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Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner-von Neumann potential. (English) Zbl 1372.47059

In this paper, the author considers the operator \[ L_{\alpha}:L_2(\mathbb R_+)\cap H_{\mathrm{loc}}^2(\mathbb R_+)\to L_2(\mathbb R_+) \] as \(L_{\alpha}y=Ly\), where \[ Ly:=-y''+q(x)y+q_{WN}(x,\gamma)y+q_1(x)y,\quad x\in \mathbb R_+:=[0,\infty), \] and \[ y(0)\cos\alpha-y'(0)\sin\alpha=0. \] Here, \(q\in L_1(0,a)\) is periodic with period \(a\), \[ \begin{aligned} q_{WN}(x,\gamma)=\begin{cases} c\sin(2\omega x+\delta)/x^{\gamma},\quad &\text{if }\gamma\in(\frac12,1),\\c\sin(2\omega x+\delta)/x+1, &\text{if }\gamma=1,\end{cases}\end{aligned} \] \(c,\omega,\delta\in\mathbb R\), \(2a\omega/\pi\not\in\mathbb Z\), \(q_1\in L_1(\mathbb R_+)\), \(\alpha\in[0,\pi)\). The author studies the properties of the spectral density of the operator \(L_{\alpha}\). The main results are given in Theorem 1.1.

MSC:

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