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1-D Schrödinger operators with periodic singular potentials. (English) Zbl 0992.47021

The authors consider one-dimensional Schrödinger operators with potentials from \(W^{-1}_{2,\text{unif}}(\mathbb R)\). That class consists of distributions \(f\) such that \(f\varphi_n\in W_2^{- 1}(\mathbb R)\) for all \(n\) and \(\sup\limits_{n\in \mathbb Z} \|f\varphi_n\|_{W_2^{-1}(\mathbb R)}<\infty\), where \(\varphi_n=\varphi (t-n)\), \[ \varphi (t)=\begin{cases} 2(t+1)^2,&\text{if \(t\in [-1,-1/2)\),}\\ 1-2t^2,&\text{if \(t\in [-1/2,1/2)\),}\\ 2(t-1)^2,&\text{if \(t\in [1/2,1]\),}\\ 0,&\text{if \(|t|>1\).}\end{cases} \] It is shown that for this situation a selfadjoint semibounded operator can be defined that depends continuously on the potential in the uniform resolvent sense. In the periodic case a pure absolute continuity and a band structure of the spectrum are also established.

MSC:

47E05 General theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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