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Belokolos, E. D.; Enolskii, V. Z.

doi:10.1023/a:1011983313249

Cited by 30 publications

(18 citation statements)

References 44 publications

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“…On the other hand, there are points with any number of frequencies arbitrarily close to any point in the phase space. Other solutions with m < n frequencies can be constructed using the reduction theory for hyperelliptic functions 46 . Note also that degenerate solutions with m frequencies contain ones with fewer frequencies and can be further reduced to m − 1, m − 2 etc.…”

Section: Degenerate Solutionsmentioning

confidence: 99%

Solution for the dynamics of the BCS and central spin problems

Yuzbashyan

Altshuler

Кузнецов

et al. 2005

J. Phys. A: Math. Gen.

138

217

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We develop an explicit description of a time-dependent response of fermionic condensates to perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem -the central spin model, which describes a localized spin coupled to a "spin bath". Here, we focus on presenting the solution and describing its general properties, but also mention some applications, e.g. to nonstationary pairing in cold Fermi gases and to the issue of electron spin decoherence in quantum dots. A typical dynamics of the BCS and central spin models is quasi-periodic with a large number of frequencies and stable under small perturbations. We show that for certain special initial conditions the number of frequencies decreases and the solution simplifies. In particular, periodic solutions correspond to the ground state and excitations of the BCS model.

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Section: Degenerate Solutionsmentioning

confidence: 99%

Solution for the dynamics of the BCS and central spin problems

Yuzbashyan

Altshuler

Кузнецов

et al. 2005

J. Phys. A: Math. Gen.

138

217

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“…Редукция может быть описана непосредственно в терминах римановой матрицы периодов (см. [31]; недавнее изложение и приложения даны в [32], [33]). Будем говорить, что риманова матрица Π = A B размера 2g × g допускает редукцию, если существует числовая комплексная матрица λ размера g × g 1 максимального ранга, числовая комплексная матрица Π 1 размера 2g 1 × g 1 и целочисленная матрица M размера 2g × 2g 1 также максимального ранга такие, что…”

Section: )unclassified

$\operatorname{SU}(2)$-монополи, кривые с симметриями и наследие Рамануджана

Браден¹,

Braden²,

Enolskii³

2010

Матем. сб.

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SU(2)-монополи, кривые с симметриями и наследие Рамануджана Предложено обобщение конструкции Эрколани-Синха SU(2)-монополей на случай семейства пятипараметрических центрированных монополей с зарядом три. В частности, показано как разрешить трансцендентные условия на спектральную кривую. Для класса кривых с симметриями эти трансцендентные условия приводят к некоторой теоретико-числовой проблеме, которая может быть решена при помощи недавно доказанного равенства Рамануджана. Библиография: 36 названий. Ключевые слова: неабелевы монополи, гипергеометрические соотношения Рамануджана, алгебраические кривые с автоморфизмами.

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“…From Theorem 5.9, we obtain the decompositions of the functions ℘ 1,1 , ℘ 1,3 − α 2 β 2 , and ℘ 3,3 into the products of meromorphic functions (Remark 5.10), a solution of the KdV-hierarchy in terms of functions constructed by the elliptic functions ℘ E 1 and ℘ E 2 (Remark 5.11), and the expressions of the coordinates of the Kummer surface in terms of the Weierstrass elliptic functions ℘ E 1 and ℘ E 2 (Remark 5.12). In Section 6, we compare our results with the results of [6,19,25].…”

Section: Introductionmentioning

confidence: 99%

“…99, 100], [25] are described in terms of coordinates on the Kummer surfaces. In this paper, by an approach different from [6,7,17,19,25], we derive relationships between the hyperelliptic functions associated with the curve of genus 2 and the Weierstrass elliptic functions.…”

Section: Introductionmentioning

confidence: 99%

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

Ayano¹,

Buchstaber²

2022

SIGMA

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The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve V of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves E i , i = 1, 2, and morphisms of degree 2 from V to E i . We construct hyperelliptic functions associated with V from the Weierstrass elliptic functions associated with E i and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with V to the appropriate subspaces in C 2 are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with E i . Further, we express the hyperelliptic functions associated with V on C 2 in terms of the Weierstrass elliptic functions associated with E i . We derive these results by describing the homomorphisms between the Jacobian varieties of the curves V and E i induced by the morphisms from V to E i explicitly.

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Cited by 30 publications

References 44 publications

Solution for the dynamics of the BCS and central spin problems

Solution for the dynamics of the BCS and central spin problems

$\operatorname{SU}(2)$-монополи, кривые с симметриями и наследие Рамануджана

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

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