zbMATH — the first resource for mathematics

Algebraic integrability for the Schrödinger equation and finite reflection groups. (English. Russian original) Zbl 0805.47070
Theor. Math. Phys. 94, No. 2, 182-197 (1993); translation from Teor. Mat. Fiz. 94, No. 2, 253-275 (1993).
Summary: Algebraic integrability of an \(n\)-dimensional Schrödinger equation means that it has more than \(n\) independent quantum integrals. For \(n=1\), the problem of describing such equations arose in the theory of finite- gap potentials. The present paper gives a construction which associates finite reflection groups (in particular, Weyl groups of simple Lie algebras) with algebraically integrable multidimensional Schrödinger equations. These equations correspond to special values of the parameters in the generalization of the Calogero-Sutherland system proposed by Olshanetsky and Perelomov. The analytic properties of a joint eigenfunction of the corresponding commutative rings of differential operators are described. Explicit expressions are obtained for the solution of the quantum Calogero-Sutherland problem for a special value of the coupling constant.

47N50 Applications of operator theory in the physical sciences
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text: DOI
[1] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ?Nonlinear equations of KdV type, finite-gap linear operators, and Abelian varieties,?Usp. Mat. Nauk,31, 55 (1976). · Zbl 0326.35011
[2] I. L. Burchnall and T. W. Chaundry, ?Commutative ordinary differential operators,?Proc. London Math. Soc., Ser. 2,21, 420 (1923). · JFM 49.0311.03 · doi:10.1112/plms/s2-21.1.420
[3] I. M. Krichever, ?Methods of algebraic geometry in the theory of linear equations,?Usp. Mat. Nauk,32, 180 (1977). · Zbl 0372.35002
[4] P. A. Chalykh and A. P. Veselov, ?Commutative rings of partial differential operators and Lie algebras,?Commun. Math. Phys.,126, 597 (1990). · Zbl 0746.47025 · doi:10.1007/BF02125702
[5] O. A. Chalykh and A. P. Veselov, ?Integrability in the theory of Schrödinger operator and harmonic analysis,? Preprint FIM (ETH, Zürich) (1992) (to appear inCommun. Math. Phys.). · Zbl 0767.35066
[6] N. Bourbaki,Éléments de Mathématique, Vol. 34,Groupes et Algèbres de Lie, Hermann, Paris (1968).
[7] F. Calogero, ?Solution of the one-dimensionaln-body problem with quadratic and/or inversely quadratic pair potential,?J. Math. Phys.,12, 419 (1971). · doi:10.1063/1.1665604
[8] B. Sutherland, ?Exact results for a quantum many-body problem in one dimension,?Phys. Rev. A,4, 2019 (1971);Phys. Rev. A,5, 1372 (1972). · doi:10.1103/PhysRevA.4.2019
[9] M. A. Olshanetsky and M. M. Perelomov, ?Quantum completely integrable systems connected with semisimple Lie algebras,?Lett. Math. Phys.,2, 7 (1977). · Zbl 0366.58005 · doi:10.1007/BF00420664
[10] M. A. Olshanetsky and A. M. Perelomov, ?Quantum integrable systems related to Lie algebras,?Phys. Rep.,94, 313 (1983). · Zbl 0366.58005 · doi:10.1016/0370-1573(83)90018-2
[11] J. Moser, ?Three integrable Hamiltonian systems connected with isospectral deformations,?Adv. Math.,16, 1 (1978).
[12] H. Airault, H. McKean, and J. Moser, ?Rational and elliptic solutions of the KdV equation and a related many-body problem,?Commun. Pure Appl. Math.,30, 94 (1977). · Zbl 0338.35024 · doi:10.1002/cpa.3160300106
[13] I. M. Krichever, ?Rational solutions of the Kadomtsev?Petviashvili equation and integrable systems of particles on a line,?Funktsional. Analiz i Ego Prilozhen.,12, 76 (1978). · Zbl 0408.70010
[14] M. A. Olshanetsky and A. M. Perelomov, ?Quantum systems associated with root systems and the radial parts of Laplacians,?Funktsional. Analiz i Ego Prilozhen.,12, 57 (1978).
[15] A. P. Veselov and O. A. Chalykh, ?Explicit expressions for spherical functions of symmetric spaces of the type AII,?Funktsional. Analiz i Ego Prilozhen.,26, 74 (1992). · Zbl 0784.47045 · doi:10.1007/BF01077087
[16] G. J. Heckman and E. M. Opdam, ?Root systems and hypergeometric functions. I,?Compositio Math.,64, 329 (1987). · Zbl 0656.17006
[17] G. J. Heckman, ?Root systems and hypergeometric functions. II,?Compositio Math.,64, 353 (1987). · Zbl 0656.17007
[18] E. M. Opdam, ?Root systems and hypergeometric functions. III, IV,?Compositio Math.,67, 21, 191 (1988). · Zbl 0669.33008
[19] E. M. Opdam, ?Some applications of hypergeometric shift operators,?Invent. Math.,98, 1 (1989). · Zbl 0696.33006 · doi:10.1007/BF01388841
[20] G. J. Heckman, ?An elementary approach to the hypergeometric shift operators of Opdam,?Invent. Math.,103, 341 (1991). · Zbl 0721.33009 · doi:10.1007/BF01239517
[21] V. V. Kozlov, ?Polynomial integrals of systems of interacting particles,?Dokl. Akad. Nauk SSSR,301, 785 (1988).
[22] J. Feldman, H. Knörrer, and E. Trubowitz, ?There are no two-dimensional Lamé equations,? Preprint FIM (ETH, Zürich) (1991). · Zbl 0763.58027
[23] M. Adler and P. van Moerbeke,Algebraic Integrable Systems: A Systematic Approach. Perspectives in Math., AP, Boston (1989).
[24] O. A. Chalykh, ?A construction of commutative rings of differential operators,?Mat. Zametki,53, No. 3 (1993).
[25] K. L. Styrkas, ?Commutative rings of differential operators of a Lie algebra and finite reflection groups,?Diploma Thesis [in Russian], Moscow State University (1992).
[26] G. J. Heckman, ?A remark on the Dunkl differential-difference operators,?Proc. of the Bowdoin Conference on Harmonic Analysis on Reductive Groups (1989).
[27] F. A. Berezin, ?Laplacians on semisimple Lie groups,?Tr. Mosk. Mat. Ob-va,6, 371 (1957).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.