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Asymptotic estimates for the spectral gaps of the Schrödinger operators with periodic \(\delta'\)-interactions. (English) Zbl 1132.34063
The operator \[ H:=-\frac{d^2}{dx^2}+\sum_{l=-\infty}^{\infty} (\beta_1\delta'(x-2\pi l)+\beta_2\delta'(x-\kappa -2\pi l)) \] in \(L^2(\mathbb{R})\), is studied where \(\beta_1, \beta_2 \in \mathbb{R}\backslash \{0\}\) and \(\kappa/\pi\in \mathbb{Q}\). This is a special case of a Sturm-Liouville operator on \(\mathbb{R}\) with an infinite number of transfer conditions (an exceptionally difficult problem). The components of the set \(\mathbb{R}\backslash \sigma(H)\) are called the spectral gaps of \(H\), and can be labelled \(G_0,G_1,\dots\), where \(G_0\) is of the form \((-\infty,\gamma)\) for some \(\gamma\in\mathbb{R}\) and, for \(r\in\mathbb{N}\), if \(a\in G_r, b\in G_{r+1}\) then \(a<b\). Detailed asymptotics are given for the length of \(G_r\) as \(r\to \infty\).
MSC:
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
47E05 General theory of ordinary differential operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L25 Scattering theory, inverse scattering involving ordinary differential operators
34E05 Asymptotic expansions of solutions to ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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