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Эта публикация цитируется в 4 научных статьях (всего в 4 статьях)
On the Spectra of Real and Complex Lamé Operators
William A. Haese-Hilla, Martin A. Hallnäsb, Alexander P. Veselova a Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
b Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
Аннотация:
We study Lamé operators of the form
\begin{gather*}
L = -\frac{d^2}{dx^2} + m(m+1)\omega^2\wp(\omega x+z_0),
\end{gather*}
with $m\in\mathbb{N}$ and $\omega$ a half-period of $\wp(z)$. For rectangular period lattices, we can choose $\omega$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices.
Ключевые слова:
Lamé operators; finite-gap operators; spectral theory; non-self-adjoint operators.
Поступила: 4 апреля 2017 г.; в окончательном варианте 21 июня 2017 г.; опубликована 1 июля 2017 г.
Образец цитирования:
William A. Haese-Hill, Martin A. Hallnäs, Alexander P. Veselov, “On the Spectra of Real and Complex Lamé Operators”, SIGMA, 13 (2017), 049, 23 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1249 https://www.mathnet.ru/rus/sigma/v13/p49
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