Geodesic
Sigma Functions and Lie Algebras of Schr\"odinger Operators
Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 3-16.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 $Q_0, Q_2, \dots, Q_{4g-2}$ 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 $Q_0, Q_2, \dots, Q_{4g-2}$ for $g = 1,2,3,4$.
Keywords: Schrödinger operator, differentiation of Abelian functions with respect to parameters.
Mots-clés : polynomial Lie algebra 
@article{FAA_2020_54_4_a0,
     author = {V. M. Buchstaber and E. Yu. Bunkova},
     title = {Sigma {Functions} and {Lie} {Algebras} of {Schr\"odinger} {Operators}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {3--16},
     publisher = {mathdoc},
     volume = {54},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a0/}
}
TY  - JOUR
AU  - V. M. Buchstaber
AU  - E. Yu. Bunkova
TI  - Sigma Functions and Lie Algebras of Schr\"odinger Operators
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2020
SP  - 3
EP  - 16
VL  - 54
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a0/
LA  - ru
ID  - FAA_2020_54_4_a0
ER  - 
%0 Journal Article
%A V. M. Buchstaber
%A E. Yu. Bunkova
%T Sigma Functions and Lie Algebras of Schr\"odinger Operators
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2020
%P 3-16
%V 54
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a0/
%G ru
%F FAA_2020_54_4_a0
V. M. Buchstaber; E. Yu. Bunkova. Sigma Functions and Lie Algebras of Schr\"odinger Operators. Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 3-16. http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a0/

[1] V. M. Bukhshtaber, D. V. Leikin, “Uravneniya teploprovodnosti v negolonomnom repere”, Funkts. analiz i ego pril., 38:2 (2004), 12–27 | DOI | MR | Zbl

[2] D. Mamford, Lektsii o teta-funktsiyakh, Mir, M., 1988 | MR

[3] T. Shiota, “Characterization of Jacobian varieties in terms of soliton equations”, Invent. Math., 83:2 (1986), 333–382 | DOI | MR | Zbl

[4] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, “Hyperelliptic Kleinian functions and applications”, Solitons, Geometry, and Topology: On the Crossroad, Adv. Math. Sci., Amer. Math. Soc. Transl., Ser. 2, 179, Providence, RI, 1997, 1–34 | MR

[5] V. M. Bukhshtaber, “Polinomialnye dinamicheskie sistemy i uravnenie Kortevega–de Friza”, Sovremennye problemy matematiki, mekhaniki i matematicheskoi fiziki. II, Sbornik statei, v. 294, Tr. MIAN, MAIK, M., 2016, 191–215 | DOI

[6] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, Multi-dimensional sigma-functions, arXiv: 1208.0990

[7] V. M. Buchstaber, V. Z. Enolski, D. V. Leykin, “$\sigma$-functions: Old and new results”, Integrable Systems and Algebraic Geometry, v. 2, LMS Lecture Note Series, 459, Cambridge Univ. Press, Cambridge, 2019, 175–214 ; arXiv: 1810.11079 | MR

[8] V. M. Bukhshtaber, D. V. Leikin, “Polinomialnye algebry Li”, Funkts. analiz i ego pril., 36:4 (2002), 18–34 | DOI | MR | Zbl

[9] V. I. Arnold, Singularities of Caustics and Wave Fronts, Mathematics and its Applications, 62, Kluwer Academic Publisher Group, Dordrecht, 1990 | DOI | MR | Zbl

[10] J. C. Eilbeck, J. Gibbons, Y. Onishi, S. Yasuda, Theory of heat equations for sigma functions, arXiv: 1711.08395

[11] H. F. Baker, “On the hyperelliptic sigma functions”, Amer. J. Math., 20:4 (1898), 301–384 | DOI | MR | Zbl

[12] V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, “Kleinian functions, hyperelliptic Jacobians and applications”, Rev. Math. Math. Phys., 10:2 (1997), 3–120 | MR | Zbl

[13] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Reprint of 4th (1927) ed., v. 2, Transcendental functions, Cambridge Univ. Press, Cambridge, 1996 | MR | Zbl

[14] E. Yu. Bunkova, “Differentiation of genus 3 hyperelliptic functions”, Eur. J. Math., 4:1 (2018), 93–112 ; arXiv: 1703.03947 | DOI | MR | Zbl

[15] V. M. Bukhshtaber, D. V. Leikin, “Differentsirovanie abelevykh funktsii po parametram”, UMN, 62:4(376) (2007), 153–154 | DOI | MR | Zbl

[16] V. M. Bukhshtaber, D. V. Leikin, “Reshenie zadachi differentsirovaniya abelevykh funktsii po parametram dlya semeistv $(n,s)$-krivykh”, Funkts. analiz i ego pril., 42:4 (2008), 24–36 | DOI | MR | Zbl

[17] V. M. Bukhshtaber, E. Yu. Bunkova, “Algebry Li operatorov teploprovodnosti v negolonomnom repere”, Matem. zametki, 108:1 (2020), 17–32 | DOI | MR | Zbl

[18] F. G. Frobenius, L. Stickelberger, “Über die Differentiation der elliptischen Functionen nach den Perioden und Invarianten”, J. Reine Angew. Math., 92 (1882), 311–337 | MR