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...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2017 |
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Інститут математики НАН України
2017
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| Online Zugang: | 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 |
| Zitieren: | 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862714122249961472 |
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| author | Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| author_facet | Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| 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. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| 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.
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| first_indexed | 2025-12-07T17:49:29Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-148577 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:49:29Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
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| spelling | 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 |
| spellingShingle | On the Spectra of Real and Complex Lamé Operators Haese-Hill, W.A. Hallnäs, M.A. Veselov, A.P. |
| title | 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_short | On the Spectra of Real and Complex Lamé Operators |
| title_sort | on the spectra of real and complex lamé operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148577 |
| work_keys_str_mv | AT haesehillwa onthespectraofrealandcomplexlameoperators AT hallnasma onthespectraofrealandcomplexlameoperators AT veselovap onthespectraofrealandcomplexlameoperators |