential expression NEWLINE\[NEWLINEl= - {{d^{2}} \over dx^{2}} +q(x) + {c\sin(2\omega x +\delta) \over {x+1}} + q_{1}(x)NEWLINE\]NEWLINE and by the boundary condition NEWLINE\[NEWLINE\psi(0) \cos{\alpha} + \psi'(0) \sin {\alpha} = 0,\quad \alpha \in [0, \pi).NEWLINE\]NEWLINE Here, \(q\) is a periodic function summable on its period, \({c\sin(2\omega x +\delta) \over {x+1}}\) is a Wigner-von Neumann type potential and \(q_{1}\) is a summable function. One proves that the spectral density of the operator \(\mathcal{L}_{\alpha}\) has power like zeroes at each of the resonance points (i.e., the absolutely continuous spectrum has pseudogaps). In the particular case \(q \equiv 0\), the authors' result follows from \textit{D. B. Hinton, M. Klaus} and \textit{J. K. Shaw}'s paper [``Embedded halfbound states for potentials of Wigner-von Neumann type'', Proc. Lond. Math. Soc. (3) 62, No. 3, 607--646 (1991; Zbl 0689.34018)]. But the method used in the paper under consideration allows to avoid the use of oscillatory integrals and it could be used for the study of other Schrödinger operators.