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

Abstract

The behaviour of the lengths of spectral gaps $\{{\gamma }_n(q){\}}_{n=1}^{\infty }$ of the Hill-Schrödinger 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 $\{\hat{q}(n){\}}_{n=-\infty }^{\infty }$ of the potentials $q(x)$ with respect to the weighted sequence spaces $h^{s,\phi }$, $s>-1$, $\phi \in \mathrm{SV}$. The case $q(x)\in L_{1-per}^2(\mathbb{R})$, $s\in {\mathbb{Z}}_{+}$, $\phi \equiv 1$ corresponds to the Marchenko-Ostrovskii Theorem.

10 pages

References in corpus (1)

• Singularly perturbed periodic and semiperiodic differential operators