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Wigner-von Neumann perturbations of a periodic potential: spectral singularities in bands. (English) Zbl 1170.34355

In 1929, Wigner and von Neumann showed that the Schrödinger operator on the semi-axis given by \[ - \frac{d^2}{dx^2} + \frac{A \sin(2x)}{1+x} \] possesses for \(A>2\) and suitable boundary conditions an eigenvalue \(E =1\) that is embedded into the absolutely continuous spectrum. More generally, potentials whose asymptotic behaviour at \(\pm\infty\) is given by \[ \sum_{i=1}^N \frac{c_i\sin(2 \omega_i x + \phi_i)}{x} \] may generate eigenvalues at the energies \(E_i = \omega_i^2\), \(i=1,2,\dots,N\). In the paper under review the authors study the effect the perturbation by a Wigner-von Neumann potential may produce for a periodic background operator. Namely, they consider the Schrödinger operator \[ H = - \frac{d^2}{dx^2} + Q_{\mathrm{per}}(x) + V_0(x) + \sum_{i=1}^N \frac{c_i\sin(2 \omega_i x + \phi_i)} {(|x|+1)^{\gamma_i}} \] on the whole line, where \(Q_{\mathrm{per}}\) is a locally integrable periodic potential of period \(T\), \(V_0 \in L^1(\mathbb{R})\), and \(\gamma\equiv \min\gamma_i >\tfrac12\), and answer the question, under what conditions there may appear eigenvalues inside the spectral bands of the periodic background operator \[ H_0 = - \frac{d^2}{dx^2} + Q_{\mathrm{per}}(x). \]
The authors investigate the asymptotics of generalized eigenfunctions and show that a subordinate solution at the energy level \(E\) (and thus an embedded eigenvalue at \(E\)) may exists only if \(\omega_iT/\pi \not\in\mathbb{Z}\) and at least one of the quantization conditions \[ \frac{\omega_i T + \theta}{\pi} \in \mathbb{Z} \qquad \text{or} \qquad \frac{\omega_i T - \theta}{\pi} \in \mathbb{Z} \] holds; here \(\theta= \theta(E)\) is the quasi-momentum. Whether or not the subordinate solution is square integrable is determined by the spectrum of the so-called principal homogenized interaction matrix, which is a Cesaro average of the Wigner-von Neumann perturbation using the periodic background \(Q_{\mathrm{per}}\). Although these embedded eigenvalues are highly unstable, their possible location is independent of the \(L^1\)-perturbation \(V_0\).

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47A11 Local spectral properties of linear operators
47A55 Perturbation theory of linear operators
47E05 General theory of ordinary differential operators
47N50 Applications of operator theory in the physical sciences
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
35J10 Schrödinger operator, Schrödinger equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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