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Zeroes of the spectral density of the periodic Schrödinger operator with Wigner-von Neumann potential. (English) Zbl 1257.34072

The authors consider the Schrödinger operator \(\mathcal{L}_{\alpha}\) defined on the positive half-line by the differential expression \[ l= - {{d^{2}} \over dx^{2}} +q(x) + {c\sin(2\omega x +\delta) \over {x+1}} + q_{1}(x) \] and by the boundary condition \[ \psi(0) \cos{\alpha} + \psi'(0) \sin {\alpha} = 0,\quad \alpha \in [0, \pi). \] 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 D. B. Hinton, M. Klaus and 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.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47A10 Spectrum, resolvent
47E05 General theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
34L05 General spectral theory of ordinary differential operators

Citations:

Zbl 0689.34018
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References:

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