Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials

Details Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials Spectral gaps of the one-dimensional Schrödinger operators with singular periodic potentials

2008-05-14

arxiv.org/abs/0805.2136v3

Methods Funct. Anal. Topology 15 (2009), no. 1, 31-40

Mathematics

Spectral Theory

10 pages

Scientific paper

The behaviour of the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ of the Hill-Schr\"odinger operators S(q)u=-u''+q(x)u,\quad u\in \mathrm{Dom}(S(q)) with real-valued 1-periodic distributional potentials $q(x)\in H_{1{-}per}^{-1}(\mathbb{R})$ is studied. We show that they exhibit the same behaviour as the Fourier coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\varphi}$, $s>-1$, $\varphi\in \mathrm{SV}$. The case $q(x)\in L_{1{-}per}^{2}(\mathbb{R})$, $s\in \mathbb{Z}_{+}$, $\varphi\equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem.

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