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Funktsional. Anal. i Prilozhen., 2020, Volume 54, Issue 4, Pages 3–16 (Mi faa3837)  

This article is cited in 1 scientific paper (total in 1 paper)

Sigma Functions and Lie Algebras of Schrödinger Operators

V. M. Buchstaber, E. Yu. Bunkova

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $g > 0$, a system of $2g$ multidimensional Schrödinger equations in magnetic fields with quadratic potentials was defined. Such systems are equivalent to systems of heat equations in a nonholonomic frame. It was proved that such a system determines the sigma function of the universal hyperelliptic curve of genus $g$. A polynomial Lie algebra with $2g$ Schrödinger operators $Q_0, Q_2, …, Q_{4g-2}$ as generators was introduced.
In this work, for each $g > 0,$ we obtain explicit expressions for $Q_0$, $Q_2$, and $Q_4$ and recurrent formulas for $Q_{2k}$ with $k>2$ expressing these operators as elements of a polynomial Lie algebra in terms of the Lie brackets of the operators $Q_0$, $Q_2$, and $Q_4$.
As an application, we obtain explicit expressions for the operators $Q_0, Q_2, …, Q_{4g-2}$ for $g = 1,2,3,4$.

Keywords: Schrödinger operator, polynomial Lie algebra, differentiation of Abelian functions with respect to parameters.

Funding Agency Grant Number
Russian Science Foundation 20-11-19998


DOI: https://doi.org/10.4213/faa3837

Full text: PDF file (635 kB)
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English version:
Functional Analysis and Its Applications, 2020, 54:4, 229–240

Bibliographic databases:

UDC: 515.178.2+517.958+517.986
Received: 21.08.2020
Revised: 21.08.2020
Accepted:03.09.2020

Citation: V. M. Buchstaber, E. Yu. Bunkova, “Sigma Functions and Lie Algebras of Schrödinger Operators”, Funktsional. Anal. i Prilozhen., 54:4 (2020), 3–16; Funct. Anal. Appl., 54:4 (2020), 229–240

Citation in format AMSBIB
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\by V.~M.~Buchstaber, E.~Yu.~Bunkova
\paper Sigma Functions and Lie Algebras of Schr\"odinger Operators
\jour Funktsional. Anal. i Prilozhen.
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\vol 54
\issue 4
\pages 3--16
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\jour Funct. Anal. Appl.
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\vol 54
\issue 4
\pages 229--240
\crossref{https://doi.org/10.1134/S0016266320040012}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. M. Bukhshtaber, E. Yu. Bunkova, “Giperellipticheskie sigma-funktsii i polinomy Adlera–Mozera”, Funkts. analiz i ego pril., 55:3 (2021), 3–25  mathnet  crossref
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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