On the Spectra of Real and Complex Lamé Operators
We study Lamé operators of the form L=−d²/dx²+m(m+1)ω²℘(ωx+z₀), with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a b...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2017 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2017
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/148577 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the Spectra of Real and Complex Lamé Operators / W.A. Haese-Hill, M.A. Hallnäs, A.P. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. |
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Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. 2019-02-18T16:10:46Z 2019-02-18T16:10:46Z 2017 On the Spectra of Real and Complex Lamé Operators / W.A. Haese-Hill, M.A. Hallnäs, A.P. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34L40; 47A10; 33E10 DOI:10.3842/SIGMA.2017.049 https://nasplib.isofts.kiev.ua/handle/123456789/148577 We study Lamé operators of the form L=−d²/dx²+m(m+1)ω²℘(ωx+z₀), with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 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. We are grateful to Jenya Ferapontov, John Gibbons and Anton Zabrodin for very useful and encouraging discussions, and especially to Boris Dubrovin, who many years ago asked one of us (APV) about the position of open gaps in the spectra of Lam´e operators. We would like to thank Professor Gesztesy for his interest in our work and for pointing out further relevant references, including [1] and [9]. The work of WAH was partially supported by the Department of Mathematical Sciences at Loughborough University as part of his PhD studies. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Spectra of Real and Complex Lamé Operators Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
On the Spectra of Real and Complex Lamé Operators |
| spellingShingle |
On the Spectra of Real and Complex Lamé Operators Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| title_short |
On the Spectra of Real and Complex Lamé Operators |
| title_full |
On the Spectra of Real and Complex Lamé Operators |
| title_fullStr |
On the Spectra of Real and Complex Lamé Operators |
| title_full_unstemmed |
On the Spectra of Real and Complex Lamé Operators |
| title_sort |
on the spectra of real and complex lamé operators |
| author |
Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| author_facet |
Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| publishDate |
2017 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We study Lamé operators of the form
L=−d²/dx²+m(m+1)ω²℘(ωx+z₀),
with m∈N and ω a half-period of ℘(z). For rectangular period lattices, we can choose ω and z0 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/148577 |
| citation_txt |
On the Spectra of Real and Complex Lamé Operators / W.A. Haese-Hill, M.A. Hallnäs, A.P. Veselov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 32 назв. — англ. |
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2025-12-07T17:49:29Z |
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2025-12-07T17:49:29Z |
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1850872713028042752 |