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On the Spectra of Real and Complex Lamé Operators
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2017, том 13
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On the Spectra of Real and Complex Lamé Operators
Інший ID:
2010 Mathematics Subject Classification: 34L40; 47A10; 33E10
DOI:10.3842/SIGMA.2017.049
DOI:10.3842/SIGMA.2017.049
Посилання:
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.
Підтримка:
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.
Дата:
2017
Переглядів:
507
Завантажень:
250
Анотація:
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.
