an:01139552
Zbl 0901.58021
Braverman, A.; Etingof, P.; Gaitsgory, D.
Quantum integrable systems and differential Galois theory
EN
Transform. Groups 2, No. 1, 31-56 (1997).
00045918
1997
j
37J35 37K10 81Q30 37N99 14H99 12H05 13N10
\(D\)-module; quantum completely integrable systems; eigenvalue problem; differential Galois group; algebraic integrability
Summary: 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 corresponding \(D\)-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.