# Algebraic integrability of Schrodinger operators and representations of Lie algebras

Pavel Etingof; Konstantin Styrkas

Compositio Mathematica (1995)

- Volume: 98, Issue: 1, page 91-112
- ISSN: 0010-437X

## Access Full Article

top## How to cite

topEtingof, Pavel, and Styrkas, Konstantin. "Algebraic integrability of Schrodinger operators and representations of Lie algebras." Compositio Mathematica 98.1 (1995): 91-112. <http://eudml.org/doc/90396>.

@article{Etingof1995,

author = {Etingof, Pavel, Styrkas, Konstantin},

journal = {Compositio Mathematica},

keywords = {highest weight modules; Calogero-Sutherland operator; algebraic integrability; matrix Schrödinger operators; complex simple Lie algebra},

language = {eng},

number = {1},

pages = {91-112},

publisher = {Kluwer Academic Publishers},

title = {Algebraic integrability of Schrodinger operators and representations of Lie algebras},

url = {http://eudml.org/doc/90396},

volume = {98},

year = {1995},

}

TY - JOUR

AU - Etingof, Pavel

AU - Styrkas, Konstantin

TI - Algebraic integrability of Schrodinger operators and representations of Lie algebras

JO - Compositio Mathematica

PY - 1995

PB - Kluwer Academic Publishers

VL - 98

IS - 1

SP - 91

EP - 112

LA - eng

KW - highest weight modules; Calogero-Sutherland operator; algebraic integrability; matrix Schrödinger operators; complex simple Lie algebra

UR - http://eudml.org/doc/90396

ER -

## References

top- [C] Calogero, F.: Solution of the one-dimensional n-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys.12 (1971), 419-436. Zbl1002.70558MR280103
- [Ch] Cherednik, I.: Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation, preprint; Submitted to Comm. Math. Phys. (1994). Zbl0826.35100MR1329203
- [CV1] Chalykh, O.A. and Veselov, A.P.: Integrability in the theory of Schrödinger operator and harmonic analysis, Comm. Math. Phys.152 (1993), 29-40. Zbl0767.35066MR1207668
- [CV2] Chalykh, O.A. and Veselov, A.P.: Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys.126 (1990), 597-611. Zbl0746.47025MR1032875
- [E] Etingof, P.I.: Quantum integrable systems and representations of Lie algebras, hep-th 9311132, Journal of Mathematical Physics (1993), to appear. Zbl0861.17002MR1331279
- [EK1] Etingof, P.I. and Kirillov Jr., A.A.: Representations of affine Lie algebras, parabolic differential equations, and Lamé functions, hep-th 9310083, Duke Math. J., vol. 74(3), 1994, pages 585-614. Zbl0811.17026MR1277946
- [EK2] Etingof, P.I. and Kirillov,, Jr. A.A.: A unified representation-theoretic approach to special functions, hep-th 9312101, Functional Anal. and its Applic.28 (1994), no. 1. Zbl0868.33010
- [G] Grinevich, P.G.: Commuting matrix differential operators of arbitrary rank, Soviet Math. Dokl.30 (1984), no. 2, 515-518. Zbl0598.47049MR765610
- [HO] Heckman, G.J. and Opdam, E.M.: Root systems and hypergeometric functions I, Compos. Math.64 (1987), 329-352. Zbl0656.17006MR918416
- [H1] Heckman, G.J.: Root systems and hypergeometric functions II, Compos. Math.64 (1987), 353-373. Zbl0656.17007MR918417
- [KK] Kac, V.G. and Kazhdan, D.A.: Structure of representations with highest weight of infinite dimensional Lie algebras, Advances in Math.34 (1979), no. 1, 97-108. Zbl0427.17011MR547842
- [Kr] Krichever, I.M.: Methods of algebraic geometry in the theory of non-linear equations, Russian Math. Surv.32:6 (1977), 185-213. Zbl0386.35002
- [O1] Opdam, E.M.: Root systems and hypergeometric functions III, Compos. Math.67 (1988), 21-49. Zbl0669.33007MR949270
- [O2] Opdam, E.M.: Root systems and hypergeometric functions IV, Compos. Math.67 (1988), 191-207. Zbl0669.33008MR951750
- [OOS] Ochiai, H., Oshima, T. and Sekiguchi, H.: Commuting families of symmetric differential operators, (preprint), Univ. of Tokyo (1994). Zbl0817.22010MR1272672
- [OP] Olshanetsky, M.A. and Perelomov, A.M.: Quantum integrable systems related to Lie algebras, Phys. Rep.94 (1983), 313-404. Zbl0366.58005MR708017
- [Osh] Oshima, T.: Completely integrable systems with a symmetry in coordinates, (preprint), Univ. of Tokyo (1994). Zbl0965.37055MR1734134
- [OS] Oshima, T. and Sekiguchi, H.: Commuting families of differential operators invariant under the action of the Weyl group, (preprint) UTMS 93-43, Dept. of Math. Sci., Univ. of Tokyo (1993). Zbl0863.43007
- [S] Sutherland, B.: Exact results for a quantum many-body problem in one dimension, Phys. Rev.A5 (1972), 1372-1376.
- [Sh] Shapovalov, N.N.: On bilinear form on the universal enveloping algebra of a simple Lie algebra, Funct. Anal. Appl.6 (1972), 307-312. Zbl0283.17001
- [VSC] Veselov, A.P., Styrkas, K.A. and Chalykh, O.A.: Algebraic integrability for the Schrödinger equation and finite reflection groups, Theor. and Math. Physics94 (1993), no. 2. Zbl0805.47070MR1221735