The one-dimensional Schrödinger operators with singular periodic potentials

Details The one-dimensional Schrödinger operators with singular periodic potentials The one-dimensional Schrödinger operators with singular periodic potentials

2008-05-07

arxiv.org/abs/0805.1000v1

Methods Funct. Anal. Topology 14 (2008), no. 2, 184-200

Mathematics

Spectral Theory

17 pages, to appear in Meth. Funct. Anal. Top

Scientific paper

The one-dimensional Schr\"odinger operators S(q)u:=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic singular potentials $q(x)\in H_{per}^{-1}(\mathbb{R})$ are studied on the Hilbert space $L_{2}(\mathbb{R})$. An equivalence of five basic definitions for the operators $S(q)$ and their self-adjointness are established. A new proof of spectral continuity of the operators $S(q)$ is found. Endpoints of spectral gaps are precisely described.

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