Abstract
This paper is devoted to a systematic study of quantum completely integrable systems (i.e., complete systems of commuting differential operators) from the point of view of algebraic geometry. We investigate the eigenvalue problem for such systems and the correspondingD-module when the eigenvalues are in generic position. In particular, we show that the differential Galois group of this eigenvalue problem is reductive at generic eigenvalues. This implies that a system is algebraically integrable (i.e., its eigenvalue problem is explicitly solvable in quadratures) if and only if the differential Galois group is commutative for generic eigenvalues. We apply this criterion of algebraic integrability to two examples: finite-zone potentials and the elliptic Calogero-Moser system. In the second example, we obtain a proof of the Chalyh-Veselov conjecture that the Calogero-Moser system with integer parameter is algebraically integrable, using the results of Felder and Varchenko.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AK] A. Altman, S. Kleiman,Introduction to Grothendieck Duality Theory, Lecture Notes in Math., 146, Springer-Verlag, 1970.
[Am] S. Amitsur,Commutative linear differential operators, Pacific Journal of Math.8 (1958), 1–11.
[Ar] В. И. Аролъд,Маемамические мемоды классической механки, Москва, Москва, Наука, 1974. English translation: V. Arnold,Mathematical Methods of Classical Mechanics, Graduate Texts in Math., 060, Springer-Verlag, 1978.
[BD] A. Beilinson and V. Drinfeld,Quantization of Hitchin's Hamiltonians and Hecke eigensheaves, Preprint (1996).
[Bo] A. Borel et al.,Algebraic D-modules, Progress in Mathematics, Birkhäuser, Boston 1988.
[CV1] O. Chalykh, A. Veselov,Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys.126 (1990), 597–611.
[CV2] O. Chalykh, A. Veselov,Integrability in the theory of Schrodinger operator and harmonic analysis, Comm. Math. Phys.152 (1993), 29–40.
[De]P. Deligne,Catégories Tannakiennes, Grothendieck Festschrift2 (1991), Birkhäuser, Boston, 111–195.
[DMN] B. Dubrovin, V. Matveev, S. Novikov,Nonlinear equations of KdV type, finite zone linear operators, and abelian varieties, Uspekhi Mat. Nauk31 (1976), no. 1, 55–136 (in Russian). English translation: B. Dubrovin, V. Matveev, S. Novikov,Nonlinear equations of KdV type, finite zone linear operators, and abelian varieties, Russian Math. Surv.31 (1976), no. 1, 59–146.
[Dr] V. Drinfeld,Commutative subrings of certain noncommutative rings, Funktsional. Anal. i Ego and Prilozh.11 (1977), no. 1, 11–14 (in Russian). English translation: V. Drinfeld,Commutative subrings of certain noncommutative rings, Funct. Anal. Appl.11 (1977), no. 1, 9–11.
[EFK] P. Etingov, I. Frenkel and A. Kirillov, Jr.,Spherical functions on affine Lie groups, Duke Math. J.80 (1995), no. 1, 59–90.
[FV] G. Felder and A. Varchenko,Three formulas for eigenfunctions of integrable Schrödinger operators, Preprint (1995).
[Ka] I. Kaplansky,An Introduction to Differential Algebra, Hermann, Paris, 1957.
[Katz] N. Katz,On the calculation of certain differential Galois groups, Inv. Math.87 (1987), 13–62.
[Kr] I. Krichever,Methods of algebraic geometry in the theory of non-linear equations, Uspekhi Mat. Nauk32 (1977), no. 6, 182–214 (in Russian). English translation: I. Krichever,Methods of algebraic geometry in the theory of nonlinear equations, Russian Math. Surv.32 (1977), no. 6, 183–208.
[ML] L. Makar-Limanov,The automorphisms of the free algebra with two generators, Funktsional. Anal. i Ego Prilozh.4 (1970), no. 3, 107–108 (in Russian). English translation: L. Makar-Limanov,The automorphisms of the free algebra with two generators, Funct. Analysis and Appl.4 (1970), no. 3, 332–333.
[OP] M. Olshanetsky and A. Perelomov,Quantum integrable systems related to Lie algebras, Phys. Rep.94 (1983), 313–404.
[VSC] A. Veselov, K. Styrkas, O. Chalykh,Algebraic integrability for the Schrödinger equation and finite reflection groups, Teor. i Mat. Fiz.94 (1993), no. 2, 253–275 (in Russian). English translation: A. Veselov, K. Styrkas, O. Chalykh,Algebraic integrability for the Schrödinger equation and finite reflection groups, Theor. and Math. Phys.94 (1993), no. 2, 182–197.
[WW] E. T. Whittaker, G. N. Watson,Course of Modern Analysis, 4th Edition, Cambridge Univ. Press, 1958.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Braverman, A., Etingof, P. & Gaitsgory, D. Quantum integrable systems and differential Galois theory. Transformation Groups 2, 31–56 (1997). https://doi.org/10.1007/BF01234630
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01234630