Weyl-Titchmarsh-type formula for periodic Schrödinger operator with Wigner-von Neumann potential (Q2837116)

The paper discusses the differential operator NEWLINE\[NEWLINE L_\alpha:=-\frac d{d x^2}+q(x)+\frac{c\;\!\sin(2\;\!\omega x+\delta)}{(x+1)^\gamma}+q_1(x) NEWLINE\]NEWLINE in the Hilbert space \(L^2(\mathbb R_+)\) subject to the self-adjoint boundary condition NEWLINE\[NEWLINE \psi(0)\cos(\alpha)-\psi'(0)\sin(\alpha)=0NEWLINE\]NEWLINE with \(c,\omega,\delta\in\mathbb R\), \(\alpha\in[0,\pi)\), \(\gamma\in(1/2,1]\), \(q\) an \(a\)-periodic function such that \(q|_{(0,a)}\in L^1(0,a)\), and \(q_1\in L^1(\mathbb R_+)\). That is, a Schrödinger operator on \(\mathbb R_+\) with a potential which is the sum of a periodic function, a Wigner-von Neumann function, and a \(L^1\)-function. In a previous paper [the authors, Math. Proc. Camb. Philos. Soc. 142, No. 1, 161--183 (2007; Zbl 1170.34355)] it has been shown that the absolutely continuous spectrum of \(L_\alpha\) coincides as a set with the purely absolutely continuous (band) spectrum of the corresponding periodic operator on \(\mathbb R\) NEWLINE\[NEWLINE L_{\mathrm{per}}=-\frac d{d x^2}+q(x). NEWLINE\]NEWLINE The main result of this paper (Theorem 1.1) is a Weyl-Titchmarsh formula which provides a relation between the Bloch solutions \(\psi_\pm(x,\lambda)\) for the operator \(L_{\mathrm{per}}\), the spectral density \(\rho_\alpha'\) of the operator \(L_\alpha\), and the asymptotic behavior of the solution \(\varphi_\alpha(x,\lambda)\) of the Cauchy problem NEWLINE\[NEWLINE L_\alpha\;\!\varphi_\alpha=\lambda\;\!\varphi_\alpha, \quad\varphi_\alpha(0,\lambda)=\sin(\alpha),~\varphi_\alpha'(0,\lambda)=\cos(\alpha). NEWLINE\]NEWLINE Namely, if \(2\alpha\;\!\omega/\pi\notin\mathbb Z\), then it is shown that for almost all \(\lambda\) in the spectrum of \(L_{\mathrm{per}}\) there exists \(A_\alpha(\lambda)\in\mathbb C\) such that NEWLINE\[NEWLINE \varphi_\alpha(x,\lambda) =A_\alpha(\lambda)\;\!\psi_-(x,\lambda)+\overline{A_\alpha(\lambda)}\;\!\psi_+(x,\lambda)+o(1) \quad\text{as \(x\to+\infty\)} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \rho_\alpha'(\lambda) =\frac1{2\pi\;\!|W\{\psi_+(\;\!\cdot\;\!,\lambda),\psi_-(\;\!\cdot\;\!,\lambda)\}| \;\!|A_\alpha(\lambda)|^2}\;\!, NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEwith \(W\{\;\!\cdot\;\!,\;\!\cdot\;\!\}\) the Wronskian. The proof relies on the derivation of uniform asymptotics of the solution \(\varphi_\alpha\).